On the Volterra and Other Nonlinear Models of Interacting Populations

A detailed investigation of various aspects of the populations of interacting biological species is made, using Volterra's and other models. For a system with two species, a detailed analysis is possible, and we plot the time variation of the population of two species for a set of values of the parameters. For a many-species system, it is shown that the time average of the population ${N}_{i}$ of the $i\mathrm{th}$ species is equal to its steady state value ${q}_{i}$. For time averages of other functions of ${N}_{i}$ and ${N}_{i}{N}_{j}$, only the equations satisfied by them are derived. The population growth of a species is studied by assuming that the effect of other species is to introduce a random function of time in the growth equation. The resulting Fokker-Planck equation is shown to have the same form as the Schr\"odinger and Bloch equations. The "potential" of the Schr\"odinger equation depends upon the form of the number-dependent term in the growth function. For the Gompertz form, it is a simple harmonic oscillator potential and, for the Verhulst form, the Morse potential. For both the forms, the Fokker-Planck equation is explicitly solved to obtain the probability distribution $P(N, t)$ of the population as a function of time. It is shown that considerable simplification is achieved in the calculation of the steady state concentrations (${q}_{i}'\mathrm{s}$) of the component species if Pfaffians are used. Since ${q}_{i}$ is equal to the time average of the population of the $i\mathrm{th}$ species, a great deal can be said about the stability of the population in the sense of the rarity of explosion or extinction of one or more species. If the interactions between various species and their growth coefficients (in the absence of interactions) are known, we show that a priori one can determine whether the population will be stable or not and, if not, which of the species will disappear. Also, the stability of the population is discussed when several new species are introduced. We show that the stability is dependent on how the newly introduced species interact with each other and with the population into which they are introduced. If the information about the detailed interactions between various species is absent, as is usually the case, then a statistical mechanical treatment of the population is desirable. We show that for small deviations from steady state populations, the necessary and sufficient condition for such a treatment is that the number of species is large. For arbitrary deviations the latter is a necessary condition and may not be sufficient. This statistical mechanical treatmetn provides an empirical method for calculating the interaction between two species and the stability of the population. A measure of the stability of an ecology is defined which could be used to compare the relative stabilities of two ecologies with the same macroscopic properties. The effect of changes in the interactions between var\'{\i}ous species, due to changes in temperature, humidity, age distribution, etc., is studied by assuming the rate constants to be random. A master equation for the probability distributions of ${N}_{i}'\mathrm{s}$ is derived. It is shown that the stationary population distribution is Poisson only if the variation in the rate constants is not too rapid. A brief outline of another stochastic model for the population growth in terms of the probabilities of birth and death of the individuals is given. Since the members of the population do not react instantaneously to any change in the environment, the prey-predator interaction does not affect the population of either prey or predator instantaneously, and the egg is not converted into an adult instantaneously, the effects of the time lags in the above processes on the behavior, in particular the stability, of the population are studied. Further generalizations of the Volterra model are discussed, and a brief review of other systems of interacting species, e.g., systems of biochemical oscillators, nervous systems, multimode lasers, systems of simultaneously growing bacteria, etc., is given. Finally, a sampling of the experiments which throw some light on the validity of Volterra's model and the statistical mechanical treatment is given.