Turing pattern of a reaction-diffusion predator-prey model with weak Allee effect and delay

In this paper, we establish a reaction-diffusion predator-prey model with weak Allee effect and delay and analyze the conditions of Turing instability. The effects of Allee effect and delay on pattern formation are discussed by numerical simulation. The results show that pattern formations change with the addition of weak Allee effect and delay. More specifically, as Allee effect constant and delay increases, coexistence of spotted and stripe patterns, stripe patterns, and mixture patterns emerge successively. From an ecological point of view, we find that Allee effect and delay play an important role in spatial invasion of populations.

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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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Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model

Abstract and Applied Analysis ◽

10.1155/2013/487810 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 9

Author(s):

Weiming Wang ◽

Yongli Cai ◽

Yanuo Zhu ◽

Zhengguang Guo

Keyword(s):

Pattern Formation ◽

Allee Effect ◽

Reaction Diffusion ◽

Positive Equilibrium ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Diffusion Controlled ◽

Model Dynamics ◽

And Diffusion

We investigate the spatiotemporal dynamics induced by Allee effect in a reaction-diffusion predator-prey model. In the case without Allee effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee effect is not simple, but rich and complex.

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Global bifurcation for a reaction–diffusion predator–prey model with Holling-II functional response and prey–taxis

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.110975 ◽

2021 ◽

Vol 147 ◽

pp. 110975

Author(s):

Demou Luo

Keyword(s):

Functional Response ◽

Global Bifurcation ◽

Reaction Diffusion ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Holling Ii Functional Response

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The dynamics of a Leslie type predator–prey model with fear and Allee effect

Advances in Difference Equations ◽

10.1186/s13662-021-03490-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

S. Vinoth ◽

R. Sivasamy ◽

K. Sathiyanathan ◽

Bundit Unyong ◽

Grienggrai Rajchakit ◽

...

Keyword(s):

Local Stability ◽

Allee Effect ◽

Jacobian Matrix ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model ◽

Lyapunov Coefficient ◽

Points Of View ◽

Two Parameter

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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