Spatiotemporal dynamics of delayed discrete Lotka-Volterra competitive patch models in heterogeneous environments

A patch network is introduced to describe the spatiotemporal dynamics of a delayed Lotka-Volterra competition model in heterogeneous environments. The species are subject to general dispersal patterns and spatial resource variation. It is shown that the model admits a positive equilibrium, and the infinitesimal generator associated with the linearized system has two pairs of purely imaginary eigenvalues when there are no losses of individuals during the dispersal. Furthermore, we study the stability of this positive equilibrium and the associated Hopf bifurcation when the dispersal rate is large. Moreover, the differences in the Hopf bifurcation values between no losses and losses of individuals during dispersal are considered in a special case.