A general two-species competition model with time-varying rates

The classical mathematical model for the behaviour of the populations of two competing biological species has previously been generalized by the author, by assuming that the coefficients of intrinsic growth, self-inhibition and interaction were all functions of time and, for a certain class of the governing differential equations, the exact solutions were obtained: an example was given in which the coefficients were periodic functions. In the differential equations of this model (as well as in the autonomous Lotka-Volterra equations), each of the Malthusian growth-rates was assumed to be diminished by a linear function of the populations of the two species, without there being any rigorous justification for this assumption. We here generalize the differential equations by assuming for these diminution functions general nonlinear forms having time-varying coefficients. The exact solutions are given for four classes of the resulting strongly nonlinear non-autonomous differential equations. Various conclusions about the growth modes of the two populations and their asymptotic behaviour are drawn, both when specific and when arbitrary forms are assumed for the coefficient functions. Cases are examined in which the Competitive Exclusion Principle holds and others in which it does not.