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In this paper I am concerned with the number of species that will be held in stable equilibrium in a community of competing organisms, using the general form of the Lotka-Volterra competition equations for m species. Defining Ki as the saturation density for the ith species and αij as the competition coefficient between species i and j, and Ni as the equilibrium density of species i, the number of species will be determined by N, K, $$\overline{\alpha}$$, var (K), the covariances among the α's, and the covariance between α and N. In particular, the number of species increases as K increases but as N, $$\overline{\alpha}$$, cov (α), cov (α,N) and variance of K decrease.

In homomorphic self-incompatibility (SI) systems of plants, the loci controlling the pollen and pistil types are tightly linked, and this prevents the generation of compatible combinations of alleles expressing pollen and pistil types, which would result in self-fertilization. We modeled the initial invasion of the first pollen and pistil alleles in gametophytic SI to determine whether these alleles can stably coexist in a population without tight linkage. We assume pollen and pistil loci each carry an incompatibility allele S and an allele without an incompatibility function N. We assume that pollen with an S allele are incompatible with pistils carrying S alleles, whereas other crosses are compatible. Ovules in pistils carrying an S allele suffer viability costs because recognition consumes resources. We found that the cost of carrying a pistil S allele allows pollen and pistil S alleles to coexist in a stable equilibrium if linkage is partial. This occurs because parents that carry pistil S alleles but are homozygous for pollen N alleles cannot avoid self-fertilization; however, they suffer viability costs. Hence, pollen N alleles are selected again. When pollen and pistil S alleles can coexist in a polymorphic equilibrium, selection will favor tighter linkage.

We present a computer simulation that examines the dynamics of cooperative behav- ior in a patchy environment. A new defecting strategy called "ROVER" is introduced, which uses a rule analogous to the "marginal value" theorem to determine how long to stay in a patch. A wide range of parameter values allow ROVERs to invade a metapopulation of tit-for-tat strategists (TFT) and spread either to fixation or to a stable equilibrium in which ROVER is the majority strategy. Our model, along with previous models that invoke different mechanisms, suggests that even guarded forms of cooperation such as TFT are often vulnerable to invasion and that a stable mixture of cooperators and exploiters might be common in nature.

We develop an approach for studying population dynamics resulting from mutualism by employing functional responses based on density-dependent benefits and costs. These functional responses express how the population growth rate of a mutualist is modified by the density of its partner. We present several possible dependencies of gross benefits and costs, and hence net effects, to a mutualist as functions of the density of its partner. Net effects to mutualists are likely a monotonically saturating or unimodal function of the density of their partner. We show that fundamental differences in the growth, limitation, and dynamics of a population can occur when net effects to that population change linearly, unimodally, or in a saturating fashion. We use the mutualism between senita cactus and its pollinating seed-eating moth as an example to show the influence of different benefit and cost functional responses on population dynamics and stability of mutualisms. We investigated two mechanisms that may alter this mutualism's functional responses: distribution of eggs among flowers and fruit abortion. Differences in how benefits and costs vary with density can alter the stability of this mutualism. In particular, fruit abortion may allow for a stable equilibrium where none could otherwise exist.

In deterministic approaches to modeling, two species are generally regarded as capable of coexistence if the model has a stable equilibrium with both species in positive numbers. Temporal environmental variability is assumed to reduce the likelihood of coexistence by keeping species abundances away from equilibrium. Here we present a contrasting view based on a model of competition for space among coral reef fishes, or any similarly territorial animals. The model has no stable equilibrium point with both species in positive abundance, yet both species persist in the system provided environmental variability in birth rates is sufficiently high. In general the higher the environmental variability the more likely it is that coexistence will occur. This conclusion is not affected by one species having a mean advantage over the other. Not all kinds of environmental variability necessarily lead to coexistence, however, for when the death rates of the two species are highly variable and negatively correlated, th...

We derive a model that has one predator and one prey species and includes antipredator behavior. The prey are allowed to increase their investment in antipredator behavior, thereby decreasing their chance of being captured by the predator. However, they must pay for this protection by a cost exacted through decreased fecundity or increased mortality caused by factors other than predation. The population-dynamic consequences of antipredator behaviors are explored by comparing systems in which the efficiencies of the antipredator behavior differ; as the antipredator behavior becomes more efficient, the prey need to invest less in order to achieve the same level of protection from the predator. We assume that for any degree of efficiency, the prey choose their level of investment in antipredator behavior in order to optimize their expected reproductive fitness. By assuming only that the predators and prey coexist and that there is a stable equilibrium, we show that increased efficiency of antipredator behavi...

Simple models of the mutualistic interaction between ants and the diaspores of some spring herbs (myrmecochory) predict that a stable, feasible equilibrium is unlikely. The two-species model is expanded to examine the effects of a predator or competitor because, in nature, myrmecochorous herbs often suffer heavy seed predation from small rodents or compete with nonmyrmecochores. Low to moderate levels of predation stabilize mutualism that is unstable in isolation. Numerical simulations indicate that the locally stable equilibrium point "attracts" reasonable initial population sizes of the three species. Competition between the myrmecochore and a nonmyrmecochore decreases the probability of a stable coexistence compared to the isolated mutualism. It is suggested that many mutualisms are stabilized, and not merely bounded by a third species. Predation is the most likely stabilizing interaction.

A model of nutrient-limited phytoplankton growth, based on Michaelis-Menten uptake kinetics, is offered as an explanation of the so-called paradox of the plankton. The model suggests that an assemblage of coexisting phytoplankton may be limited by several nutrients, each species principally limited by the availability of a different nutrient. The principal requirement of the model is that each species be less effective at obtaining at low concentration the nutrient of which it requires larger amounts for continued growth. The model offers an explanation for the continued coexistence of associated phytoplankters in a stable equilibrium. It also offers a possible explanation for the decreased diversity commonly observed in eutrophic lakes and is consistent with observations that a larger number of different nutrients are in short supply in oligotrophic waters.

A community which would not reach a stable equilibrium may nevertheless persist if there is temporal variation and nonlinear dynamics. A procedure is introduced for taking time averages of the rates of change. Since the average of a nonlinear function is not the function of the average, higher terms such as the variances of resources or covariances among species and environmental factors enter into the coexistence conditions. These measures behave as if they were resources. Therefore the number of consumer species cannot exceed the number of resources plus distinct nonlinearities. The nonlinearities arise from predator saturation, learning, group hunting, multiple nutritional requirements, or seasonally variable feeding rates. It is shown that there is no long term correlation between the abundance of a species and its rates of increase.

The result of compensation in family size has been investigated, both for the case of selection against a dominant and recessive. In both cases a stable equilibrium may result. The range of values of compensation and selection for which equilibria do result has been shown. The methods developed are used to estimate the strength of selection in the specific case of acholuric jaundice.

Deterministic models are employed to investigate the evolutionary consequences of selection due to imprinted mating preferences in relation to the domestic pigeon Columba livia and to sexually monomorphic and dimorphic species of ducks. It is found that in large populations with absolute imprinting, the population tends to split into two noninterbreeding groups each characterized by a different allele, whereas with relative mating preferences a stable equilibrium tends to become established. The terms pegmatype and pegmatypic mating are introduced to describe such mating preferences and such a mating system.

The effect of intra-population selection on gene frequency in a large random mating population has been examined. In the absence of other forces when the genotypic selective values are independent of gene frequencies, a stable equilibrium value of gene frequency yields a maximum value of the average fitness of the population and an unstable equilibrium yields a minimum average fitness. This relationship is best illustrated by plotting both the W curve and the Δ q curve on parellel q axis of the same scale. In more involved cases, it is often possible to define a modified average fitness of the population so that the above conclusion holds. Only relative magnitudes of the intra-population selective values are relevant. It was emphasized that they cannot be used as a basis for comparison between two separate populations under two different environments. A comparison of absolute adaptiveness of two populations cannot be done until "absolute adaptiveness" is defined and measured.

When genetic change is taken into account, ecological principles must be reexamined. For example, even if total numbers of a prey population are essentially resource-limited, the prey may, through its genotypic composition, regulate a predator at relatively low levels, below those for which self-damping terms or effects of the predators on overall prey numbers need be taken into account. The system considered would not persist without genetic feedback. It is further shown that a situation described by a "series" model, rather than a "parallel" one, is one in which a genetic feedback mechanism for predator population regulation is more likely to be successful. Exact conditions on the selection coefficients are given for the existence of a stable equilibrium. Theorem 2 also provides a natural procedure for the choice of acceptable parameters for simulations. The model has been developed in accordance with a model suggested by Pimentel (1961), and thus certain of its quantitative predictions are subject to c...

One of the outstanding problems of genetics is that of the origin of new heritable variations. With the passage of time, and the accumulation of exact experimental data, it becomes increasingly clear that this factor is the basic one in evolutionary change, whether progressive or retrogressive.Pearl, 1917. There is no a priori reason why the substances of the germ cells should not be as easily altered as any other mass of chemicals. But in practice it turns out that most agents which produce chemical alterations of the germ cells at the same time kill the organism. Further, the germ cells, like other living substances, have a great tendency to compensate for any disturbances induced in them, their condition is one of stable equilibrium, in which an alteration is followed by a return to the original condition.-Jennings, 1920.

A general model which accommodates different fitness values of an individual in association with each other type of individual is studied. The model in conjunction with population composition, in which individuals are assumed to be dispersed at random, determines the relative fitness values of individuals. Expressions for the change in gene frequency and for the mean are derived for one locus with two alleles in a random mating population. Equilibrium and max/min mean compositions generally require solving third-degree equations, but explicit expressions are found for models with general levels of dominance and some of their variants. Protected polymorphisms, genetic loads, inbreeding depressions, and the cost of a gene substitution are also considered. Many additional features of the statics and dynamics of populations are introduced with the effects of population composition on fitnesses. A single globally stable equilibrium can exist without any dominance. There may be multiple equilibria, up to three,...

The compensation on the part of rh-negative mothers who tend to replace their lost (heterozygous) children by having more (recessive) births cannot lead to a stable equilibrium value of rh frequency. The compensation coefficient (t) has been estimated to be roughly 0.36 to 0.40 for American Whites, being much larger than selection intensity (.025 to .10). Although it seems to be on the increase at present, the future direction of change in rh frequency and its ultimate fate in the population is still indefinite, depending upon the future status of selection and compensation.