Dynamical transition for a 3-component Lotka-Volterra model with diffusion

Abstract

The main objective of this article is to investigate the dynamical transition for a 3-component Lotka-Volterra model with diffusion. Based on the spectral analysis, the principle of exchange of stability conditions for eigenvalues is obtained. In addition, when $ \delta_0 \delta_1 $, the first eigenvalue is real. Generically, the first eigenvalue is simple and all three types of transition are possible. In particular, the transition is mixed if $ \int_{\Omega}e_{k_0}^3dx\neq 0 $, and is continuous or jump in the case where $ \int_{\Omega}e_{k_0}^3dx = 0 $. In this case we also show that the system bifurcates to two saddle points on $ \delta 0 $, and bifurcates to two stable singular points on $ \delta > \delta_1 $ as $ \tilde{\theta}