Turing instability for a ratio-dependent predator–prey model with diffusion

Abstract

Ratio-dependent predator-prey models have been increasingly favored by field ecologists where predator-prey interactions have to be taken into account the process of predation search. In this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction-diffusion model in which predator mortality is neither a constant nor an unbounded function, but it is increasing with the predator abundance. We show that analytically at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion). We also show that the stationary solution becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises. A numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented. Numerical examples are also included.

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

“…As far as Turing instability is concerned, the necessary conditions for occurrence of Turing instability in the diffusive predator–prey system are obtained by many researchers . In this paper, we can determine the sufficient and necessary condition for occurrence of Turing instability in the β − e 2 plane for fixed other parameters.…”

“…For the diffusive Leslie–Gower predator–prey system with Dirichlet boundary condition, Zhou and Shi investigated the existence of the positive steady‐state solutions. We would also like to mention that the study on the dynamics of Lotka–Volterra predator–prey system with diffusion has also recently recovered some interest .…”

Bifurcation analysis of a diffusive predator–prey system with a herd behavior and quadratic mortality

“…The dynamics of the diffusive predator-prey model with Lotka-Volterra interaction term has been extensively investigated by many researchers using the bifurcation method. For instance, the stability and Hopf bifurcations have been studied in [21][22][23][24][25][26] for the Holling type functional response, in [27] for the system incorporating a prey refuge, and in [28][29][30][31][32][33] for ratio-dependent response. However, there are little work on this diffusive predator-prey system (1.2) with a herd behavior and quadratic mortality.…”

Effect of Prey Refuge on the Spatiotemporal Dynamics of a Modified Leslie-Gower Predator-Prey System with Holling Type III Schemes

“…Lately, Camera [27] has specified the spatiotemporal dynamics of Equation (1) with diffusion of species. Meanwhile, a large amount of literatures mainly study this theme in reaction-diffusion systems since Turing [28] pointed out that this kind of system could yield many complex patterns, which are usually consistent with a wide variety of phenomena that have been observed in chemistry, physics and biology [29][30][31]. Thus, Equation (2) with the spatial factor can be described as following:…”