Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms

Abstract

<p style='text-indent:20px;'>In this paper, we present results about existence and non-existence of coexistence states for a reaction-diffusion predator-prey model with the two species living in a bounded region with inhospitable boundary and Holling type II functional response. The predator is a specialist and presents self-diffusion and cross-diffusion behavior. We show the existence of coexistence states by combining global bifurcation theory with the method of sub- and supersolutions.</p>

Cross-Diffusion-Induced Turing Instability in a Two-Prey One-Predator System

“…1 ) for some i, then det(A(µ i )) > 0 by (2), and, consequently, A 0i = − det(A (i) ) < 0. The number of sign of changes in the characteristic polynomial (9) ρ i (λ) = λ 3 + A 2i λ 2 + A 1i λ + A 0i is either one or three.…”

“…Theorem 1. The unique positive equilibrium ū is globally asymptotically stable for the ODE system (2).…”

“…Recently, due to the most interesting qualitative feature of pattern formation induced by the cross-diffusion effect, there have been some works on the diffusion-driven instability (Turing instability [1]), bifurcation theory and the existence of a non-constant stationary solution; please refer to [2][3][4][5][6][7][8][9][10][11][12][13] and the references cited therein. As we know, the problem of cross-diffusion was proposed first by Kerner [14] and first applied to competitive population systems by Shigesada et al [15].…”

“…The main purpose of this paper is to study the Turing instability, which is driven solely by the effect of nonlinear cross-diffusion, using mathematical analysis and numerical simulations. The rest of this paper is organized as follows: In Section 3, we first present the global asymptotic stability of the unique positive equilibrium of the ODE system (2). We also prove that the positive equilibrium remains linearly stable in the presence of selfdiffusion.…”