Abstract
A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating timeperiodic solutions are investigated and a normal form of Bogdanov–Takens bifurcation is determined as well.
Highlights
The interaction of predator and prey has abundant dynamical features the investigations on predator-prey models has improved and lasted for several decades, which are based on the pioneering works of Lotka and Volterra [34]
We study the Bogdanov–Takens bifurcation by regarding the cross-diffusion term β as one of bifurcation parameters
This paper presents the existence of the non-constant positive steady states of system (1.4)
Summary
The interaction of predator and prey has abundant dynamical features the investigations on predator-prey models has improved and lasted for several decades, which are based on the pioneering works of Lotka and Volterra [34]. The quantity v is not influenced by any cross diffusion in the sense that the coefficient β in the second equation of (1.4) vanishes, that is, we ignore the population migration of predators due to the presence of preys In this situation, Li et al [20] considered the following reaction-diffusion system in the one-dimensional space domain Ω = (0, π):. For a predator-prey system, what we are interested in is whether the various species can exist and takes the form of non-constant time-independent positive solutions. Researchers have paid more attention to Hopf bifurcation and steady state bifurcation (cf [9,10,15,18,19,36,42,46]), and investigated some predator-prey models without cross diffusion term.
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