Spatiotemporal Patterns of a Homogeneous Diffusive Predator–Prey System with Holling Type III Functional Response

Nonexistence of nonconstant positive steady states of a diffusive predator-prey model

“…(3.2). Moreover, our result supplements the results in [27] and implies that each global bifurcating branch of steady state solutions of model (1.3) obtained in [27] is a bounded loop, which connects at least two different bifurcation points, (see Section 3).…”

“…The dynamics of the ODE systems can be found in [4,11,12] and references therein. Recently, Wang [27] studied the following nondimensionalized diffusive predatorprey model with Holling type III functional response and no flux boundary conditions,…”

Spatial pattern formation in activator-inhibitor models with nonlocal dispersal

“…Turing's pioneer work [50] suggested that different diffusion rates of activator and inhibitor in a biological system can lead to the generation of spatially inhomogeneous patterns, and such diffusion-induced instability (Turing instability) has been credited as the driving mechanism of pattern formations in chemistry [25,37], developmental biology [14,22,45,46], and ecology [19,40,41]. Mathematical theory of linear stability and symmetry-breaking bifurcation have been applied to the such reaction-diffusion models to rigorously establish the existence and stability of spatial patterns, see [16,17,27,35,38,52,53,54] and references therein. In the framework of reaction-diffusion model, it is well-established that the condition for spatial pattern formation in two-species model is to have a slow diffusion rate for activator and a fast diffusion rate for inhibitor [14,23].…”

Stochastic spatiotemporal diffusive predator-prey systems

“…m > max{d, (a 2 −1)d}. [20] gives a stability result regarding the equilibrium (κ, v κ ) and (1, 0) is under some conditions respectively. But for the stochastic case, because we formulate system (2) by stochastic perturbations σ 1 uẆ 1 and σ 2 vẆ 2 directly, there is no positive equilibrium point as a solution.…”

“…The relation of two species can be described by competition, predator-prey, auspiciousness and so on. Among them, the predator-prey interaction is a significant one, which was introduced by Lotka and Volterra( [10]) and has been developing rapidly in the last decades( [4,21,5,20,8]). In this paper, we consider a stochastic homogeneous spatiotemporal diffusive predator-prey system with functional response.…”