Persistence, Stability and Hopf Bifurcation in a Diffusive Ratio-Dependent Predator–Prey Model with Delay

Abstract

In this paper, we study the persistence, stability and Hopf bifurcation in a ratio-dependent predator–prey model with diffusion and delay. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system and global stability of the boundary equilibrium. The local stability of the positive constant equilibrium and delay-induced Hopf bifurcation are investigated by analyzing the corresponding characteristic equation. We show that delay can destabilize the positive equilibrium and induce spatially homogeneous and inhomogeneous periodic solutions. By calculating the normal form on the center manifold, the formulae determining the direction and the stability of Hopf bifurcations are explicitly derived. The numerical simulations are carried out to illustrate and extend our theoretical results.

Delay-induced Hopf bifurcation in a diffusive Holling–Tanner predator–prey model with ratio-dependent response and Smith growth

“…In this section, we investigate the stability of these Hopf bifurcations and bifurcating direction by using the normal formal theory of partial differential equation [21][22][23]. Without loss of generality, denote any of these critical values by τ 0 , at which the characteristic equation (1.2) has a pair of simple purely imaginary roots iω 0 .…”

“…Following the standard procedure in [21,23], especially [22], we can obtain the following normal form on the center manifold:…”

“…In the survey papers [12,[17][18][19], the authors deduced that delay difference equations are much more complicated than ordinary differential equations, because time delay can lead to unstable equilibrium and induce bifurcation. Many results on the Hopf bifurcation of diffusion system can be found in [20][21][22][23][24][25][26][27]. The Hopf bifurcation of the diffusive predator-prey system with delay subject to homogeneous Neumann boundary conditions was studied in [20][21][22][23].…”

“…Many results on the Hopf bifurcation of diffusion system can be found in [20][21][22][23][24][25][26][27]. The Hopf bifurcation of the diffusive predator-prey system with delay subject to homogeneous Neumann boundary conditions was studied in [20][21][22][23]. The Hopf bifurcation for a delayed predator-prey diffusion system with Dirichlet boundary condition wasonsidered in [24].…”

Stability and Hopf Bifurcation Analysis of a Gene Expression Model with Diffusion and Time Delay

“…For standard notations and classical results on partial functional differential equations, please refer to [12,13,17]. More details on techniques for computing the normal form can also be found in recent work [18]. Now, normalizing by the time-scaling → / , then (12)-(13) can be rewritten as…”