Qualitative behaviour of positive solutions of a predator—prey model: effects of saturation

Abstract

We study a predator–prey system with Holling–Tanner interaction terms. We show that if the saturation rate m is large, spatially inhomogeneous steady-state solutions arise, contrasting sharply with the small-m case, where no such solution could exist. Furthermore, for large m, we give sharp estimates on the ranges of other parameters where spatially inhomogeneous solutions can exist. We also determine the asymptotic behaviour of the spatially inhomogeneous solutions as m → ∞, and an interesting relation between this population model and free boundary problems is revealed.

Allee effect and bistability in a spatially heterogeneous predator-prey model

“…In both [DD] and [DHs], only one attracting state seems possible, contrasting the possible bistability for (1.2). On the other hand, when the environment is homogeneous (a(x) ≡ a), the set of steady state solutions of (1.2) was studied in [DL3] for the large m case (see also [BB,CEL,DL1,DL2] for Dirichlet problems), and for small m, it follows from results in [dMR] that the system has at most one positive steady-state, and it is a constant solution and is globally attracting (when it exists). In this paper, we consider a general m > 0, and most of our results holds for all positive m. In particular we show that for intermediate m (neither small nor large), (1.2) may have a global branch of steady state solutions whose shape is roughly a reversed S (see Figure 1 (b)).…”

Qualitative analysis for a diffusive predator–prey model

“…For non-constant positive steady states (i.e. stationary patterns), we refer to [11][12][13]. On the other hand, under homogeneous Dirichlet boundary conditions, many authors have discussed the existence of positive steady states which involves difficult a priori estimates, see e.g., [14][15][16].…”

Qualitative analysis of a strongly coupled predator–prey system with modified Holling–Tanner functional response

“…Starting with Turing's seminal paper [34], diffusion and cross-diffusion have been observed as causes of the spontaneous emergence of ordered structures, called patterns in a variety of non-equilibrium situations. They include the Gierer-Meinhardt model [35][36][37][38], the Sel'kov model [26,39], the Lotka-Volterra competition model [40][41][42] and the Lotka-Volterra predator-prey model [20,23,24,[43][44][45] and so on.…”