Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments

Abstract

Previously in [ 14 ], we considered a diffusive logistic equation with two parameters, \begin{document}$ r(x) $\end{document} – intrinsic growth rate and \begin{document}$ K(x) $\end{document} – carrying capacity. We investigated and compared two special cases of the way in which \begin{document}$ r(x) $\end{document} and \begin{document}$ K(x) $\end{document} are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra competition-diffusion system with general intrinsic growth rates and carrying capacities for two competing species in heterogeneous environments. We establish the main result that determines the global dynamics of the system under a general criterion. Furthermore, when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional — such ratios can also be interpreted as the competition coefficients — this criterion reduces to what we obtained in [ 18 ]. We also study the detailed dynamics in terms of dispersal rates for such general case. On the other hand, when the two ratios are not proportional, our results in [ 14 ] show that the criterion in [ 18 ] cannot be fully recovered as counterexamples exist. This indicates the importance and subtleties of the roles of heterogeneous competition coefficients in the dynamics of the Lotka-Volterra competition-diffusion systems. Our results apply to competition-diffusion-advection systems as well. (See Corollary 5.1 in the last section.)