Spatial heterogeneity of resources versus Lotka-Volterra dynamics
This paper considers the Lotka-Volterra dynamical system \(\dot{u}=u(1-u-bv)\), \(\dot{v} = v(1-cu-v)\). To study this, the authors consider the following system of reaction-diffusion equations \[ \mu \Delta \widetilde{u}+\widetilde{u}[1+\gamma \beta(x) - \widetilde{u}] =0,\quad \nu \Delta \widetilde{v}+\widetilde{v}[1+\gamma \beta(x) - \widetilde{v}] =0, \] with zero Neumann boundary conditions. The authors investigate the stability of the equilibria and the existence of coexistence solutions with emphasis on cases where the spatial variation of the environment becomes large. The authors discuss the stability of \((\widetilde{u}, 0)\) and \( ( 0, \widetilde{v})\) respectively. The authors also discuss the implications of their results for the principle of competitive exclusions and for the equation of the evolution of diffusion discussed in [\textit{J. Dockery}, \textit{V. Hutson}, \textit{K. Mischaikow} and \textit{M. Peruarowski}, J. Math. Biol. 37, 61-83 (1998; Zbl 0921.92021)].
scientific article; zbMATH DE number 1884196