Stability and bifurcation analysis of a two-patch model with the Allee effect and dispersal

Yue Xia; Lijuan Chen; Vaibhava Srivastava; Rana D. Parshad; Yue Xia; Lijuan Chen; Vaibhava Srivastava; Rana D. Parshad

doi:10.3934/mbe.2023876

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 11: 19781-19807. doi: 10.3934/mbe.2023876

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Research article

Stability and bifurcation analysis of a two-patch model with the Allee effect and dispersal

1.
School of Mathematics and Statistics, Fuzhou University, Fuzhou, Fujian 350108, China
2.
Department of Mathematics, Iowa State University, Ames, IA 50011, USA

Academic Editor: Yang Kuang

Received: 18 June 2023 Revised: 16 October 2023 Accepted: 19 October 2023 Published: 30 October 2023

In the current manuscript, a two-patch model with the Allee effect and nonlinear dispersal is presented. We study both the ordinary differential equation (ODE) case and the partial differential equation (PDE) case here. In the ODE model, the stability of the equilibrium points and the existence of saddle-node bifurcation are discussed. The phase diagram and bifurcation curve of our model are also given as a results of numerical simulation. Besides, the corresponding linear dispersal case is also presented. We show that, when the Allee effect is large, high intensity of linear dispersal is not favorable to the persistence of the species. We further show when the Allee effect is large, nonlinear diffusion is more beneficial to the survival of the population than linear diffusion. Moreover, the results of the PDE model extend our findings from discrete patches to continuous patches.
- nonlinear dispersal,
- Allee effect,
- stability,
- saddle-node bifurcation,
- patch model,
- reaction-diffusion system
Citation: Yue Xia, Lijuan Chen, Vaibhava Srivastava, Rana D. Parshad. Stability and bifurcation analysis of a two-patch model with the Allee effect and dispersal[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19781-19807. doi: 10.3934/mbe.2023876

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Abstract

In the current manuscript, a two-patch model with the Allee effect and nonlinear dispersal is presented. We study both the ordinary differential equation (ODE) case and the partial differential equation (PDE) case here. In the ODE model, the stability of the equilibrium points and the existence of saddle-node bifurcation are discussed. The phase diagram and bifurcation curve of our model are also given as a results of numerical simulation. Besides, the corresponding linear dispersal case is also presented. We show that, when the Allee effect is large, high intensity of linear dispersal is not favorable to the persistence of the species. We further show when the Allee effect is large, nonlinear diffusion is more beneficial to the survival of the population than linear diffusion. Moreover, the results of the PDE model extend our findings from discrete patches to continuous patches.

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