*(English)*Zbl 0872.35054

The role of spatial heterogeneity and dispersal in the dynamics of populations has been an important research subject. As *M. E. Gilpin* and *F. J. Ayala* [Proc. Nat. Acad. Sci. USA 70, 3590-3593 (1973; Zbl 0272.92016)] pointed out, the Lotka-Volterra systems are the linearization of the per capita growth rates ${\dot{N}}_{i}/{N}_{i}$ at the equilibrium. In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala claimed that a slightly more complicated model was needed and proposed the following competition model:

Here ${N}_{i}$ are the population densities, ${r}_{i}$ are the intrinsic exponential growth rates, ${K}_{i}$ are the carrying capacities in the absence of competition, and ${\theta}_{i}$ are the parameters to modify the classical Lotka-Volterra model. In this paper, we incorporate spatial diffusion in the Gilpin-Ayala competition model. Stability investigations of the nontrivial equilibrium of the Gilpin-Ayala competition model are given. A generalized Gilpin-Ayala model with diffusion, where the interactions are assumed to be also nonlinear, is proposed and the stability of the nontrivial equilibrium is studied.