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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 {\it M. E. Gilpin} and {\it 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: $${dN_i\over dt}= r_iN_i\Biggl(1- (N_i/K_i)^{\theta_i}- \sum^m_{\Sb j=1\\ j\ne i\endSb}\alpha_{ij}(N_j/K_j)\Biggr),\quad i=1,\dots,m.$$ 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)
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##### References:
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