Chapter VII

Individual Measures

and Population Parameters

Adaptive Fitness and Environmental Selection

by Hugh M. Lewis

It is my central hypothesis that most basic aspects of evolutionary theory, including natural selection patterns and speciation, can be accounted for in systemic terms based on this model, and do not need to employ "prime mover" type theories. From the standpoint of evolution, prime mover events can be adequately explained as "trigger" mechanisms that can precipitate state alterations in evolutionary systems, but contribute mostly to its overall sense of entropy and not to its defining sense of recurrent order.

It is true that the concepts of fitness and selection are closely related, so closely related in fact that they constitute a kind of functional tautology. Something is selected for if it is fit and something is fit if it is selected for. At the same time, these concepts are defined and used out side of strict population genetics in ways that are usually imprecise and leave a lot of room for unasked questions and uncertainties in our generalizations we derive from them.

Before beginning this digression, it is important to point out one presumed natural tendency of all life forms. It is a point that usually seems obvious, but evolution cannot be explained or understood without it. In ideal conditions, any population should show natural increase at a rate that is specific to the species, such that it will increase along a predictable curve until the population will eventually outstrip the limits of its basic resource pool. It is natural that any healthy organism should maximize its own reproductive capacity to the limit. This is a normal function and natural outcome of an organism's life imperative, biological imperative and evolutionary imperative all operating in tandem.

If the resource base were unlimited, the population would grow infinitely. Populations will naturally tend to over reach their built-in limits to biological increase, and thus face over-population. This is known as the Malthusian parameter, and its basic formula is referred to as the "intrinsic rate of natural increase" (r) and is taken as the measure of instantaneous rate of population change per individual. It is expressed as numbers per unit of time and is defined as units of 1/time. This is defined in a closed population as the instantaneous birth rate (b) minus the instantaneous death rate (d).

r = (b - d)

In an open population, this is defined as:

r = (b + immigration) - (d + immigration)

When population birth rates exceed death rates (b > d) the population is increasing and r is positive.

When population death rates exceed birth rates (b < d) the population is decreasing and r is negative.

Population growth is normally calculated by iteration using Euler's implicit equation where e is the base of the natural logarithms and x subscripts age:

∑_x e^-vx l_xm_x = 1

If the net reproductive rate (R₀) is close to 1, r can be estimated by the following formula:

r ˜ log_eR₀/T

where T equals the generation time derived by the following formulas:

T = ^w∑_x=axl_xm_x or T = ∫_a^w xl_xm_xdx

Where (a) is age of first reproduction

(w) is age of last reproduction

(m_x) is the number of offspring produced by an average organism of age x during that age period

(l_x) is the probability that an average newborn will survive to an age x

The net reproductive rate is defined as the average number of age -class zero offspring produced by an average newborn organism during its entire lifetime. When R₀ is greater than 1, the population is increasing, and when equals one, the population is stable. If R₀ is less than 1, the population is decreasing. R₀ is also known as the replacement rate of the population. It is the product of age specific survivorship and fecundity schedules, or all ages at which reproduction occurs:

R₀= ^x∑_x=0l_xm_x

For a stable population to stay at equilibrium, death rates must be equivalent to birth rates. If death rates are high, birth rates must also be high, and vice versa.

Under optimal conditions, when R₀ is as high as possible, the maximal rate of natural increase is attained, designated R_max

The intrinsic rate of increase is inversely related to generation time T.

R_max varies widely between different kinds of species. Small species that are short lived tend to have high rates of increase, while larger, longer-lived species tend to have lower rates of increase.

A population that grows linearly with time would have a constant population given by:

Population growth rate (r) = N_t - N₀/t - t₀ = ∆N/∆ t= constant

Where N_t is the number at time (t),

N₀ is the initial number,

t₀ is the initial time

Now it is understood clearly that at any fixed positive value of r, the per capita rate of increase is constant and the population will grow exponentially, given by the following:

dN/dt = bN-dN= (b-d)N = rN

Where (r) is constant and

dN/dt is calculus shorthand for the instantaneous rate of change of N at t

These equations are normally depicted in a linear form, but it is known that population growth curves usually follow non-linear trajectories.

If we plot a normal linear growth line at any time (T) and then return after a period (T') then we might plot the constant expected increase ∆N/∆ t where t = T' - T, we would find that the area below the mid-point of the plot t would be less than our expected rate, and that greater than the mid-point would be greater. ∆N/∆ t will approximate the true rate of increase at t as these values are made smaller and smaller. When ∆ of N and t reach 0, it most closely equals dN/dt.

The number of organisms (N) at time (t), N_t during exponential increase is a function of the initial number at time zero N₀ , r, and the time available for growth since time (t). It is given as:

N_{t =} N₀ e^rt

Where e is the base of the natural logarithms. We get:

Log e N_t = Log e N₀ + Log e e^rt = Log e N₀ + rt

If we set N₀ equal to 1 such that the population is initiated at one individual, then after a single generation (T) the population is equal to the net reproductive rate of that individual, or R₀

Substituting in the previous formula, we get:

Log e R₀ = Log e 1 + rT

Because (Log e 1) equals 0, the formula above is equal to the formula for the net reproductive rate. A related parameter is the finite rate of increase, (v) defined as the rate of increase per individual per unit of time. It is given as:

R = log e v or v = e^r

No population can increase exponentially forever in a limited world, before it soon overstretches its natural boundaries or limits. Unless the average rate of increase is zero, in the long run a growing population must either decrease to extinction or else increase to the extinction of other populations. This is known as the Malthusian Dilemma.

If we know the age structure of any discrete population, then a great many population parameters can be derived from the above considerations, which parameters have significant bearing on our understanding of the central problem. Any individual may be ascribed discrete values, like height, weight, age, or "fitness." A population can also be ascribed statistical descriptors like mean, or average, variance, mode, etc., which constitute the population parameters of that group. While evolution plays out daily in real terms of the life, success and death of the individual organism, it is the role and membership of the organism to the group that makes the critical difference in evolutionary outcomes. The relative fitness of the individual helps to determine the overall fitness of the group, hence indirectly of all the other members of the group, and vice versa.

Thus, values of fitness and selection are understood in terms of the individual, but are described in terms of the population. This is a fundamental paradox of evolutionary theory that to some extent seems to stand in the way of a clear generalist synthesis of evolutionary theory at the level explaining speciation. We know that speciation is a function of what happens to individual organisms in their natural life-history, but it shows its affects and is measured only in net terms of the total group to which that individual belongs by biological definition and heredity. It is worthwhile therefore to further consider conventional models of individual and group fitness

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Fitness in evolutionary theory is defined conventionally in terms of a model of genetic reproduction. It is defined as an individual's ability to perpetuate its genes in the gene pool of its host population, or kin-group. This is known as reproductive success, or reproductive fitness. The standard definition is as the net reproductive rate (the average number of offspring produced by an individual times the probability that the individuals will survive to reproductive age) as in:

w = n(s)

Where w equals reproductive success (fitness), n equals net reproductive rate (average number of offspring produced per surviving individual) and s equals the probability that an individual present at the start of the generation will survive to reproductive age.

In population genetics, we determine the relative fitness of different genotypes within a given population by dividing the net reproductive rate by the reproductive rate of the genotype with the greatest fitness:

Fr = Nab/Nxy

Where Fr = relative fitness, Nab = net reproductive rate of genotype ab, Nxy = net reproductive rate of the genotype with the greatest fitness (xy)

By this definition, 1 is the highest possible fitness achievable with Nab = Nxy, and 0 is the lowest possible fitness, when ab = 0.

According to this standard model, natural selection mechanisms work indirectly against fitness to alter the distribution of genotypes in a population. The intensity of selection is generally measured in terms of the relative fitness of the individual.

Fitness values are relative when they are assigned to genotypes. The standard symbol of fitness is w and is counted by every hundred offspring. It is common to assign one class of genotype as the normal standard of unity, for instance AA, where the fitness of the other types can be calculated as ratios of offspring averages, such that:

AA - Aa - aa

AA = w₁ = 1

Aa = w₂, and aa = w₃

Aa < 1; w₃ does not equal w₂ < 1

The selection coefficient, s, is given as the complement of w, or:

w = 1 - s

s= 1 - w

The selection coefficient measures the extent to which less fit genotypes deviate from a fitness of 1, such that:

s = 0 when w = 1

Natural selection is defined in this model as "the differential reproductive success of genotypes in a population," resulting in the changes of genotypic frequencies of the population reflected in observable alterations of phenotypic ratios. Selection works in complex ways that tend to promote structural, behavioral or functional adaptation of the individual organism and the group as a whole. Selection is held to operate directly upon phenotypic traits of the individual that are the complex expression of genotypes. Selection may work in different ways. Population geneticists recognize several kinds of selection patterns:

1. Selection against deleterious alleles or the culling of weak genes from the gene pool.

2. Balancing selection affecting genotypic polymorphisms.

3. Stabilizing selection toward a "standard phenotype"

4. Directional selection that shifts the "mean" of the population in the direction of adaptation to environmental changes.

5. Disruptive or diversifying selection that results in the favoring of two or more genotypes concurrently.

6. Sexual selection that usually results in differential morphologies of male and female, or sexual dimorphism,

7. Co-evolutionary selection, or counter-adaptational selection, or the mutual changing of two or more interacting populations of different species.

Beyond these types of selection, other selection patterns can be found. Differential selection is a variety of sexual selection or balancing selection, affecting different subgroups in different ways. Competitive selection may be another form of common selection. While all forms of fitness and adaptation may be construed in some sense as competitive, it is evident that some forms of competition are more clearly marked and selective than others. Specializing selection can be considered a form of diversifying selection that affects specific traits or trait complexes within a population on basic feeding or breeding adaptation. Generalizing selection can be considered a form of homogenizing or stabilizing selection that selects for generalized trait complexes that permit organisms to expand the range of their niche adaptations. Oscillating selection, an important concept in evolutionary ecology, reflects the tendency for selective patterns to reverse themselves over the long term. Hybridizing selection would be selection that favors the mixing or hybridization of subspecies or species.

Natural selection does not change a genotype. This occurs primarily through genetic mutation, either as point or frame-shift mutation, transposition, recombination or structural modification of the chromosome. Selection merely changes, through natural process, the relative frequencies of genotypes found in population. Other dynamic factors also influence genotypic ratios, including migration, which introduces "gene flow" between populations. Random genetic drift is held to be the natural chance fluctuations in gene pool frequencies, all other factors being neutral. In gene flow, two other effects are considered important: the Founder effect is the introducing of a particular genotype from a single founder or a small colony, leading to subsequent expansion of the population, significantly altering genotypical ratios between the parent population and the founded population. The other affect, closely related to the founder effect, is the "bottlenecking" of a population by its reduction in size, altering the ratios of gene frequencies in the resulting population.

This model must be understood as emphasizing the genotypic aspects of the population. From the standpoint of population genetics, this is referred to as the "Mendelian population" (or deme, as in demography) which comprises a gene pool, consisting of all genes in discrete frequencies which are described in terms of the famous Hardy-Weinberg Law.

In general, any gene pool has continuity in both space and time, and all organisms of a common population are potentially capable of interbreeding. This describes a species as the maximum size of any community, and any subspecies population within it. This law predicts that genetic frequencies of a population will remain constant about a mean from generation to generation, if factors of selection, mutation, migration, etc., are not present. The basic formula as given as the expansion of the standard binomial:

(p + q)² = p² + 2pq + q²

Where p is the proportion of a gene "A" and q is the proportion of the associated gene "a" in the possible combinations of alleles as "AA" (pp), "Aa" (pq) and "aa" (qq). If complete randomness of reproduction is hypothesized, and all other things being equal, then the total genotypic frequencies of these alleles remain the same for each successive generation. The extension to multiple alleles is direct.

If we assume three basic phenotypically distinct genotypes occurring in a population, such that AA = x, Aa = y, and aa= z, with random mating is assumed. Where the sum of two alleles p and q equals 1, (p + q) = 1, and where there is no natural selection, then the resulting distribution pattern will look like the following:

The frequency of hetero-zygote mating will dominate for each type of mating. The proportion of frequencies should be the same for all types of mating occurring. We can portray the distribution of progeny genotypes from 9 possible combinations shown above, listed by genotype, with the sum of each genotype at the bottom:

If the resulting summed expressions are factored for each genotype, then the following results are obtained:

Having calculated the progeny genotype expression, we can rewrite the total frequencies making use of standard type allele relationships so long as all three genotypes are phenotypically distinct and thus countable. The frequency of any one allele can be obtained by finding the proportion of homozygous plus the proportion of 1/2 heterozygous.

We can rewrite the following:

Frequency AA = x² + xy + 1/4y² = (x + 1/2y)² = p²

Frequency Aa = xy² + 2xz + 1/2y² + 2yz = 2 (x + 1/2 y)(1/2y + z)= 2pq

Frequency a = 1/4 y² + yz + z²= (1/2y + z) ²= q²

Thus the apparent genotype frequencies of the original generation have become after one generation:

p² + 2pq + q²

The new frequency of each alelle after one generation remains unchanged. This defines a basic equilibrium of genotypic frequencies from one generation to the next, given that no natural selection or drift or gene flow has occurred. Equilibrium of course is not expected among natural populations where there is deviation from random mating. Inbreeding will result in increasing homo-zygotes and decreasing hetero-zygote frequencies. Out-breeding will result in increased hetero-zygote frequencies. Assorted mating or preferential mate chose based on resemblance of features, stature, or specific traits, will tend to result in increased homozygous frequencies.

Drift is defined as the random fluctuation of genetic frequency distributions of a population about a norm, that is like a sampling error, leading to the chance elimination of genes in small groups. This is treated quantitatively if neither selection nor migration is assumed and mating is entirely random. Then drift will be the positive or negative fluctuation of the value of the frequency of q with each generation, in which the magnitude of the fluctuation ∆q cannot be predicted in advance. On can construct a distribution of expected changes with a mean of 0 and a variance of:

s² = pq/2n

where s is the normal symbol of population variance and n refers to the number of reproductively capable individuals.

The emphasis upon drift is the basis of a "neutralist" theory of evolution that largely discounts the effects of other forms of natural selection. Drift is found in interaction with other mechanisms of change, namely natural selection, migration or gene flow, and mutation to produce evolutionary outcomes.

We can take a simplifying set of conditions to explore the effects of selection on genetic frequencies of populations, with two sets of alelles p= frequency A and q = frequency a, and equilibrium is assumed at the start such that:

AA = 1

Aa = 1 - s₁

aa = s₂

Given original genotype frequencies of p² + 2pq + q² after each subsequent generation the proportion of these genotypes will change, obtained by multiplying frequencies by new fitness values:

(p² * 1) + 2pq (1 - s₁) + q²(1 - s₂)

Where the sum of these frequencies does not equal one any longer, because all the genotypes are no longer contributing an equal number of progeny. If we rearrange values to equal 1 of the new generation, then we get the sum of genotype proportions:

p² + 2pq - 2pq s₁ + q² s₂ = 1 = 1 - 2pq s₁ - q² s₂

We can then calculate the actual genotype frequencies that can be expressed as ratios:

Aa = 2pq/(1 - 2pq s₁ - q² s₂)

Aa = q²(1 - s₂) /(1 - 2pq s₁ - q² s₂)

Thus we can calculate the change of frequency of any one allele, p or q as the result of selection. We can calculate the frequency of an allele as always one half the frequency of the hetero-zygote that is distinguishable from either the frequency of the homo-zygote dominant or recessive:

∆q = q₁ - q = 2pq (1 - s₁) + q²(1 - s₂) - q/(1 - 2pq s₁ - q² s₂)

= pq [q(s₁ - s₂) - s_1p]/(1 - 2pq s₁ - q² s₂)

Delta q is 1/2 the frequency of Aa allele plus the frequency of aa - q, and it is directly proportional to the product of pq. Delta q is largest if pq is lower. Selection will slow down if q diminishes and delta q becomes smaller. In other words, one allele, q, is progressive excluded at the expense of the other. Delta q is directly proportional to the average fitness of individuals in the selected group, denoted ẅ, such that:

(1 - 2pq s₁ - q² s₂)

A mean fitness value less than one means that delta q will be relatively large and change is rapid, but when mean fitness is high, delta q will be low and selective change will be low. Change will work to maximize the average fitness of each population.

In this way, w can be derived to fit a variety of circumstances. If selection is only operating against recessive homozygous alleles, such that AA = s₂, and Aa = 1 then s₁ = 0 - s₂ and ∆q = s₁ = 0

∆q = -pq²s₂/ (1 - q²s₂) = -pq²/ (1 - q²) = -(1-q) q²/ (1 - q) (1 + q)

If delta q is negative, then the frequency of the a allele is reducing, and -pq² will become small as q decreases toward 0, and delta q will drop faster even when selection is complete and homozygous recessive alleles are removed, the small a allele will linger in the population. Any normal population will carry any number of recessive traits minimally for a very long time at low frequencies. Some recessive traits will drift away by chance.

It is evident that for any sexually reproductive form of life, there is direct genetic variation such that in each generation there is unavoidable production of individuals that are ill fit for survival. This, as J. Haldane wrote, is the "price that a population has to pay for the privilege of evolving." Each generation harbors a store of genetic variation that results in the loss of unfit individuals as the price of selection. A proportion of genetic variations is retained in heterozygous state without finding expression. Mutant alleles can be concealed for many generations by this means. Poorer genotypes can be reproduced repeatedly that reduce the net average fitness of the entire population. This is termed the genetic "load" imposed on any population. It refers to the potential or actual reduction of fitness of a population by the presence of genetic variation. J. Crow defined load as "the proportion by which population fitness is decreased in comparison with optimum genotype by presence of deleterious variation."

The optimum cannot be predetermined by non-arbitrary standards, but it generally refers to the most "normal" prevailing type, or else to a "wild" homozygous allele occurring normally in any generation, at least according to a classical model of evolutionary thinking. This has led to an abandonment of a classical model of evolution based on "drift" about a norm for a model of "shifting balance" of types that does not recognize a single "wild" type but an array of alleles that are essentially heterozygous in their matrix. In a normal range of environments many different alleles may be fit or normal. Thus the adaptive genetic norm for any population is polymorphous and heterozygous.

Load is expressed as L:

L = w_max - wˉ/ w_max

Where w_max is the fitness value of the best genotype and wˉ is the averagel population fitness.

Thus, if L = 1 - wˉ, then 1 is the relative fitness.

Three types of load are associated with mutation, natural selection and gene flow.

Mutation load is the fraction of the total population load consisting of recessive alleles usually present at low frequencies and maintained in recurrent mutations. Its effect is not apparent in hetero-zygotes but renders the recessive homo-zygotes less fit and leading to an aggregation of such an allele in a population. This is associated with diversifying selection. It can be modeled thus:

A > a at µ - ∆p = µp (mutation rate of p)

Where ∆p is the mutation rate times the frequency of p.

If p₀ is the frequency of large A allele in a population, and p₁ is the decreased frequency of this allele in the next generation and ∆p is the initial population frequency multiplied by the initial rate, then:

∆ = µp₀

and then:

p₁ = p₀ - ∆p = p₀ - µp₀ = p₀ (1 - µ)

After nth generations, p_n is substituted for p₁ and p_n-1 is substituted for p₀. Pn can be expressed in terms of initial frequencies of the first generations (p_n-2 ) such that:

p_n = p_n - ₁ (1 - µ) = p₀ (1 - µ)ⁿ

If back mutation can be assumed, then there is a more complex representation:

a > A = v

∆p = µp - vp

∆p = -µp + vq

∆p = µp - vq

Load associated with natural selection addresses the recessive homo-zygote, aa, in which a change in gene frequency is expected ∆q that is the result of selection and not of mutation:

∆q = -sq²p/(1 - 2q²)

This holds when selection weeds out the recessive homo-zygote. The -s is the coefficient of selection removing the small a allele, while mutation should have the opposite effect, by definition. Recurrent mutation could result in the retention of the allele despite continuous selective removal, which could lead to an equilibrium such that:

∆q = ∆p

If we set the former equation at equilibrium, we get:

pµ - qv = -sq²p/(1 - sq²)

If frequency q is responsible for the detrimental phenotype, it is likely to be minor and thus we can drop terms such that qv is close to 0 and (1 - sq²) is approximately 1, then we get:

pµ = sq²p

and

µ = sq²

q² = µ /s

The last equation represents the proportion of affected alleles. If A were dominant, then:

q² = 2µ /s

In this context, average fitness wˉ is 1 - sq², which is less than 1. In the case of selection against recessive homo-zygotes, wˉ = 1. When selection coefficients are not equal to zero, then hetero-zygotes are not an an advantage, such that:

wˉ = 1 - 2pqs₁ - q²s₂

And wˉ = 1 - 2 q² is optimum fitness. If wˉ = 1 - s2 q² then 1 - wˉ = sq² and therefore

L = sq²

And L (load) is equal to the mutation rate, such that if wˉ = 1, then

L = 1 - wˉ = sq² = µ

If the selection coefficient is set equal to one, all recessive homo-zygotes will be removed and q² will equal the mutation rate. These set the limiting values of mutation load relevant for populations. The case for the dominant allele can be worked out in similar fashion, such that in all cases L will lie between mµ and 2mµ.

Another form of selection load is referred to as balanced or segregational load that results from one or another form of balancing selection favoring the homozygous alleles at the expense of the heterozygous. This is counter-balanced by stabilizing or normalizing selection in which either allele as a homo-zygote is less fit, thus favoring heterosis or the fitness of the hetero-zygote.

We can work out average fitness for any population frequencies undergoing balancing selection or heterosis such that:

wˉ = 1 - p²s₁ - q²s₂

In the case where s₁ equals s₂ and both equal .1, then the mean fitness at equilibrium is .95 with .05 percent of zygotes lost in each generation. If most organisms are poly-morphic at many loci simultaneously, the effects of fitness will be cumulative and the load becomes unbearable in order to retain overall reduction of fitness. Average fitness is too small. This is Haldane's dilemma. The alternative explanation is that most variation is simply neutral in effect on fitness, and a kind of steady state is reached, with a dynamic balance maintained between selection, drift and chance mutations. There is neutral accumulation without loss of fitness.

The third form of load, substitutional load, is associated with directional selection and environmental shifts and leads to a favoring of the recessive homo-zygote at the expense of the dominant homo-zygote.

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But "fitness" as both an individual measure and a population parameter, must be understood and defined only in the context of that individual's total environment. This means that it cannot be interpreted exclusively from the standpoint of population genetics. Population genetic arguments for natural selection have been criticized for the functional tautology of defining such selection in terms of fitness, and fitness in terms of selection. Furthermore, in its pure form, when emphasizing balancing selection characteristic of cladogenesis, it has not been able to overcome "Haldane's dilemma" which is basically that rapid genetic evolution, especially in conditions of balancing selection, requires too great a load in too small of a population. It would wipe out the population by the overall reduction of fitness. On the other hand, if change were spread out generationally over a long term, genetic change would be so slow as not to lead to evolution at all.

Fitness is a relative measure of "adaptation" by an individual, and hence, by derivation, of the group, to an open-ended range of variables in the life-environment of that individual or group.

As mentioned in the previous chapter, it is clear that issues of survival, and life itself, underlie and in part predetermine issues of reproductive success, thus we need to seek a deeper and expanded understanding of fitness as adaptation, and by extension and relation, selection.

I have sought therefore to distinguish between models of adaptive fitness and reproductive fitness, as defined above, as well as to explore the relationships between these two kinds of models. To understand adaptive fitness, it is necessary to understand the individual as an organism that must function effectively within certain limiting constraints in order to survive.

Adaptation can be defined in biophysical terms as the measure of conformity between an organism and its environment. Adaptive fitness would be this measure of conformity/disconformity that can be defined in terms of physiological requirements and associated resources, and functional behavior. A "niche" especially understood as a fundamental unit, can be considered to be the expression of the relative adaptive fitness of an organism expressed in environmental terms.

For any given environment, there can be calculated a given optimum state of perfect adaptation for any given organism of a specific kind. For any given organism of a specific kind, there can be calculated to be some ideal set of environmental conditions that is ideal for that individual. In general, there is never a perfect fit between an organism and its environment. Individual organismic adaptation in general tends to follow changing conditions in the environment.

By extension, we can hypothesize that for any given eco-system, there are a set of adaptive states between the individuals that comprise that eco-system and their environments that is ideally optimal for that eco-system. The measure of actual to "ideal" adaptation can be understood as the degree of equilibrium achieved by that ecosystem. Equilibrium can be defined therefore as the net measure of adaptive fitness of an ecosystem, as determined by the adaptive fitness of the members of that system in interrelationship to the group's common environmental context.

Evolutionary entropy assures us that these ideal conditions will at best be temporarily approximated, much less permanently achieved. In general, because ecosystems are partially open systems, optima of adaptational equilibrium within the system will always tend to follow these exogenous changes. Furthermore, by extension, changes of the ecosystem will tend to lead to changes in the adaptational regimes of the individual organisms within that ecosystem, such that the optima of adaptive fitness of any individual within the system will tend to follow or track the continuous changes of the overall system.

Any given ecosystem with defined sets of limits offers a theoretical optimum that can be achieved in basic terms of adaptive equilibrium along the lines of those limits. In fact, there is probably a range of potential optima that can be achieved by any ecosystem, depending on the biological constituents and alternative pathways of development that can be taken by those groups contained within it.

Climax communities tend to be the most complex that a given ecosystem can support. Such climax communities are relatively heterogeneous. Any given area can potentially support an infinite number of alternative kinds of ecosystems, each with its own optimal climax states. Climax communities are correlated with the extreme physical limiting factors. They tend to be as complex and heterogeneous as the region can support. In optimum areas, where biological limiting factors are most important, climax communities are the most complex.

To rephrase E.P. Odum's classic statement: "The survival and reproductive success of an organism or a group of organisms is dependent upon a complex set of conditions."

In seeking a more precise definition of adaptational fitness, especially as this relates to the individual organism, the concept of "performance curves" and limiting factors are generally employed. A limiting factor is conventionally any identifiable factor that sets limits to the size of an individual or in the numbers of a population. In a more general sense it can include behavioral and functional limiting factors that set limits to the functioning or behavior of the individual.

A limiting factor is any condition that approach or exceeds the limits of tolerance of an organism or a group of organism. The limits of physiological tolerance for any organism along any dimension usually describe bell-shaped unimodal performance curves. For any such curve, there is an optimal range of performance that includes the central regions of the curve. The tails of such tolerance curves represent limits of tolerance. Distinction is made between broad, low curves (eury-) and narrow, high curves (steno-) in the description of performance curves and limiting factors.

Along any adaptational dimension, there is usually a range of variability of performance that is represented by the members of a reproductive population within a given ecosystem. The population parameters of this group describe average curves of performance and limiting factors, to which individual members may deviate or approximate.

In general, it can be said that changing external conditions can alter performance curves of an individual (acclimation or aclimitization). If an individual migrates into a new territory or region, all other things being equal, that individual will have to adapt to a new set of limiting factors and variables that describe new optima of performance along different adaptational dimensions.

Thus, for any individual in any given environmental context, there can be said to be a number of limiting factors affecting that individual's adaptability. There are both extrinsic and intrinsic limiting factors. Intrinsic limiting factors may include morphological and physiological design constraints. In a forest, a heavy animal adapted to the ground cannot easily climb trees. Birds whose beaks may be adapted to eating fruit, will likely find eating seeds more difficult.

Intrinsic limiting factors include basic energy requirements of an organism that are related to the metabolic function, size and mass of the individual organism, as well as such things as shape of the body. For instance, endo-therms have certain minimal body size and higher energy requirements than do comparable ecto-therms. These requirements in turn affect the feeding patterns of such animals. It affects as well their morphology and their patterns of movement and migration. In terms of energy requirements for animals we may posit the following kind of squared table:

In general, for any organism along any given multiple adaptational dimensions, there can be said to be a trade-off in tolerance limits and performance curves, such that to change towards higher optima along one dimension, will mean moving towards lower optima along other, related dimensions. In general, most performance curves are sensitive to two or more environmental variables. This is the principle of allocation in tradeoffs in tolerance limits and performance curves, based upon a limited supply of energy or other basic resources that are required for physiological, behavioral or functional adaptation. To increase the range of one's tolerance limits along a curve often means to push the curve to lower optima along another dimension.

The "law of the minimum" states that growth or population will be dependent upon the minimum amount of nutrients available to it. The law of the minimum is important in understanding the influence of complex limiting factors upon a population or an individual. Whatever factor or set of factors that set the greatest limits upon an organism or population, will tend to be the defining limiting factors in terms of that organism's adaptation and development.

Limiting factors affect population dynamics in complex ways. We may distinguish between "density-dependent" factors that are determined by the population density of a given area. Density-dependent factors set external limits to the carrying capacity of a region for any given population. Density-independent factors are those that occur in a region or area regardless of the specific population density.

The law of the minimum can be extremely important upon a population, especially when this is a basic resource requirement. The effect of this may be only seasonal or periodically felt, and yet it could drive the entire evolutionary system forward. In saturated systems, whether such a factor is density independent, affecting all members of the population more or less equally, or density dependent, makes little difference. Extreme restriction can result in mass deaths and a subsequent bottlenecking of the population to a small number of new founders that may be more adapted to the conditions of the minimum.

We can see minimum determining factors often connected to fairly discrete trait factors or complexes that have straight-forward functional consequences in the lives of the members of the population, such that minute variations may make a critical difference in net outcomes. These traits often link to issues of either feeding or breeding. These kinds of functional trait adaptations provide vital clues to thinking about and constructing the fossil record and taxonomic patterns of speciation. Traits can almost be seen in this framework like tools of survival or reproduction that allow success to the individual who controls them. When we study a set of traits of a specimen, we must ask ourselves, what function could this trait have served in the adaptive survival or reproductive success of the individual in times long past?

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Fitness is expressed in evolutionary terms of trait adaptation of the individual. Traits may be begging the question a bit, but it fixes the meaning of fitness in more concrete terms. This implies an important evolutionary relationship between an organism's trait complex and its survival. In general, traits evolve for functional reasons, that they serve some purpose in the adaptive fitness of the individual. They help the individual to survive. It is true that some traits are vestigial--either they are not true traits, or they are coincidental traits that are the by-product of trait development, or else they are residual survivals of a bygone organism within which such traits served some function.

Understanding adaptive fitness, underlying reproductive fitness, as "traits" gives us a clear genetic connection between the issue of adaptation and selection. In this regard, a trait can be defined as some mechanism conferring adaptive fitness along some performance curve (or curves). It is a measure of the organism's ability to adapt in the dimensions represented by such curves.

Each organism constitutes a unique set of traits with a unique set of values that can only be measured in their consequence in some environmental context. There is of course great variability of trait expression, such that no two individuals are exactly alike, except perhaps for homozygous twins.

Trait plasticity can be accounted for on various levels. The polymorphic and pleiotropic character of genes that affect multiple traits in different complexes, suggest that genes function as transcription and transformation algorithms, within which process there is some room for variability of geno-typical expression. Phenotype varies also because during the course of development of the organism, environmental limiting factors do play a part in its epi-genetic expression. Furthermore, genotype and phenotype do not fully determine the extra-somatic behavior of the living organism, and it is this behavior that can lead to positive or negative selection.

Trait plasticity is often in nature finely controlled in fitness and selection. Almost any trait, over the long run, is nearly infinitely plastic, such that the hands of nature through selection may gradually molds a trait in almost any direction it may see fit. Thus in time, fins can be come legs, and legs can become arms and hands, and arms and legs can become wings, and wings can become legs again and legs can return to being fins once again. Trait plasticity appears to be fairly continuous. The fact of "jumps" in the fossil record are the result of the lacunae in the deposition of these fossils, such that the intermediate forms are lost. The process of trait variation that leads to its plasticity may increase or decrease in rate and frequency over time. New suites and forms of traits may emerge rather rapidly in the record under the right conditions.

We might hypothesize a kind of rule that states: where ever we find wide variation of a trait, we are also likely to find rapid evolution of that trait as the result of its great plasticity. To some extent, the load of variation carried by a population in terms of certain traits or their complexes is off set dramatically by the possibilities of this wonderful biomorphological plasticity that permits, in the long run, the adaptive success of a new species.

We might hypothesize another related rule that states: trait complexes that are driven by adaptation often blindly seek and find an optimal solution to the set of problems posed by adaptation, and this can be referred to as trait streamlining. Part of the wonderful progressive intelligence of blind evolution is its ability to find the best fitness pattern for a particular adaptational context or range of contexts, given enough time. Thus, many trait complexes recur in the fossil record as parallel "convergences" of pattern. The degree of convergent evolution between different kinds of species has been truly remarkable. This "intelligence" can be seen as the end-product of a vast exploration of possible solutions to a common problem of adaptation by an on-going population, as the result of selective fitness that is tied to almost infinite trait plasticity.

It leads us to speculate about another kind of phenomena related to trait-plasticity, and that is the frequent appearance of hyper-tropisms in the fossil record. Hyper-tropisms can be described as specialized traits that are very exaggerated and marked in their form, for instance, the teeth of the saber-toothed cats. These are the "end of the line" products of trait plasticity. Huge canines may have been effective for these large cats in felling large prey, but they probably made it very difficult to deal with smaller prey packages. Extreme hyper-tropisms in general appear to be the result of overspecialization to a particular adaptive niche. They may also be the indication of a species that has gone too far out on a limb of the evolutionary tree.

I propose the following generalization:

Individual variability is to population parameters what organismic adaptability or fitness is to selection.

In this understanding, I conclude that all natural selection must be construed first as trait selection, and that the kinds of selection that are conventionally described are in fact only the description of different pathways of such trait selection. We can say that trait selection tends toward the optimization of certain suites or complexes of traits in a number of different directions simultaneously. Selection statistically favors some trait complexes over others. It does this by selecting in or out the entire individual trait complex. Selection increases fitness by increasing trait adaptation of an individual.

Each individual organism embodies a unique set of traits with a unique set of values. Each species embodies a unique range of traits with a large range of possible values.

We can say thus:

All traits are genetically determined in the individual organism.

All traits are pheno-typically expressed in the individual organism.

The expression of all traits is environmentally influenced.

The outcomes of trait selection that works on an individual level can only be determined within population trait parameters. Every group has a finite range of variability along a trait continuum. The relationship of fitness to a trait complex leads to selection for or against that complex. Trait selection specifically links both adaptive and reproductive fitness of the individual to group selection and survival. Trait selection therefore tends toward optimization of fitness along a number of different trait-adaptive dimensions simultaneously. It explores this landscape and seeks complex solutions.

Traits may be any aspect of an individual, its functioning and behavior that may be described in discrete, measurable terms. Thus, there is in traits an inherent comparability between organisms and between different kinds of organisms. This idea of trait comparability has important implications in understanding evolutionary taxonomy and history.

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Trait-selection is not the same for all possible traits, nor for all organisms. Of special importance here are what can be considered to be key traits that account for basic patterns of selection, speciation and evolution. Some traits are held to confer special advantages to species that acquire them. Especially in this consideration it is those traits that characterize stadial or graded developments in the evolutionary record that are of importance. Thus we may look for basic traits of brain development, sociability, mobility, morphological "independence", or certain specific mechanical specializations as having special importance in understanding evolutionary history, and these traits occur not as individually identified features, so much as they appear in basic trait configurations or patterns.

I will thus hypothesize two different general kinds of trait-selection:

Generalized trait selection tends toward general selection of overall organismic trait configurations and patterns, involving multiple traits.

Specialized trait selection tends toward discrete selection of single sets of traits alone narrow dimensions.

In this regard, I propose that two of the most basic and important forms of key trait selection are for rate of reproduction, what I will call reproductive selection and for body size, what I will call size-selection. Size is clearly linked to rates of reproduction, so much so that there is a nearly perfect fit of positive linear regression between increasing body size and lower rates of reproduction.

In consideration of these kinds of basic trait-selection patterns, I propose the following paradigm based on what I call the "Evolutionary Clock:"

1. Trait mutation is basically random except in some unusual conditions, and therefore, in the largest context, tends to occur at fairly uniform rates that fit a natural bell shaped curve.

2. All other things being equal, the absolute rate of genetic change, in the long run, is fairly uniform and constant. The mutational clock ticks at the same rate, overall, for all species, but this clock is selectively tied to size, such that for each species the evolutionary clock ticks at a different relative speed.

The evolutionary clock is therefore relative to the species.

3. Different species evolve at different speeds because they reproduce at different rates.

4. Cellular division and differentiation for all life forms conform to an average rate. Periodicity between reproduction events varies widely for all life forms.

5. Smaller size life forms tend to reproduce faster than larger size forms, and therefore evolve faster.

6. Larger size life forms may be selected under some conditions for a variety of reasons, involving predatory exclusion, feeding pattern, competitive advantage or mate selection.

7. Larger size forms tend, on average, to be more differentiated than smaller life forms, and therefore tend to incorporate a wider range of traits than smaller ones. We may say that they have greater degrees of freedom of trait plasticity and adaptability, while smaller forms have a narrow range of constraints.

8. There is a trade-off between larger size and faster rates of reproduction, such that larger, more differentiated species gain in wide-spectrum trait adaptability, but lose in their rate of reproduction.

a. Smaller life forms accomplish through speciation what larger life forms accomplish through organismic trait adaptation.

9. Larger organisms exhibit therefore a greater range of trait-variability and plasticity than smaller organisms, and therefore are subject to more different kinds of trait-selection.

10. Once larger life forms have evolved, their rates of reproduction and evolution are slower.

With this model of basic trait selection in mind, I will suggest the following kind of basic paradigm where specialized/generalized traits are contrasted with larger and smaller sized species:

In consideration of this kind of table, I will speculate on the following:

All species naturally tend towards opportunism in their trait-adaptation. For survivorship and reproductive success, they must take advantage of situations as they happen. This naturally leads to a context where, if possible, populations will increase to the natural limits of their environmental niches. In other words, in the best of possible worlds, populations will tend to work towards their carry-capacities.

Thus, I believe it is a misconception to think that species tend toward equilibrium with their environment. Individuals and populations cannot through their normal patterns attain homeostatic equilibrium with their ecosystem, though in the long run this might be the most advantageous kind of adaptation to achieve.

States of equilibrium are always temporary and relative, and must only be construed from the standpoint of environmental adaptations of individuals and populations in eco-systems. It can be said that eco-systems evolve towards periods of stasis, or relative complex equilibrium as the result of give and take within the system between different forms of life at different trophic levels. They also develop as the result of the long-term trait adaptation of host populations to larger oscillatory cycles that may occur in the geo-physical context. Homeostasis is only a relative conception, mostly of the exception and the very short-lived. Equilibrium within evolutionary eco-systems tends, in the long run, to be quite dynamic.

It can be more accurately said therefore, that populations that are in equilibrium, are trait-adapted in balanced eco-systems such that the selection regime they follow tends towards equilibrium in the long run, all other things being equal.

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In order to understand the significance of the preceding table and discussion, it is necessary to return to our formulas about rates of reproduction and population and to discuss the natural regulation of population growth. In ideal circumstances, all populations will tend to naturally increase until their niche-defined carrying capacity is reached. In fact, under the best of possible circumstance, any population would naturally overshoot its carrying capacity resulting in the proliferation of wider variation of traits.

In general, equilibrium of a population is considered to be at its carrying capacity, or K. It is the optimal state of adaptation for any population, above which there will only be decrease and environmental deterioration. Now it must be seen that K is a dynamic that tracks changes in the overall eco-system. In a sense it is the measure of the size and health of an eco-system. Thus K is not a constant value, but a variable that is always fluctuating.

In the following sets of equations, the intrinsic rate of reproduction (r) and the net rate of reproduction (R₀) are allowed to vary with population density. In this regard, carrying-capacity (K) is defined as the density of organisms in a given area at which R₀ equals unity (1) and r equals 0. If there were zero-density, a population of only one parent, then R₀ is maximal and r is maximized (r_max). Both the net rate of reproduction and intrinsic rate of reproduction will decrease as a population grows until the population ceases to grow at its carrying capacity. If the population is larger than K, then the population is expected to decrease to K.

It must be noted that this issue concerns the total individual, for which size is taken as the principal common trait, such that for each individual, there is a definite size, and for each population, there is a definite average size. It therefore comprehends the total trait complex of the individual, as well as the total range of trait variation of the population.

In these considerations, r_a is defined at the actual instantaneous rate of increase that is zero at K, positive below K and negative above K. It is given by the following formula:

r_a = dN/dt - 1/N

where d is the death rate of a population

It is assumed that r_a increases linearly with N and is zero if N equals K.

If we assume:

1. all individuals are equivalent.

2. r_max and K are constant

3. and no time lag exists between rate of increase and changes in N

Then the result is the Verhulst-Pearl Logistical equation:

DN/dt = rN-rN(N/K) = rN -rN²/K = rN(1-N/K) = rN(K-N/K)

If r/K is set equal to z, then we get:

DN/dt = rN - zN²

In this formula, rN(N/K) or zN² is the density-dependent reduction in rate of population increase. If N equals zero, this dN/dt is exponential, or if N equals K, then it is zero and the population is in a steady state at its carrying-capacity (equilibrium). These kinds of equations lead to s-shaped (sigmoid) growth curves typical of population curves.

The actual rate of increase, r_a, which is a function of r, N, and K, can be solved in the preceding equation by factoring out N such that:

r_a = dN/Ndt = r(K-N/K) = r - (N/K) r

It can be concluded that r_a is always equal to or less than r_max

Per individual,

r_a= b - d

And in ideal conditions, b is maximized and d minimized such that r_a approximates r_max that can be only realized at minimal density.

If b and d are subscripted, we get:

r_N = b_N - d_N

and, therefore

r_max = b₀ - d₀

Which means that the population attains equilibrium and the actual rate of increase and dN/dt equals zero, when b_N equals d_N

Therefore both b_N and d_N covary linearly with N:

b_N = b₀ - xN

d_N = d₀ + yN

where x and y are the slopes of a line plot

In the last formula, the instantaneous death rate incorporates both density dependent components (yN) and density independent components (d₀).

Considering the equality of subscripted b and d at equilibrium, therefore also:

d₀ + yN = b₀ - xN

N_e may be substituted for N at equilibrium, and r for (b₀ - d₀) we get:

R = (x + y) N_e

And

N_e = r/(x + y)

The sum of the slopes of b and d (x + y) equals z or r/K, which is a density dependent constant that is analogous to the density independent constant r_max.

It is true though that in nature, none of these presumptions strictly hold, such that there is substantial variability between individuals, K is constantly fluctuating and complexly variable, and there is always a variable time lag to be expected in adjustments of rate of population increase and total resulting population. It is therefore more realistic to assume curvilinear relationships between rate of increase and population density, which lead into astronomically complex equations, especially considering inherent complexity and composite variability of the variables themselves.

The Verhulst-Pearl equation is considered most representative for small changes in population growth that occur near equilibrium and over short periods of time during which relative linearity can be assumed.

The issue of density-dependency and density-independence of limiting factors in the environment, coupled with the law of the minimum and principle of allocation, determines that we probably can never clearly sort out what are truly density-dependent from density-independent variables. Density independent factors affect all organisms to the same proportion regardless of density. Climatologic variations, on average, are considered density-independent factors in any given area, though the affects of such variations vary continuously across such areas, thereby affecting differentially different organisms of the area, based on density-distributions. If a flood sweeps across an entire defined area, taking away all organisms of that area, then we have a truly density-independent factor of selection. But most often, especially at the level of population or species ecosystems, floods to not impact equally in all areas simultaneously. Only factors like length of day can be said to approximate a truly density-independent relationship.

I propose that density dependency is a relative value, for any given limiting factor, or suite of limiting factors along a continuum constituted by an ecosystem, such that we may plot any specific factor or set of factors somewhere along a curve that only approximates perfect density-dependence or independence:

This suggests a basic log-linear formula for the relationship between density, where density dependence (x) and density independence (y) are variables of any given limiting factor or set of limiting factors, and x = N - y(z)

The relative importance of density-dependent and density-independent factors varies considerably between populations and eco-systems, and fluctuates continuously over time. The closest we can come to asserting control is to estimate the relative degree of density-independence of a factor that is held to be a minimal determining factor.

Density dependence and independence also co-vary in another way. Density dependence has a built-in assumption of a carrying capacity, such that the further from the hypothetical carrying capacity one is, the more density-independence can be assumed to play a defining role.

We can picture a unimodal bell shaped curve at which the mid point is a presumed optimal level of equilibrium (carrying capacity) and at which density dependency plays the greatest role. At some point, along either tail, a cut-off must be made arbitrarily at which density-independent factors must be presumed to play a greater role. This links the notion of density dependence/independence directly to the population equations we have been considering.

Another way of looking at this is to assume that in any given area of a definable carrying capacity, such that low-density values, or density values exceeding this optimal threshold, become more susceptible to density independent variables. If there is only one reproductive individual in a region with an estimated carrying capacity of 1000, it can be assumed that all environmental limiting factors affecting that individual, however minimal, will be equal. On the opposite extreme, it can be considered that in an area containing 10,000 individuals that has a carrying capacity of only 10, that all factors will be almost equally constraining, and therefore essentially density independent.

Now, a constant that is density independent is the maximum rate of natural increase of a population (r_max. The value of z, or the sum of the slopes of the death and birth rates, or r/K, is held to be a density-dependent constant that is defined by the carrying capacity of the population and the actual rate of instantaneous increase.

If this is true, then at maximum disequilibrium of population, it can be expected that maximum rates of reproduction will be expressed.

Death rates can be assumed to be tied directly to adaptive trait selection, and birth rates are tied to reproductive trait selection. While it appears that death rates are determined ultimately by birth rates, (only those born can die), such that birth rates are independent variables and death rates are dependent variables, it is also often the case that in fact birth rates track or follow fluctuations in rates of death. This kind of quandary links back to the hen and egg dilemma, and it must be seen in our original evolutionary formula that in fact birth rates, as the principle expression of the reproductive imperative, are dependent variables on death rates, which are assumed to be the expression of the life-imperative.

Both death rates (which imply negative selection) and birth rates (which imply positive selection) are normally construed to be density-dependent variables, such that as density increases death rates are held to increase and birth rates decrease in a natural manner. Death rates incorporate truly density independent factors, while birth rates are ultimately constrained by these factors.

It is also understood that death rates and birth rates are at least indirectly interdependent in ways that are density-independent. Whereas at equilibrium, K, death rates and birth rates are equal, it can be seen that birth and death rates co-vary such that increasing death rates equals increasing birth rates, and increasing birth rates leads to increasing death rates. Death rates and birth rates can be said to be proportionately equal at unity, such that:

b/d = 1

where b = 1/d and d = 1/b

This is a very stable formula that assures that populations will tend to approach and maintain themselves at equilibrium, all other things being equal. This complex relationship between birth rates and death rates rests on the assumption that populations naturally reach and maintain themselves at equilibrium with the carrying capacity.

It can always be assumed that this relationship is indirect, because there is always a lag between increase in birth rate and increase in death rate, and hence also between increase in death rate and consequential increase in birth rate. The lag between death and birthrates vary in direct proportion to the longevity or average life-span and length of gestation between conception and birth. These variables are directly related to the average size of the organism, such that we can assume the following:

Smaller organisms on average have a shorter lag-time between death and birth rates.

Larger organisms on average have a longer lag-time between death and birth rates.

In this regard, it can be understood that at any one time, the rate of births will be unequal to the rate of deaths, because the presumption of the lag between the two rates in balance. If the rate of death is high, at equilibrium, it can be expected that the rate of birth will eventually increase in a proportionate manner after a given period of time. If the rate of birth is high, it can be expected that the rate of death will increase, after a given period of time.

In general, it can be said that the lag between the initial change of the rate of death and the resulting change of the rate of birth is shorter on average (db) than the initial change in the rate of birth and the subsequent change in the rate of death (bd).

It can be assumed therefore that the relative density-dependency of birth rates and death rates are similar to the relative values of density-dependency and independency in the first place. In this way, density dependent factors play a bigger role at equilibrium, whereas density independent factors play a bigger role at states of maximal disequilibrium between death and birth rates (i.e., where the population N is greatly unequal to K).

It must be concluded that the two relationships (death rate as a dependent variable of birth rate, and birth rate as a dependent variable of death rate) are not necessarily equal and the same. What happens as a result of death-instigated processes (departures from the curve of carrying capacity of a pre-established population) is fundamentally different than what happens when as a result of birth-instigated population processes (establishment of a new population or replacement of an old population).

In the first place, it can be seen that most populations fluctuate quite dramatically about the variable line of equilibrium, in the long run. A natural population will tend in good times to overshoot the mark widely, leading to over-population. In hard times, mass death will call almost equally at every organism's door. In hard times, birth rates and death rates necessarily vary inversely such that birth rates will remain low while death rates become high. Death rates and birth rates therefore are not directly tied to one another as this equilibrium equation assumes.

Furthermore, death rates are a form of direct selection that is adaptive directed to any organism in a population. There is genuine relative equality of opportunity in death. The paradox of this is that the most adaptively fit by definition would be the oldest survivors, who would also tend, on average, to be beyond their reproductive ages. Death can be expected to visit the young disproportionately, such that most organisms are selected out even before they are able to reproduce, often regardless of their innate fitness.

On the other hand, birth rates are the result of reproductive selection, and are preferential and differential, unlike death rates. The whimsical hand of nature often does not need to control the rate of birth, as it is to a large measure self-controlling and equally uncontrolled. In boom times, there can be a mad frenzy of sexual reproduction when rates of birth would be expected to slough off as expected by an equilibrium equation.

Equilibrium models presume a kind of stability of pattern in nature--a steady-state, that may disguise a great deal of chance variation of pattern, or chaotic happenstance and longer term oscillations in the background of the environmental context. It is therefore an over-deterministic model of an underdetermined system.

The alternate model is to see life caught in a blind cycle of "boom, bloom and doom." A boom and doom cycle is defined by the extreme limits of adaptation, rather than by some intermediate line of equilibrium. Thus carrying capacity is something ideal that may rarely be realized except in a very approximate and temporary way. Setting upper and lower thresholds for a population in a given environmental context, determines the limits beyond which it cannot stray unless it is headed for extinction. We can see this as the cut-offs in a normal curve of distribution of population that fluctuates in size and shape over time. The carrying capacity would be some population parameter that defines the overall character of the curve at any given time. While the maximum and minimum limits would be the range within which a population can safely adapt and evolve. To push a curve above or below these limit-lines is disastrous, and this confers a robustness about evolutionary processes in the long run, as it can be assumed that in most contexts, in most instances, life has fairly wide-margins in which it can range.

The density-dependence of death and birth rates is rooted in the equilibrium presumption, that once a population reaches carrying capacity, deaths and births become equal. Some populations can be observed to approach this standard of equilibrium more readily or approximately than other populations. Some populations typically follow more "opportunistic" patterns of replacement, that resemble "birth-instigated" patterns, whereas other populations typically follow more "equilibrium" based patterns of replacement, that resemble "death-instigated" patterns. It can be unequivocally demonstrated that these are fundamentally size-dependent relationships. In other words, they are patterns that are rooted in size-selection.

If we go back to our theoretical presupposition of an evolutionary clock, we can deduce that the absolute or fundamental rates of reproduction for a given size (not number) of a population varies tremendously, by orders of magnitude, between different kinds of life-forms, and this is primarily a size-dependent relationship.

If rates of reproduction can be assumed to be high for small species, it can be concluded that rates of death will also be high. Such species achieve equilibrium far less on average, and experience relatively wide and rapid fluctuations of population.

If rates of reproduction can be assumed to be low for large species, it can be concluded that resulting rates of death also tend to be low. Such species achieve equilibrium far more often, on average, and experience relatively small overall fluctuations of population.

If we speculate that the Ra (absolute reproductive rate) underlies and determines maximum rate of reproduction in any given environmental framework r_max, and we assume that this value is very high for a small species (s) and very low for a large species (B), it can be also assumed that both bd and db lag times are shorter for the smaller species than for the larger species.

The result is to imagine that for a short-lived, small species that rapidly replaces itself, the density-dependence/independence curve for carrying capacity has a very high peak (x-axis relative to y-axis) and is very narrow and has very long tails. The density-dependence/independence curve for carrying capacity of a large, long lived and slowly reproductive species is proportionately speaking very low in its x-axis compared to the y-axis, with platy-kurtic principles, such that the movement away from the midline of equilibrium is more gradual and the resulting tails much shorter.

Many small species are in fact so small, that they can be said never to achieve carrying capacity of their environment, unless their environment is strictly delimited by some external factors. Thus, in essence, every factor that affects their population tends to be relatively density-independent. In a sense, very small life forms never usually achieve carrying capacity, because there is no capacity to achieve.

On the other extreme, it can be said that some species may become so large, that no matter how small their population or how large their environmental vacuum, they reach a state of relative equilibrium with their environment, such that every limiting factor becomes in essence a density-dependent factor. In a sense, very large-bodied populations are always usually at or close to their carrying capacity, regardless of their birth or death rates.

This interpretation is not quite complete or yet correct, as we know that even relatively small forms of life typically exhibit patterns that appear to be density dependent. Typically, they will form colonies that increase to a certain size before they appear to fission off. Indeed, there is no species that is completely density-dependent or completely density-independent. Small sized species often have so few dimensions of freedom of trait variability, with such a narrow adaptive range, that factors that are essentially density-independent may come to function or appear as if they were density-dependent. Thus small species will come to exhibit periodic population fluctuations about some mean that is an intrinsic characteristic of their limited trait range.

On the other extreme, once again, large species can exhibit such a fundamental range of trait variability, that factors that are essentially density-dependent may appear to function as if they are density-independent limiting factors. In other words, such populations may come to approximate the line of equilibrium so closely that their margins or extreme limits of adaptation may follow a very close range about equilibrium, such that even minor environmental fluctuations can have dramatic resonances upon the population in terms of death-instigated responses.

I will call this difference one of internalized adaptive trait-variation of a species that results in differential patterns of trait-selection tied to population saturation of a given niche. In general, the larger the species, the greater the internalized adaptive trait-variation.

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The systematic variations between populations that typically exhibit wide fluctuations about a hypothesized optimum of equilibrium, are said to be opportunistic populations. Populations that remain close within the boundaries of equilibrium established by a carrying capacity of a niche are said to be equilibrium populations. These two types are end-points of a continuum that is defined over time by both the line of equilibrium and the maximum and minimum limits of population. These lines are all fluctuating primarily due to exogenous environmental factors, and the trait-fitness and selection pattern that a population follows will vary accordingly.

In contexts of great density-independence, mass death will appear to have little to do with trait-selection, adaptation or absolute size of the population. Where greater density-dependence of relationships can be presumed about some optima of equilibrium, there appears less frequent mass death, and adaptive-selection appears to affect trait-selection on an individual level.

Opportunistic organisms do not deplete their natural resources in the large in the way that equilibrium populations can easily do. In an extensively rarefied competitive vacuum, opportunistic species achieve very rapid maximal birth rate--the maximal instantaneous birth rate approaches infinity in fact. In such contexts, competition is not a very significant variable. In equilibrium populations, they are more often at or near the optima of saturated equilibrium. In such contexts, density dependent factors are more important and rates of reproduction may tend to become minimized, guided by rates of adaptive selection and death. Competition in such contexts is greater, and such competition tends to favor larger organisms that require greater per capita energy.

These two opposing selection patterns have been designated as r-selection and K-selection, derived from the terms of the logistic equation. Just as no organism is completely on the density dependent or density-independent end of the spectrum, no organism can be said to be complete r or K selected. These represent two ends of a continuum of variability along which all organisms can be situated.

The characteristics of opportunistic, r-selected species are as follows:

They occur in climatologic contexts that tend to be variable, extreme and unpredictable. They tend to suffer catastrophic mortality that is selectively undirected and density-independent. They follow what is known as type III survivorship patterns. Population varies in size over time, with extremes of disequilibrium, and often occur well below carrying capacity in unsaturated communities in ecological vacuums that face periodic or annual re-colonization. Competition is often relaxed or variable, and selection tends to favor rapid development, high maximal reproductive rates, early reproduction, small body size, single reproduction, multiple offspring, short longevity, leading to high reproduction and early stages in succession.

By contrast, K selected species tend to have the following characteristics: They tend to live in stable climates with high predictability, greater intrinsic biodiversity and hetero-geneousness, with more directed and density dependent mortality, following type I or II survivorship patterns with a stable constant population near carrying capacity, in saturated communities that rarely experience re-colonization. Competition is usually intense and constant, and they feature slower development, greater adaptability, delayed reproduction, larger size, repeated offspring, fewer progeny that are larger, higher efficiency usage of energy, reproduction, and occupy climax stages in succession cycles.

Opportunistic species frequently become "fugitive" --that suggests that selection mechanism operating on such species tends toward their peripheralization and dispersion away from some central area occupied by a larger competitor. It is my contention that such species tend to exhibit inherent diversifying selection patterns that represent frequent bottlenecking and founder effects of population disruptions and relative isolation. This can be found even in thickly and uniformly concentrated environments, such as with the paradox of the plankton.

Consideration of these differences between r and K selected species, suggests that even rapidly evolving and diversifying r-species may undergo some measure of K-selection within their adaptive frameworks, such that they are led toward a more stable equilibrium with their environments. The next chapter will take up some of these questions in detail. The issue here is to suggest that all populations of all species exhibit natural population cycles that are intrinsic to the species. Though the causal complexity of these cycles are multi-factorial, it is apparent that their long term periodicity can be quite regular. A similar periodicity recurs in the cycles of mass-extinctions during evolutionary epochs.

It is obvious that these cycles are complex. While large species have cycles that may play out in years, or centuries or even millennia, small species may have cycles that play out in hours, days or weeks.

But it is important to emphasize that the explanation for the periodicity of these cycles must come from a systemic perspective, such that there is no one prime mover but that such unicausal explanations may represent a catalytic or trigger-effect in reversing the cyclical process. I believe that the central explanation for these processes are inherent to the population dynamics of all species.

As mentioned earlier, there are two sets of considerations to take into account in this regard. First, birth-instigated patterns are fundamentally different from death-instigated patterns. Secondly, whether a species appears relatively r- or K-selected, all species occupy a point somewhere along a relative r- K- continuum that is appropriate for the niche that a species normally occupies. Any species is always moving somewhere along that continuum, either due to shifting values relating to carrying capacity, or shifting trait-patterns relating to adjustments of adaptation, or some combination of both.

It follows from these points that all species follow normal cycles of patterning that lead that species from a position of relative r-selection to one of relative K-selection, and that under these shifting conditions the species will shift from birth-instigated to death-instigated patterns. A point of relative super-saturation will be reached by any species, at which time the effective carrying capacity is over-shot.

I believe that death-instigated patterns that are the result of relative over-population in a given area, no matter how relative and density-independent will tend to have catastrophic consequences for the entire population. The rates of deaths will tend to rapidly outstrip the rates of birth, and the lag time between these two will be sufficient enough to affect all or most of the population equally.

Rates of reproduction may be fluctuate seasonally, or may fluctuate predictably as the result of shifting death rates. Thus the periodicity of reproductive patterns may be different somewhat from the periodicity of shifting death rates. Thus an extended period can feasibly occur in regular cycles in which rates of death remain relatively high, while rates of birth remain relatively low during the same period or in a subsequent period.

Super-saturation can be described as a condition in which the optima of equilibrium shifts at the same time that adaptive trait variability shifts in the opposite direction along the density dependence-independence curve.

The result would be a relative condition where the population is primed for massive die-off due to relatively minor perturbations of density-independent factors. Death rates would be high at a time when birth rates would be low.

Such a state would be preceded by a local climax that would be followed rapidly by a sudden catastrophic anti-climax. Such a relative state might be expected to recur at regular intervals in a larger framework where external density-independent factors would be relatively stable.

After such a state of rapid dying off, the system would be expected to return to a state of openness or ecological vacuum at which point birth rates and death rates would catch up with one another and a new birth-instigated cycle would follow.

It must be seen that such periodic cycles would recur in the context of larger systems that have greater long-term stability and that influence the cyclical pattern in critical ways. Particularly, the fluctuation of what, for the smaller system, can be considered to be density-independent factors. In this sense, on one level, for a subsystem, what occurs as essentially a density independent variable, can become, in the context of the larger system at a higher level, a density-dependent variable. Vice-versa, what can be a density-dependent variable in a smaller system, can become, by virtue of local super-abundance, essentially a density-independent variable.

Some intermittent considerations:

Smaller organisms are inherently more prone to density-independent factors than larger organisms. To the extent that density-independent factors tend to be physical limiting factors, smaller organisms are inherently more prone as a population to environmental fluctuations of minimal limiting factors.

Larger organisms are inherently more prone to density dependent factors than smaller organisms, such that even minor fluctuations in resource allocation or availability may have major consequences.

Smaller organisms have shorter lag times between birth and death rates.

It can be seen that birth rates and death rates necessarily affect one another and co-evolve together. Reproductive trait selection closely follows adaptive selection, and in turn leads to changes in adaptational trait selection patterns.

A species maximal instantaneous rate of increase r_max is regarded as a good indicator of an organism's potential for increase and reproductive fitness. It is known that there is tremendous variation between organisms in terms of maximal rate of increase plotted to generation time, such that there is a strong inverse hyperbolic relationship between rate of increase and generation time. Since actual instantaneous rate of increase averages zero over a long term, smaller organisms with high maximal rates (r-selected) are farther from realizing their maximum rate than organisms with low rates (K selected). High maximum rates also indicate greater variability and fluctuation of actual rates. It can be construed therefore as a measure of "mortality" (negative adaptive) selection associated with a species niche, such that r_max is regarded as the best indicator of an organism's relative position along the r-K continuum.

This relationship of maximum rate of reproduction and generation time is directly correlated with body size, which is highly positively correlated with generation time. In fact, the plot of generation time to body size follows a clear linear regression formula.

The evolutionary advantages of size-selection must be taken into fuller account. Obviously, greater resource investment in fewer, larger offspring confers greater survival advantages in terms of other trait-adaptational factors, even if it reduces the maximal rate of reproduction. Phyletic size increase is strongly evident in the fossil record, with the frequent increase of body size of phyletic lines.

Advantages for larger body size are one of competitive dominance, predatory exclusion and adaptational "buffering" from extreme environmental fluctuations that can be devastating for more r-selected species. This evolution of K-selected species must be understood in the context of the allocation trade-offs of limited resources, but also in terms of the shifting of environmental niches that become open to larger organisms, that lead to higher levels of intrinsic carrying-capacity. Such species tend on average to be density-dependent type populations--relatively free of density independent factors that affect the adaptational outcomes of smaller r-selected species.

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Carrying capacity is an important concept, as it defines the relative limits to the growth of a population in an area. Any given area comprised by some eco-system can be said to have some net carrying capacity that is represented by the total biomass that is possible in that area and the net productivity of that biomass over a given period of time.

In reality, no finite population will have available to it the total carrying capacity of an area. The total resource base of an area, that defines its gross carrying capacity, is generally carved up at different trophic levels and in different adaptive niches. These different trophic levels and niches may be occupied by more than one kind of animal in competing populations.

Another way of looking at this ties this discussion to the discussion of ecosystem pyramids and matrices in the previous chapter. This is to hypothesize that for any given geographically bounded eco-system, there is a matrix of possible niches open to different kinds of species, each with its own finite carrying capacity. It leads to consideration of a "periodic table of trophic niches":

Within any given ecosystem, for any given population, there is a carrying capacity defined within the trophic-niche(s) occupied by the members of that population, such that that population will tend to increase in size to reach that carrying capacity. At that stage, the niche will become saturated. A saturated ecosystem can be described as any such system in which most of the trophic-niches that fill that system are saturated, having reached its gross carrying capacity.

Implied in this kind of model, which will be taken up in the next chapter, is that evolution will tend to favor selection of individuals with larger body size that move into more stable "K" selected niches within an ecosystem. In other words, not only will populations reach their maximum density of saturation for a given niche, but the members of that population will tend to reach their own morphological maximums. Reproductive selection favoring larger individuals will tend to shift the population into higher trophic niches that can be defined as "centralizing selection." "Peripheralizing" selection will lead to displacement of smaller individuals to the periphery, of "fugitive" or floating populations.

Thus, in terms of population dynamics, it can be seen that there is a trade-off between the size of the population and the size of the individuals of that population, such that there is a total limit that is the carrying capacity of that niche. If N is the number of a population in a given niche, and S is the average size of each member of the population, and K is the carrying capacity of that niche, then we can speculate that:

NK = S

In general it can be said that K selected niches comprise populations of lower densities (N), but of greater average body size (S), whereas r-selected niches comprise populations of higher densities (N) but of lower average body size (S).

Fitness and selection are to individuals/populations what niche equilibrium & carrying capacities are to ecosystems. In this sense, it can be seen that adaptive trait variability in individual organisms are to selectional population parameters and equilibrium, what reproductive trait variability in populations are to mechanisms of selection for entire species.

We can see enshrined in reproductive trait selection the differential strategies adopted by different life forms that attempt to overcome the basic dilemma presented by the evolutionary imperative. Differential reproductive trait selection, like size-selection, must be considered a key-defining form of trait selection in the fossil record.

In considering this relationship, we can see in Fisher's (1930) formulation of reproductive allocation and reproductive effort, the working out of the dilemma of the biological imperative versus the reproductive imperative by different life forms. In other words, how much of one's limited physical resources should be devoted to life-maintenance and adaptive survival, and how much to reproductive effort. This has raised a complex question of optimal reproductive effort, and the relative proportion of resources devoted to reproductive versus non-reproductive traits. The mass reproductive efforts of octopuses that result in the mass dying off of the parent generation is an example of a near total expenditure on reproductive effort. One principle of residual reproductive effort reads like this:

Current investment in reproduction should vary inversely with expectation of future offspring.

If rates of death are expectably high and extended survivorship low, then rates of birth must also be high, and also the rate of reproductive effort in the first set of offspring should be great. This is typical of r selected species, that can be thought to invest a huge proportion of their resources to initial reproductive effort to a large number of initial multiple offspring.

If rates of death are expectably low (near equilibrium) and survivorship long, then rates of birth must be corresponding low, and the rate of reproductive effort in subsequent offspring greater or balanced with rate of effort in primary offspring. This is typical of K-selected species that can be thought to invest a smaller proportion of their resources, but to a smaller number of repeat offspring. Having fewer repeat offspring allows a greater proportional allocation of reproductive effort over time, with less net drain on adaptive effort. Optimal reproductive strategies maximize expectable lifetime reproductive resources.

Devoting more reproductive resources to a single offspring may confer an adaptive advantage to the survivorship and reproducibility of the offspring. Size selection should therefore follow patterns that lead to such itero-parous reproduction of a few offspring. There is an inherent tradeoff in reproductive effort by parents, which is held to be optimally fixed, and the realized increase in adaptive/reproductive value realized by the offspring. Quantity of offspring is traded off for quality of offspring. If total reproductive effort of offspring is presumed to be constant, then the fitness of individual progeny decreases proportionately to the increase in total number of progeny. What is optimal investment for parents, is often in conflict for the individual optima for offspring. There is often an inherent parent-offspring conflict in reproductive resource allocation. It is evident that built in expectations of death rate, mortality and survivorship guide such optimal patterning that has little to do with maximization of offspring advantages.

Such considerations lead into other issues, such as developmental delays or lags in the occurrence of traits of an allele, such that we may speculate the following table of possibilities from delayed recessional to "front-ended" effects, and beneficial to deleterious effects:

Senescence and early developmental maturation of beneficial traits have been explained in this way in entirely stochastic terms. In this argument, deleterious precessional traits cannot be selected for by means of reproductive selection, as it would lead to negative adaptational selection before reproduction. As such, these traits cannot be selected for postponement in r-type species.

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We can speculate therefore that there is a relative species-dependent carrying capacity for any area, which sets an ideal limit to that species population size, or density, in that area within the boundaries of the limiting factors applicable to that species. If a species reaches the carrying capacity of a niche within an ecosystem, then there are only a number of alternative options available to it if it were to continue to increase its population:

1. Displace through emigration some of its population.

2. Adopt mechanisms conferring equilibrium (zero population growth)

3. Suffer the consequences of over-saturation, hence deterioration of the environment leading to reduction of the population.

4. Revert to a more r-selected niche.

5. Evolve to a more K-selected niche.

These would determine the selectional pathways a species might follow. It can probably be found that some members of a species are following different pathways at the same time. In any given ecosystem, smaller individuals, populations and smaller species tend to displace to the periphery, and distribute over a larger area. Larger species tend to move to the center and to focus in a smaller area. This is a paradox, because small species are by nature confined to smaller ranges, while larger species in general need larger ranges. Larger species therefore exhibit forms of stabilizing selection, where smaller species tend to exhibit forms of diversifying selection.

Often, species under conditions of stress adopt patterns of self-limiting factors. In general, individuals or populations are better managed if they depend on self-limiting factors rather than on environmental limiting factors. In a sense, self-limiting factors are built-in traits that confer adaptive advantage to species. K-selected species are more self-regulating than r-selected species.

Limiting factors are also distinguished between biological factors, which are derived environmentally from the relationship to other organisms, and physical limiting factors, which are a built-in function of the geophysical environment. In extreme environments that are represented by peripheral zones, physical factors tend to be more limiting. Biotic limiting factors tend to distinguish K-selected species, whereas physical limiting factors tend to distinguish more r-selected species. In optimum high-biomass environments, especially in core and intermediate zones, biological, density-dependent factors tend to be more constraining. In most environments, in the biggest areas of the earth, physical, density independent factors tend to be the most constraining. Biological factors of limitation affect mostly only the most intermediate zones of the earth.

What is suggested by this model so far is the prevalence of a basic "life-curve" in biological evolution, that can be explicated in the following diagram:

Within this framework, most species can be seen to be fit within an r-selection pathway. Some species rise out of this pathway to a higher level, and as pathways of different species converge at the topmost levels, there is increasing competition and less room. Species become more selected, larger and fewer at the higher level. Reproductive fitness is traded off for greater adaptive fitness as life forms gradually climb to the K-levels. Life forms cannot move past the K-level unless some new mechanism can be hypothesized. Eventually, saturation leads to changing optima of equilibrium at all the levels, resulting in a sudden relative super-saturation, and in increased rates of death.

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This kind of accounting in terms of relative fitness and selection strongly suggests a kind of epi-genetic landscape that has clear "problem-solving" characteristics. In other words, it becomes apparent that through speciation and the development of different strategies of adaptation within different contexts of selection, that different kinds of "solutions" are achievable and evolutionary development achieves directional momentum along certain probabilistic pathways that are not utterly blind.

In direct homologous terms this can be understood in terms of the development of bigger brains as a natural long-term outcome of randomly driven but directional evolutionary development. Once brains became possible, associated with animals, factors relating to natural intelligence became increasingly operative in the basic formulas of selection and fitness. Bigger brains, in the long run, tended to win out over smaller ones. More intelligent species tended, on average, to achieve greater evolutionary success.

Natural Systems

2001