Abstract
In this paper, we considered a homogeneous reaction-diffusion predator-prey system with Holling type II functional response subject to Neumann boundary conditions. Some new sufficient conditions were analytically established to ensure that this system has globally asymptotically stable equilibria and Hopf bifurcation surrounding interior equilibrium. In the analysis of Hopf bifurcation, based on the phenomenon of Turing instability and well-done conditions, the system undergoes a Hopf bifurcation and an example incorporating with numerical simulations to support the existence of Hopf bifurcation is presented. We also derived a useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions correspond to j ≠ 0 and j = 0, respectively. Finally, all these theoretical results are expected to be useful in the future study of dynamical complexity of ecological environment.
Highlights
We considered a homogeneous reaction-diffusion predator-prey system with Holling type II functional response subject to Neumann boundary conditions
We derived a useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions correspond to j ≠ 0 and j = 0, respectively
Numerical results demonstrated that low diffusion values drive a periodic system into aperiodic behavior with sensitivity to initial conditions. [13] considered the case where densities of predator and prey are both spatially inhomogeneous in a bounded domain subject to homogeneous Neumann boundary condition, and they studied qualitative properties of solutions to this reaction-diffusion system. They showed that even though positive constant steady state is globally asymptotically stable for the ordinary differential equation (ODE) dynamics, non-constant positive steady states can coexist in a partial differential equations (PDEs) system
Summary
[13] considered the case where densities of predator and prey are both spatially inhomogeneous in a bounded domain subject to homogeneous Neumann boundary condition, and they studied qualitative properties of solutions to this reaction-diffusion system They showed that even though positive constant steady state is globally asymptotically stable for the ordinary differential equation (ODE) dynamics, non-constant positive steady states can coexist in a PDE system. The global bifurcation theory suggested the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions Based on this reference, [19] considered the possibility of the occurrence of Turing patterns and performed detailed Hopf bifurcation analysis in a diffusive predator-prey system with Holling type III functional response.
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