Spatiotemporal complexity in a Leslie-Gower type predator-prey model near Turing-Hopf point

Abstract

The Leslie-Gower type predator-prey system with the ratio-dependent Holling III functional response and Neumann boundary conditions is investigated in this paper. First, the boundedness results of both parabolic and elliptic equations are presented. Hereafter, the existence of the codimension-two Turing-Hopf point (C2THP) is identified, where the Turing and the Hopf modes intersect. To further explore the spatiotemporal dynamics near the C2THP, it is necessary to derive the amplitude equations, however, there are few results about that in the two-dimensional domain. Here the method of weakly nonlinear analysis is adopted to derive the amplitude equations. The temporal patterns, hexagonal patterns, and plane wave patterns, as well as the sufficient conditions of their existence and stability, can be presented through amplitude equations.