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Stability in Gilpin-Ayala competition models with diffusion. (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 ${\stackrel{˙}{N}}_{i}/{N}_{i}$ 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:

$\frac{d{N}_{i}}{dt}={r}_{i}{N}_{i}\left(1-{\left({N}_{i}/{K}_{i}\right)}^{{\theta }_{i}}-\sum _{{}_{\begin{array}{c}j=1\\ j\ne i\end{array}}}^{m}{\alpha }_{ij}\left({N}_{j}/{K}_{j}\right)\right),\phantom{\rule{1.em}{0ex}}i=1,\cdots ,m·$

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.

##### MSC:
 35K57 Reaction-diffusion equations 35B35 Stability of solutions of PDE 92D25 Population dynamics (general)