Global stability of spatially nonhomogeneous steady state solution in a diffusive Holling-Tanner predator-prey model

We investigate a diffusive Leslie-Gower predator-prey model with the additive Allee effect on prey subject to the zero-flux boundary conditions. Some results of solutions to this model and its corresponding steady-state problem are shown. More precisely, we give the stability of the positive constant steady-state solution, the refineda prioriestimates of positive solution, and the nonexistence and existence of the positive nonconstant solutions. We carry out the analytical study for two-dimensional system in detail and find out the certain conditions for Turing instability. Furthermore, we perform numerical simulations and show that the model exhibits a transition from stripe-spot mixtures growth to isolated spots and also to stripes. These results show that the impact of the Allee effect essentially increases the model spatiotemporal complexity.

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Spatio-Temporal Patterns of Predator–Prey Model with Allee Effect and Constant Stocking Rate for Predator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421502497 ◽

2021 ◽

Vol 31 (16) ◽

Author(s):

Xuan Tian ◽

Shangjiang Guo

Keyword(s):

Steady State ◽

Allee Effect ◽

Sufficient Conditions ◽

Steady State Solution ◽

Turing Pattern ◽

Stocking Rate ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Spatio Temporal

A diffusive predator–prey model with Allee effect and constant stocking rate for predator is investigated and it is shown that Allee effect is the decisive factor driving the formation of Turing pattern. Furthermore, it is observed that Turing pattern appears only when the diffusion rate of the prey is faster than that of the predator, which is just opposite to the condition of Turing pattern in the classical predator–prey system. Some sufficient conditions are obtained to ensure the asymptotical stability of a spatially homogeneous steady-state solution. The existence and nonexistence of positive nonconstant steady-state solutions are investigated to understand the mechanisms of generating spatiotemporal patterns. Furthermore, Hopf and steady-state bifurcations are analyzed in detail by using Lyapunov–Schmidt reduction.

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Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501451 ◽

2019 ◽

Vol 29 (11) ◽

pp. 1950145 ◽

Cited By ~ 1

Author(s):

Yu-Xia Wang ◽

Wan-Tong Li

Keyword(s):

Steady State ◽

Functional Response ◽

Spatial Patterns ◽

Bifurcation Theory ◽

Steady State Solution ◽

State Solution ◽

Predator Prey ◽

Predator Prey Model ◽

Positive Steady State ◽

Prey System

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

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Dynamics of a Predator–Prey Model with the Effect of Oscillation of Immigration of the Prey

Diversity ◽

10.3390/d13010023 ◽

2021 ◽

Vol 13 (1) ◽

pp. 23

Author(s):

Jawdat Alebraheem

Keyword(s):

Global Stability ◽

Stability Conditions ◽

Dynamic Behaviors ◽

Predator Prey ◽

Local And Global Stability ◽

Predator Prey Model ◽

Prey Model ◽

Proposed Model

In this article, the use of predator-dependent functional and numerical responses is proposed to form an autonomous predator–prey model. The dynamic behaviors of this model were analytically studied. The boundedness of the proposed model was proven; then, the Kolmogorov analysis was used for validating and identifying the coexistence and extinction conditions of the model. In addition, the local and global stability conditions of the model were determined. Moreover, a novel idea was introduced by adding the oscillation of the immigration of the prey into the model which forms a non-autonomous model. The numerically obtained results display that the dynamic behaviors of the model exhibit increasingly stable fluctuations and an increased likelihood of coexistence compared to the autonomous model.

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