Populations of Interacting Autogenous Components

General procedures are developed for finding equilibrium and maximum/minimum mean compositions for a population of autogenous components competing in a pairwise manner among individuals. The procedures, involving elementary matrix techniques, are easy to apply for any specific competitive model. Necessary and sufficient conditions are developed for the mean to be a maximum and for the equilibrium composition to be stable. The problem of a maximum mean reduces to that of evaluating a quadratic form which must be negative definite, and is easily accomplished for a specific competitive model. Stability of equilibrium depends on the eigenvalues of a first-order transition matrix relating deviations of component frequencies from their equilibrium frequencies in successive generations. For stability, all of the eigenvalues must vanish in time. These conditions, sometimes difficult to find, are illustrated for a three- and four-component model. The eigenvalues also provide considerable information about the dynamics of the population. They determine the rate of divergence from, or convergence to, equilibrium composition. Often some of them are complex, which accounts for the oscillating behavior of component frequencies. In such cases they provide estimates of the periodicity. Through them the effect of the relative magnitudes of competitive effects upon periodicity and on rate of divergence or convergence can be evaluated. Amplitudes of oscillations, when these occur, are primarily determined by initial frequencies and can be approximated by harmonic expressions derived from the transition matrix. Competitive effects are measured as the deviation of a component's competitive performance from the competitor's performance in competition with itself, a definition different from the prevalent one. Equilibria and stability are entirely a function of these competitive effects. Although the details are complicated, a general conclusion is that a preponderance of positive competitive effects is required for stable equilibria. Some models with three or more components have a multiplicity of equilibria compositions. In these cases and under certain conditions, the population moves to the equilibrium nearest to the initiating composition. Consequently, the equilibrium reached depends on the initial frequencies.