Stability and Hopf bifurcation for a delayed diffusive competition model with saturation effect

Changyong Xu; Qiang Li; Tonghua Zhang; Sanling Yuan; Changyong Xu; Qiang Li; Tonghua Zhang; Sanling Yuan

doi:10.3934/mbe.2020407

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 8037-8051. doi: 10.3934/mbe.2020407

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Stability and Hopf bifurcation for a delayed diffusive competition model with saturation effect

1.
College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, China
2.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
3.
Department of Mathematics, Swinburne University of Technology, Victoria 3122, Australia

Received: 01 September 2020 Accepted: 26 October 2020 Published: 12 November 2020

This paper presents an investigation on the dynamics of a delayed diffusive competition model with saturation effect. We first perform the stability analysis of the positive equilibrium and the existence of Hopf bifurcations. It is shown that the positive equilibrium is asymptotically stable under some conditions, and that there exists a critical value of delay, when the delay increases across it, the positive equilibrium loses its stability and a spatially homogeneous or inhomogeneous periodic solution emerges from the positive equilibrium. Then, we derive the formulas for the determination of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions. Finally, some numerical simulations are performed to illustrate the obtained results.
- competitive model,
- time delay,
- diffusion,
- stability,
- Hopf bifurcation
Citation: Changyong Xu, Qiang Li, Tonghua Zhang, Sanling Yuan. Stability and Hopf bifurcation for a delayed diffusive competition model with saturation effect[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 8037-8051. doi: 10.3934/mbe.2020407

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Abstract

This paper presents an investigation on the dynamics of a delayed diffusive competition model with saturation effect. We first perform the stability analysis of the positive equilibrium and the existence of Hopf bifurcations. It is shown that the positive equilibrium is asymptotically stable under some conditions, and that there exists a critical value of delay, when the delay increases across it, the positive equilibrium loses its stability and a spatially homogeneous or inhomogeneous periodic solution emerges from the positive equilibrium. Then, we derive the formulas for the determination of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions. Finally, some numerical simulations are performed to illustrate the obtained results.

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