Chapter X

The Circle of Life

Co-evolutionary Inter-harmonic-Periodic Oscillator Mechanisms

Only by construing evolutionary dynamics from the standpoint of recurring social cycles within larger natural cycles can we derive a more accurate systemic model of evolutionary process. These cycles may lead down different developmental pathways, whose various stages have expectable consequences within an information systems framework. Only in this way can we resolve this kind of hen or egg dilemma that has been at the background of the understanding of natural selection from the beginning.

Models of cyclical process that reflect the fundamental and general realities of evolutionary development can be built. The model I propose is that of a periodic oscillator. Any energy system that is bound to a stable state of equilibrium, such as a fully saturated ecosystem in a range of fairly stable environmental parameters, by some restoring or self-regulating force, which I take to be mechanisms of social selection based on reproductive competition, will upon disturbance from its equilibrium position, resonate at a frequency established by the reproductive rates and death rates of the populations involved. Achieved relative equilibrium of any population is a measure of its evolutionary inertia.

This oscillation tends to be driven periodically by a complex set of external forces that impinge upon the system in expectable intervals derived from the oscillation patterns of neighboring ecosystems.

*****

The preceding digression based on theories of competition demonstrates several things. In general, increasing competition between forms of life tend to lead to a pattern of exclusion, such that other kinds of relational values are excluded between such life forms. We can say that in general, as things tend toward relative K, things also tend toward increasing competition. In the extreme form of competition, total exclusion results in either extinction or marginalization.

Relational interactions that do not reflect direct competition, can be considered inherently and indirectly competitive, but are to be seen as efforts to maintain relative equilibrium in conditions that would otherwise result in disequilibrium or exclusion.

Thus complex social organization and patterns of counter-adaptational selection and co-evolutionary interdependence arise precisely in conditions where potential competition can be expected to otherwise intensify. There would be no need for social organization or for complex patterns of interdependency to arise in conditions where there is no competition as a result of saturation and relative K-states.

Thus it can be seen that competition constitutes a basic mechanism governing and leading to trait-displacement in natural selection and patterns of speciation.

Social interactions between and within groups in ecosystems tend towards increasing complexity and are difficult to generally model in realistic terms. Nevertheless, it is evident that most forms of interaction can be at least partially depicted through competition, which illustrates a basic principle. Given any two (or more) organisms (or groups) in a finite resource system, a basic density-dependent relationship is inherently established, such that increasing growth will result in competitive constraints operating between all coexisting populations. Complex patterns of symbiotic mutualism and social interaction are derivative consequences of these basic constraints. While this model describes mutual coexistence and the rise and declines of populations about some hypothesized state of optimal equilibrium, they do not describe the resulting patterns of social selection that can be expected from them.

Before proceeding, I will state that in general:

Exclusive fitness and direct social competition are positively correlated with density-dependency and relative saturation within a system.

With increasing saturation of any system, it can be expected that social selection will manifest itself in increased rates of premature (non-reproductive) death and dampened actual instantaneous rates of birth.

In highly saturated, competitive environments, some species will increase at the expense of others that will face either extinction or marginalization.

Any system must eventually become unstable if some species cannot be displaced by exclusion from the system, or the system cannot achieve a higher threshold of equilibrium.

Unstable systems will result in relative innate competition that is density independent in its function, returning the entire system through increased death rates to a lower level of saturation. We may say that a form of non-differential negative selection sets into the system.

This suggests that there is an inherent long-term instability of all ecosystems that will tend eventually towards disequilibrium in spite of relative states of achieved mutual equilibrium between members of the system.

*****

We will go back to our basic formulas, and demonstrate that any presuppositions of density-dependence results in two-way interactions between any two organisms, groups, populations or species. The following kind of "interdependency" paradigm hold generally true for any kind of social interaction we may wish to represent in time or place:

I will call this framework a discrimination table of basic interdependencies. We may hypothesize that any interaction, or any predictable set of similar interactions, between any set of individuals, groups or populations, regardless of the specificity or inequality of the compared terms, can be placed in one of the sets of squares, and in one square only. The same interaction cannot be placed in two different squares at the same time. Thus, the absolute value of the table as a whole will be equal to total number of finite interactions or relationships recordable, within a given area over a given period of time. This might be called the functional density of an area that would be a measure of the relative density-dependency of that area as well as of the relative saturation of the area and indirectly a measure of species diversity and heterogeneity.

We would of course add cells to the table in a third dimension if we which to specify relations occurring between three or more compared terms and can be represented on an enlarged squared table. The range of possible interactions can be specified for any number of terms, as well as the degrees of freedom.

This table is called a table of interdependencies because it presumes a basic principle of density-interdependence operating between any two or more organisms, groups, etc., within any finite system.

Several conditions hold in this representation:

1. It is the natural imperative of each represented group to maximize its share of resources within an ecosystem. (innate competitiveness hypothesis)

2. Each represented group will strive to minimize its loses within the ecosystem.

3. In the growth of such systems, it can be expected that eventually the gain of some will come at the expense of others.

4. Direct competition should emerge as the result of increasing densities of populations and net saturation of the system.

The center value where interactions are mutually neutral would in an absolute sense be nonexistent or incorrect, if we assume a basic assumption of innate competition. But in a relative sense it is very possible to describe the mutual coexistence of different life forms that have no direct consequence upon one another. Innate competition is probably under most circumstances a residual and negligible factor in fitness and selection patterns, unless a case can be made for total super-saturation of the area in question. At the stage where innate competition would become a factor, it can be assumed that it becomes indirectly a density-independent factor, as it would probably affect all organisms in the system in the same proportionate degree. There are many contexts in which different species are not only mutually tolerant of one another, but actually indirectly codependent upon one another.

We can say therefore that relationships tend to move away from the center of neutrality in one or another direction. We can say that maximum ideal equilibrium would be achieved in the upper left-hand corner of the table, and maximum disequilibrium in the lower right-hand corner. It will be demonstrated that probably both states are never achievable, and therefore most social relationships range between the two extremes.

*****

Before proceeding with our model, it is necessary to emphasize the concepts of relative ecological rarefaction or saturation of an ecosystem. These concepts of rarefaction and saturation are related to the notions of carrying capacity, equilibrium, density-dependency and climax within a region, but they point to the energy-dynamics and bio-geo-physical resources of the system, especially as these are stratified between tropic levels. In general, saturation of any area can be considered to be the relative degree to which the total energy budget and biological resources of any system, and therefore biomass productivity, is used up by the life forms existing within that area. A saturated system is therefore one that approaches the maximum limit of the system's total carrying capacity. A rarefied system is one that approximates some minimal level of resource utilization within the system.

The concepts of saturation and rarefaction lead to consideration of heterogeneity and species diversity found within such systems and to a complex table of allocation of systems resources distributed between different kinds of coexisting life forms. The increase of resource utilization by one life form in a system will lead to offsets in the levels of utilization by other life forms.

This table of complex resource allocation within any eco-system I will call the functional trophic-taxonomic matrix that underlies the functional dynamics of the system. Any system comprises a range of niche potential at multiple trophic levels, and becomes representative in time of a variety of different kinds of organisms that seek to inhabit various niches at different levels.

In general, the table would look like this, and is derived from the matrix and pie-of-life model developed previously:

*****

Consideration of neutral relationships invokes models of matrices and life-pies previously described about the basic pattern of relationship occurring across Kingdoms in any ecosystem. In this model, relationships occurring across the basic divisions of Kingdoms present some of the most fundamental differences that can occur between organisms sharing a common environment.

This model suggests a basic functional stability of relationships tending towards what I will call minimal r-equilibrium (or maximum r-disequilibrium) found in all ecosystems, and that underlies the evolutionary stability of the entire biosphere. The stability of all nature rests on the fundamental interdependencies that arise on this level of interaction between different primary trophic levels. At this level, competition can be expected to be minimized. Minimum r-equilibrium would represent the minimum threshold of adaptation for a population. This minimum stability underlying all ecosystems occurs at a threshold of maximum rarefaction that a system can achieve and still remain a coherent system. Thus, we cannot in reality ever presume a total or perfect ecological vacuum occurring.

At the same time, this same model sets upper limits of K for all the primary trophic orders within the total system, such that changing equilibria in one of the orders must affect the other orders somehow. This upper limit defines the upper threshold of adaptive K-equilibrium for any population.

This would also set finite limits to total carrying capacities of any one primary trophic order, as well as a sense of resonance fluctuation of trophic limits within each order and between orders that describes a cyclical feedback pattern that can be either dampening or amplifying in nature.

It is understood that all organisms share density-independent values of innate competition, and consumers share a fundamental dependency upon producers. It is possible to imagine a browser that grazes itself to extinction if it is specialized on one kind of plant, while producers indirectly depend upon both consumers and decomposers. Most models of direct competition are at this level specific to the Kingdom being represented. We expect certain forms of competition between animals, especially at the same trophic levels that we do not expect between plants and animals. Plants also compete typically with one another for sunlight and other basic resources.

Furthermore, it is the upper levels of the pyramid of life where we expect to find the greatest amounts of direct competition between species, that we conventionally stereotype as survival of the fittest. We also expect to find the greatest amounts of direct competition within trophic levels rather than between trophic levels, though it is understandable that there is significant competition between trophic levels, especially those that are contiguous with one another on the pyramid of life. There are few hard and fast rules in this modeling of social interactions between different kinds of life, because diversity of species and interaction is the rule rather than the exception. Complex food chains and cyclical systems develop within the pyramid of life such that many kinds of indirect relations are established.

The Eco-Trophic Pyramid of Life

In order to get a handle on the meaning of innate competition in its varying forms, it is necessary to distinguish between innate types of interdependent competition occurring along some kind of competition continuum that includes all possible interactions between organisms.

I will hypothesize that competition can be seen in two basic forms that are related to the selection outcomes they favor or result in. These two forms of competition I will call reproductive competition (r-competition) and adaptational competition (K-competition). I will state that r-competition leads towards reproductive success or failure of one organism or set of related organisms, in relation to that of another. I will state that adaptational competition (K-competition) leads to adaptational success or failure of one organism or set of related organisms in relation to that of another.

If we go back to our original formulas, we can see that K-competition is an independent variable in the biological imperative, and that r-competition is a dependent variable.

In any given interaction, we can always assume some minimal level of K-competition occurring between the agencies, but we do not have to assume r-competition occurring except under certain conditions.

Wherever we find r-competition occurring, we can expect some degree of K-competition also to be occurring on a more fundamental level.

In general, I will state that the more different organisms or set of organisms are in both functional and taxonomic patterns, the greater the degree of adaptational competition can be expected between them and the less the reproductive competition. Vice versa, the more similar two organisms or related set of organisms are to one another, the greater will be the degree of reproductive competition between them.

Adaptational competition can be construed as encompassing a broader spectrum of interaction in which interactions between agencies or parties do not necessarily alter reproductive fitness values of either group, but alter the adaptive fitness values of the group.

In a sense adaptational competition sets absolute limits to the carrying capacity of any unique or related grouping, compared to other groupings that are different from it, beyond which relatively density-dependent limiting factors become relatively density-independent factors. K-equilibrium is the natural expected outcome of K-competition, and is easier to establish between species that are widely divergent on the trophic-taxonomic matrix than between those that are closely related.

What exactly distinguishes reproductive r-competition from adaptational K-competition is the issue of relative exclusive fitness that serves to emphasize the selective exclusion of the individual compared to that of the entire group. In a sense, therefore, K-competition compared to r-competition is just the social interactive inversion of our notions of r-K fitness and selection values. R-competition leads to greater relative K-fitness and selection, and results from this patterning. K-competition results from and leads to greater non-K or r-fitness and selection patterns that can be said to be characterized by inclusive fitness.

It can be expected therefore that r-competition results in equilibrium between and within related species whereas K-competition tends to lead to adaptive disequilibrium between related species. K-competition only leads to equilibrium as a function of the "evolutionary distance" between the interacting species.

We can claim that reproductive-competition results in reproductive-selection which tends to narrow the intrinsic trait-variability within a population by means of exclusion and emphasis on exclusive fitness. Reproductive competition is therefore a death-instigated selection process that leads to greater r for one group while maintaining K for another group.

On the other hand, adaptational-competition would result in adaptational selection and counteradaptational selection that would tend to broaden the intrinsic trait-variability represented by a population by means of inclusion and emphasis on inclusive fitness. Successful adaptive competition is birth-instigated selective process that should result in increasing reproductive rates leading to K.

To understand this, we must seek to understand the idea of indirect social selection, and how forms of competitive exclusion can actually result in greater equilibrium and balance between different species. This is accomplished by eco-systemic compartmentalization, or the separation and reproductive isolation of similar species, where all naturally occurring systems would tend, through natural increase, towards disequilibrium anyways.

In other words, we cannot hypothesize an innate mechanism within a species that would automatically tell it to curtail its reproductive rate under conditions near equilibrium. This is in spite of increasing death rate that should offset the rate of birth at and beyond equilibrium as this is expressed in carrying capacity or relative saturation.

The presupposition in the basic population and competition formulas is that there is some internal balancing mechanism in the organism or population especially, that says to it slow down reproduction once conditions approach optimum. In general, death rates and birth rates are only indirectly interdependent. Not only is there an inherent lag and differential distribution of instances of deaths and births over time but the classical equilibrium formulas imply a causal interdependency that doesn't necessarily exist.

At the stage of equilibrium, some other set of mechanisms must begin to kick in to regulate the cycle between deaths and births. These mechanisms are not directly the density-dependent factors of stress and strain on basic adaptational resources that result in increasing rates of death. Nor are they mechanisms like predation or co-adaptation that counterbalance or offset pre-established reproduction rates.

They are mechanisms that arise intra-specifically and congenially, and in niche competition between functionally similar kinds of species, They result in the competitive separation and reproductive isolation of subgroups or organisms between the two populations. It leads to clinal distribution and divergent speciation even in sympatric contexts where no physical barriers are seen to exist.

They always are most marked in conditions approximating relative-K between the organisms or groups involved, when the resources profiles they share are the most similar and therefore the most strained. At this stage, either organism or group, in order to increase its reproductive rate, must do so at the exclusion of the other group. The group cannot otherwise continue to grow.

The obverse side of this is to consider the basic adaptational competition between to widely divergent forms of life, such that the common overlap in resource profiles between them is very narrow. Such species can tolerate high mutual densities of one another without requiring competitive exclusion.

Adaptational K-competition becomes most marked in conditions between divergent species when there is some minimal resource (or set of resources) shared between them in a profile such that density-dependency of relationship arises in what would be otherwise relatively density-independent contexts. It indirectly affects the rates of reproduction and death between the two groups. This can arise from conditions of environmental fluctuation. Other wise, it would be most marked in contexts between trophic levels such as strong predation or extreme parasitism, when the existence of one species comes to depend exclusively upon and utilize the other species as the principle and only basis for its resource. The rates of reproduction of the predatory or parasitic species or group drive the other species or group into extinction or marginalization.

In this sense, competitive exclusion can result from extreme forms of either strong r-competition or K-competition, which suggests that the most evolutionarily stable patterns are intermediate between the two extreme "strong" forms.

Reproductive competition can therefore be seen as a special form of adaptational competition that occurs when two groups greatly overlap in their resource profiles on the trophic-taxonomic matrix and are competing for reproductive advantage, or r-fitness, between one another within a shared context. It may also arise when the results of such competition are expressed in terms of relative r-fitness values between the two groups. It can occur between life-forms that are not directly intra-specific, as for instance congenic sibling species, though reproductive competition at the intra-specific level is expected to be the greatest, leading either to organismic spacing, territoriality or complex forms of social organization.

It can be seen that both kinds of hypothesized competition are in fact interrelated to one another, such that adaptational competition leads indirectly to reproductive competition, and reproductive competition is always fundamentally a form of adaptational competition.

We can say, paradoxically, that reproductive competition always leads to inter-specific patterns of exclusive fitness, whereas adaptational competition always encompasses the entire range of relative fitness values, whether it is exclusive or not.

Comparison of adaptational and reproductive competition supports the following kind of representation:

Considering this framework, we can hypothesize the following kind of generalization:

At any functional level of trophic-taxonomic classification, we can distinguish between inter-group and intra-group forms of competition. We can hypothesize that at any level there is a characteristic degree and type of adaptational competition occurring between representatives of different groups.

The closer the groups are related in both taxonomic and functional identities, the greater will be the direct reproductive competition between them. Another way of stating this is that the degree of trait-overlap or similarity on the trophic-taxonomic matrix, between any two or more comparable organisms, groups of organisms or species, the greater the inferable interdependencies between them are likely to be expressed in terms of exclusive fitness and reproductive competition.

In such contexts where very similar kinds of life come into interaction, the net result of such interaction must eventuate in some form of relative isolation or mutual exclusion between the two forms. Succession of biotic forms in certain regimes can be understood as a consequence of this operational principle. Fundamental differentiation of speciation processes underlying all evolutionary processes can be understood in this way. The result of this patterning is also to set up a variegated topography of isoclinal zonation in the distributional patterning of different forms of life.

Organisms that are sufficiently divergent from one another on the trophic-taxonomic matrix, and in which mostly adaptational competition occurs, can mutually coexist within the same habitats and environments without this adaptation leading to mutual exclusion or the creation of functional boundaries between the populations.

Complex patterns can result where it is possible for adaptational competition between two widely divergent forms of life to result in changes in reproductive competition for either form of life with some other closely related forms of life.

Social selection operating on interdependent populations must be construed from the standpoint of the long-term evolutionary consequences of such systems. Obviously, systems that drive towards extreme mutual disequilibrium reach natural limits of maximum rarefaction of interaction at which point inclusive fitness and r are maximized and density-dependence is of minimum value. In a sense, differential selection increases as the rate of reproduction increases and inclusive fitness kicks in, such that it is in the conditions of relative disequilibrium that we find the fastest rates of evolutionary development. This model is depicted below.

In such a model, it is evident that a changing rate of evolutionary development for any given line, in any given ecosystem, must be an inverse function of relative density-dependent relationships, such that increasing states of disequilibrium result in increasing rates of speciation. It would suggest that there is a fundamental lag time in this process, which is the equivalent to the lag time between birth and death rates in normal population dynamics. I also hypothesize that there is a line at which optimal selection values occur such that selection processes occurring above this line have fundamentally different consequences than selection processes occurring below this line.

We may make a distinction between differentiating and non-differential selection processes, whether they occur above or below this line, based on the pattern towards unequal negative selection as represented by differential selection, leading to speciation, or blanket negative selection which can be considered non-differential or inherently stabilizing selection.

*****

In the graphical representation of this model, we can consider the values applicable to a single normally heterozygous reproductive population, or of two different species that are closely related within a trophic-taxonomic table, such that we derive a normal unimodal bell shaped curve below that illustrates the range of variation found within either a heterozygous population or two closely related populations.

In the model below, we must see that evolutionary development is defined by the limits of minimum and maximum sustainable equilibrium, rather than by the line of optimal equilibrium, such that a population will normally oscillate between these extreme limits within a stable ecosystem. The lower limit line defines the cut-off lines of the curve of normal distribution of a population, beyond which negative selection is expected to occur. The upper limit governs the potential heigth of the curve, and thus indirectly sets the optimal line of equilibrium such that it defines the degree of relative heterogeneity or homogeneity (variance or similarity) between two populations or subpopulations.

The four vertical lines represent the limits beyond which selection is expected to occur, two at each peripheral end and two in the center, such that different selection patterns will result in movement of the lines to the left, right or center. The center-lines will converge until they come together, or spread apart until they reach the peripheral limits. The total area under the curve represents the total range of trait-variability comprised of either a single intra-specific population or

A normal curve of expected trait-distribution in any given population.

two closely related inter-specific populations. The upper horizontal line represents the maximum limit of saturation that defines the total capacity of the system. The intermediate upper horizontal line represents the limit of mutual equilibrium that can be achieved by a population or related populations. This line will raise or lower depending on the relative distributions of both overlapping curves. In reality, both lines would be oscillating and would gradually fluctuate, depending on changing external environmental conditions. The lower lines would represent the minimal level of equilibrium or maximal level of disequilibrium beyond which there is a zone of rarefaction. The bottom line represents both the total spatial distribution and the finite limit of a potential ecological vacuum achievable in such a system.

The difference between these models if it were a single specific population or two closely related populations, is that the center zone would be defined in the first instance of a single population as the region of greatest intra-specific competition, while in the second instance of two competing populations, it would be the region of greatest inter-specific competition.

If we were to map this distribution over time, we would see a fluctuation of the basic set of values governing these relationships, such that the shapes of the curves, the ranges of overlap and the lines would all change positions. The normal distribution can be considered to by a synchronous or instantaneous cross section of a population that is changing dynamically through time. In any given ecosystem, this would be but one thread in a bundle of similar kinds of threads that are bound like a rope through time.

If we were to return periodically and map our thread, we would yield different profiles of our distributional patterns. If we did this enough, we would find a predictable set of patterns that describe possible pathways of periodic recurrences of such patterns, such that the wave-patterns of the oscillation of the system would be something like the following possible patterns:

Case 1: Stabilizing selection "toward the center" that leads to narrowing the center range.

In the first case of stabilizing selection, increasing competition would lead to selection toward the center which would tend to narrow all the cut-offs toward the middle, with the result of elevating both curves up to or even above the limit of total saturation. Equilibrium is maximized in this context.

This state is considered the start state in this model because it depicts the total ecosystem in a state of saturated equilibrium, which implies several things, most important of which is the point at which reproductive social selection is held to be of greatest importance (at maximum saturation). In this system, it must be understood that changing values are not automatic and synchronous. Rather, there is always some implicit lag between changes in variable states, such that there occur resonance throughout the system.

Case 2: Directional Social Selection: Shifting to the right.

In case two, a system that has maximized itself through competitive selection and narrowed its range of variability, is primed for destabilization that begins with a directional selection either to the right or two the left. Under such conditions, one group or sub-group begins winning out over the other, leading to increasing stability of one at the loss of stability of the other.

Case 3: Disruptive Selection, "selection away from the center", results in a "collapse" of the ecosystem, which can be considered to be the state of maximum disequilibrium attained by the system.

In case 3, it is presumed that the loss of stability in either one or the other, with the gain in stability of the other, can lead to a total collapse of the system if the one group is not brought quickly to extinction or displaced out of the system. Disequilibrium sets in due to the imbalance between the two systems, resulting in the loss of stability of both populations. The result is cladogenesis or divergence of a single line into two, or else displacement of one population.

Case 4: Balancing Social Selection: emergence of two stable center lines and a movement outward of the extremities

In case 4, the collapse of the system will be followed by a reestablishment of balance in relatively rarefied ecological conditions, such that a form of balancing selection favoring two separate central loci develops. The system of selection will favor either establishment of an iso-cline between two populations, with the possibility of increasing niche-competition between them, or the redevelopment of a single heterozygous population due to displacement of one of the subgroups.

Case 5: Diversifying Selection, "reconvergence of the center," results in a period of maximum diversity within the system as a result of wide tolerance limits to either extreme and a reconvergence to a central region.

Case 6 a: Peripheralizing selection: leading to exclusion or extinction of one group or segment of the population in a state of extreme disequilibrium. This is a case of divergent cladogenesis, or else extinction of both groups, or else phyletic evolution of one group and extinction of the other group.

The system as described above can lead down different pathways.

We can describe in this model two start conditions:

Start condition 1: A single heterogeneous species

Start condition 2: Two closely related species

And three Final conditions:

Final condition 0: Extinction of both groups or subgroups.

Final condition 1: A single heterogeneous species, extinction of one group or subgroup (phyletic evolution)

Final condition 2: Two closely related species (cladogenesis)

In this model, any final outcome is possible at any start condition. Whatever the outcome, there is a return to one or the other original start conditions, or else the cessation of the system.

If we begin with a single heterozygous population in case one, we will end up with either two overlapping congenitor species (case 3), or the displacement of one species out of the system (case 6a), and the return of the system to stable balance. At the next round, we start back either to start states 1 or 2 depending on the outcome.

This model describes a very basic cycle of life in evolution that is rooted to an ecosystem. It is known that in the fossil record extinction is a common pattern. It is also known that phyletic evolution and cladogenesis are also commonly recurrent patterns.

In this model, diversifying selection is held to be a kind of "inclusive fitness" that leads to maximizing of variability of trait-pattern within or between populations, and can be taken as a period of niche radiation within an ecosystem.

Drift and selection against deleterious alleles would exist in every state within the system, describing the more or less random fluctuations of the parameters of the system. In all cases, deleterious alleles would tend to put individual in the extreme tails of the curves, leading to their being selected against regardless of the central patterns of the curve.

Negative selection of deleterious traits does not necessarily result in positive selection of adaptive traits. It can lead instead only to either stabilization or extinction.

Any population may drift about a center at any point, but the effects of drift may be more pronounced at some points, at points of the extremes of maximum equilibrium or disequilibrium, than in the intermediate regions. It is possible to imagine drift setting off a steady-state system into a period of resonance amplifying disequilibrium leading to directional selection favoring one group at the expense of the other.

What is described is a period of rapid change and adjustment, followed by relatively long trends of relative stability, represented by the following:

Exogenous changes can be described in terms of environmental fluctuations that alter the thresholds of the curves, and also as the introduction of a third extraneous species to the system. Introduction of a invader population to a system that describes a single heterogeneous population would switch the system to a two population model. Otherwise, it would extend the complexity of the model to a multi-population system that is implied in a two-population start state. Exogenous changes can be introduced at any stage in the development of the system, but it may have different consequences depending on the point at which it affects the system. Introduction of exogenous changes can cause a system in a steady state of relative equilibrium to spiral into disequilibrium.

Evolution as natural selection can be described therefore as an indirect process that is cyclical:

Adaptational competition leads to K states of saturation which leads to reproductive competition leads to r which leads back to adaptational competition.

Adaptational selection leads to inclusive fitness increasing environmental fitness maximizing trait variability and slowing evolutionary rates resulting in K which leads to saturation and reproductive trait selection.

Reproductive selection leads to exclusive fitness decreasing environmental fitness minimizing trait variability and increasing evolutionary rates which results in r which leads to rarefaction and adaptational trait selection.

********

In order to bring closure to the problem of natural systems theory in biological evolution, two issues remain to be addressed. The first is the suggestion of a set of formulas that might describe the patterns above, in terms that represent a kind of calculus of natural selection. The second issue is to describe these oscillatory patterns occurring in ecosystems in reference to other proximate and distant ecosystems that are the source of exogenous changes within the system.

I propose a model of a social mountain-island of ecosystems in a web of life that explains two sets of interrelated patterns:

1. The introduction to a system of exogenous change, and;

2. The ability of ecosystems to maintain a boundary about itself in relation to coevolving ecosystems that includes the description of this boundary as a complex pattern of zonation about core regions that structurally represent eco-systemic centers.

It can be seen that some boundary maintenance mechanism exists to confer stability to any ecosystem, but this boundary is relative and permeable, such that individual species may cross it readily, leading to introduction of exogenous sources of change into the system.

Several assumptions are made. First, since exogenous change in the total scheme is held to be essentially random from the standpoint of the internal dynamics of an ecosystem, over the long run there should be some relatively constant value of such change, which I will represent as the variable "D." If we consider a point-diversity model, we can see that the value of magnitude assigned to D will be a consequence of the size of the ecosystem. The larger the ecosystem, the greater the perimeter of its boundary and the greater the amount of exogenous change occurring across its perimeter. It presents a bigger target for invading species.

It can be concluded that for any stable ecosystem, the amount of exogenous change will be fairly uniform over time, but that the consequence of such change will be a function of the actual relative value of that change and the internal state of the system. In a system that is on the threshold of collapse, even relatively minor exogenous change values can trigger a cycle leading into disequilibrium. In a system that is very robust and of increasing stabilization near the level of saturation, even relatively major exogenous changes values can have little consequence in disturbing the system.

Nevertheless, there is hypothesized a kind of chain reaction pattern that results in periodic wave patterns of exogenous changes that can sweep through a network of ecosystems. Nature in the biosphere may organize itself at even higher levels such that a butterfly effect can be created within such a network. This wave patterns of chain reactions in interconnected ecosystems may account for a certain periodicity occurring in such patterns, and is indirectly tied to the periodicity of the internal mechanism itself.

The island-mountain model can be taken as a relatively bounded area of diversity in a sea of relative homogeneity. It is borrowed from the concept of the island that has been so central to evolutionary theory and is applied as a mountain of eco-systemic stability on an epigenetic landscape. It borrows also the concept of the mountain as the terrestrial equivalent of an island that features its own biodiversity of habitat and zonation. In this sense, even a continent can be considered a mountain on the seas. Around the mountain-island can be considered to be an intermediary zone that can be constituted by different possible ecotones. Isoclines would describe zones that range from the intermediary zone to the core region of the mountain-island.

Island models have been important constructs in evolutionary theory and experimentation. Equilibrium theory has been applied to island models.

It is known that larger islands (or island-mountain areas) support more species diversity than smaller ones. There is a linear increase in Taxon diversity with increase in island size, in which a ten-fold increase in volumetric area about an island corresponds to a doubling of the number of species in the area. A slope of linear regression through such points is designated as the Taxon's z-value in any particular island system. Z values generally range from about 0.23 to 0.33 between different taxa on different isolated islands, and this value becomes the exponent of the following formula:

S = CA^z

Where S is the species diversity,

C is a constant that varies between species and place to place

A is the area bounded by the island

Rearranging this formula with logarithms, one gets the following linear equation in which z is the slope:

log S = log C + z(logA)

Topographical diversity results in large z values and in spatial replacement of species leading to islands within islands, while low z values lead to reduced species replacement and relatively homogenous conditions.

It is known that continent mountain-islands of comparable size to true isolated islands in general support more species at higher trophic levels than true islands of equal size. The rate of increase of diversity of a continent "mountain-island" increases also with increased area, but the z-value is generally not as great as on a true island, being between 0.12 and 0.17. This difference is held to be due to the relative isolation of islands and the sampling characteristics of continental "mountain-islands" where species requiring greater area may occupy the region on a regular but discontinuous basis. Islands cannot therefore support the higher trophic levels found on continents that inherently require greater areas than isolated islands afford.

It has been conjectured that introduction of new species to an island is inversely proportional to the species diversity of an island that is tied to the relative density of the island. The rate of extinction on the island should also increase with the increasing diversity of species on an island.

The invasion of new species is linked in this theory to the extinction of old species. Equilibrium on the island will be reached when the rate of immigration equals the rate of extinction. Rate of immigration (π) and rate of extinction (Þ) largely taken the place of birth (b) and death rates (d) in the previous theories of equilibrium. The number of a species N in the original formulas is replaced by a variable of the species diversity, or S. The resulting equation describes a stable state of dynamic equilibrium.

In the initial development of the formulas, we assume linear variation of the rates of immigration and extinction, such that:

π_s = π₀ - aS

Þ = ßS

Where π₀ is the rate of change with no species on the island and a and ß represent rates of change of immigration and extinction as S increases. At equilibrium, Sˆ, the rate of extinction and immigration must be equal, such that the two formulae are equal:

π₀ - aS = ßSˆ

The number of species at equilibrium can be given as:

ßSˆ = π₀/(a + ß)

This formula is identical to the expression for carrying-capacity K in the logistical equation:

K = r/(x + y)

The average rate of immigration per species (ˉ π) and the average rate of extinction per species (ˉÞ) can be obtained by dividing by the number of species not yet on the island (P - S) and the number already on the island (S), such that:

ˉ π = π_s /(P - S) or π_s = ˉ π(P - S)

ˉÞ = Þ_s /S or Þ_s = ˉÞ S

At equilibrium, rates of immigration equal rates of extinction. This results in continuously changing composition of the island (or island-mountains) biotic profile, while the island itself will remain relative stable as an ecosystem. At equilibrium, Sˆ, we get

ˉ π (P - Sˆ) = ˉÞ Sˆ

and therefore,

Sˆ = ˉ π P/(ˉÞ + ˉ π)

Equilibrium will increase with increasing P and average rate of immigration, and decrease with the increasing average rate of extinction. The average rate of immigration is the same as the rate of change of immigration (a) and the average rate of extinction is equal to the rate of change of extinction.

Immigration rates are a function of dispersal rates, which decrease exponentially with distance. Rates of extinction are held to be unaffected by relative distance or isolation, but are related to the area. Decreasing areas should result in increasing rates of extinction because smaller areas can support lower levels of saturation and equilibria. Two islands of dissimilar area but equal distance from a source continent should experience different rates of immigration and extinction. Replacement should be more rapid on the smaller of the two areas. Increasing density of areas should also result in higher replacement rates.

Consideration of equilibrium equations tied to immigration and extinction in island models is related to a model of one way genetic flow from a continent to an island that has particular value in considering continental island-mountain models. Gene flow is considered an homogenizing force in evolution that is contrasted to the flow of drift that is the result of relative genetic isolation of two populations. The two forces are held to be counterbalancing and lead to shifting balances or dynamic equilibrium of partially closed systems as we find with all ecosystems.

The effectiveness of gene flow can be measured as the amount of migration (m) and the degree of genetic difference. If pˆ is a frequency constant of a large group and p₀ is a small isolate subpopulation and p₁ represents one generation of gene exchange, then:

∆ p = -m(p₀ - pˆ)

p₁ = (1 - m)(p₀ + mpˆ)

∆ p = (p₁ - pˆ) = (1 - m)(p₀ + mpˆ - p₀) = -mp₀ + mpˆ

The formula determines the amount of gene flow between a continent population and a subgroup population, or vice versa, with each successive generation. The first generation equals:

p₁ = (1 - m)(p₀ + mpˆ)

Subsequent generations can be easily represented in the same formula simply by changing the subscripts, such that the second generation is:

P₂ = (1 - m)(p₁ + mpˆ)

The frequency of the allele of the gene pool does not change. After "n" generations, we would get:

p_n = (1 - m)(p_n-1 + mpˆ)

This formula can be rewritten as:

(1 - m)^N = p_n - pˆ)/p₀ - pˆ)

From which we can express the rate of migration (m) as the function of gene frequencies.

Gene flow is expected to be a strong cohesive force in nature that binds populations into single evolutionary units, but gene flow can also act to spread favorable genetic combinations among populations. This leads to a shifting balance theory of evolution based on the amount of gene flow, the degree of difference between two populations, and the flow is unidirectional from an almost infinitely large continental population to a small isolated island population.

From the standpoint of island-mountain models, it can be expected that there is a relative boundary about an area that circumscribes an ecosystem that defines the relative distance between that area and other neighboring source areas. This boundary is complex and variable, rather than being one of absolute geo-physical distance.

Natural barriers are present on continental systems of course, but the kind of barriers that exist in island-mountain models must be available even in conditions of high diversity and relative homogeneity, as in tropical lowland regions. Such complex boundaries must be defined by the internal dynamics of the ecosystem, relative to the degree of saturation of the system. For any potential invading species, there is some threshold of adaptational fitness that it must pass, in order to be successfully enter and adapt to the pre-established system. This I will call the "passing threshold."

This adaptational threshold can be described as the potential counter-adaptational fitness of the preexisting species that are closest to it in resource-profile. If a niche is relatively open, or it enters at the periphery of a pre-established system, then the threshold should be relatively low. If a niche is filled by a stable, preexisting group and the niche defines the center of equilibrium for that group, then the threshold for passing must be relatively high. If a species can pass into a system, over that threshold, then it can successfully adapt, and result in the displacement of species that overlap in its trophic niches, leading either to extinction or displacement of the species.

If entering an island-mountain system requires a threshold value, then exiting the system successful might also require a threshold value, which I will call an exiting threshold. A group displaced from such a system must either go extinct or leave the system and reenter a new one. Leaving one's place in a pre-established order entails a certain loss of fitness obtained within that order. If a group cannot obtain that transitional fitness threshold, that is tied to adaptation in peripheral and transient conditions and reach the entrance threshold to another system, then it will be doomed either to extinction, or possibly to a fugitive state in some "eco-tone" between major ecosystems.

A group that is displaced from a niche in an ecosystem, that must find itself a new niche in another system, is therefore at a fundamental disadvantage in that its adaptational fitness must cross two sets of thresholds that should be considered unconnected and separate. Of course, species invade all the time, with the presupposition that such invasion is a natural response to displacement. But it is possible that many invasions are not so much a response to eco-systemic displacement as they are a result of population pressure from a densely saturated system. Any population must regularly throw of its "tail end" population, and some percentage of this must disperse some distance from the core concentration of island-mountain areas to other areas. Either way, the consequence should be the same.

The description of the island-mountain model describes a bounded ecosystem that represents a series of spatially organized habitats that coexist through time within a common area. They exhibit a sufficient level of functional coherence as to preserve the integrity of the system as relatively separate or isolated from any other system. To some extent such systems are purely a function of distance. But many other factors may impinge to create a boundary around such a system.

Island mountain models may apply to real islands in the ocean, but they are primarily intended to apply to areas that can be analytically circumscribed on continents. In such a model, I hypothesize that there is one or more core areas that define its "gravitational" center. It is the center of balance of the system, and should in theory be an area that comprises a local peak in species diversity and density of saturation. It should also be the region that comprises the greatest heterogeneity of species across trophic-taxonomic categories. In a dense, flat rain forest, a tall stand of trees, or even a single tree, may therefore effectively comprise its own small eco-system.

The core of an island-mountain may in fact be itself so hypertrophied that it can be divided into zones of an inner and outer core.

Around the edge of an island-mountain, and of varying dimensions, should be an intermediary zone of transition that is marked by great diversity of species and possibly by many small peripheral ecotones. This defines the periphery and boundary of the island-mountain, and its intermediate range describes the boundary about an island-mountain.

Between the core and the periphery would probably be one or more isoclinal succession zones that would be marked by increasing degrees of diversity and resource concentration/biotic saturation. Like the periphery that surrounds it, these zones should range from being non-existent to being very vast stretches, and may be variegated in a fashion to create a patchwork quilt of sub-systems within the larger framework. This highlights the fundamental relativity of any ecosystem in the biosphere, that it is always a part of some larger ecosystem, as well as a part of the total biosphere.

Continental island-mountains therefore are non-isolated in a fundamental sense that true islands are. They are always a part of some larger and more rarefied ecosystem, or set of ecosystems, in which it plays a part-whole role. Periodic fluctuations of waves of invasion can be expected in such ecosystem networks, or biotic webs, which leads to a secondary pattern of interference oscillation in any ecosystem of which it is a part. Such waves can be seen as evolutionary surges like hammers that sweep through areas, bring disequilibrium and change in their paths.

******

In such a manner, I have attempted to apply systems theory to an understanding of the basic mechanisms of natural selection that underlie evolutionary process in a consistent way. It is clear that the Theory of Evolution is incomplete. The multiplication of theoretical terms and concepts and the variability of their values in the elaboration of theory are a principle indication that the synthesis is yet to be complete. And this is perhaps how it should be. Biology presents an inherent dilemma of being a kind of intermediate theory between the purely physical and the even more chaotic social sciences. It has gained a great advantage in this regard, in having a grand synthesis in the first place. But when we try to nail this synthesis down theoretically to an airtight system of generalization that might be explicable for every instance observed in nature, we get increasing degrees of leakage about the seams.

Natural Systems

2001