Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response

Xiao Wu; Shuying Lu; Feng Xie; Xiao Wu; Shuying Lu; Feng Xie

doi:10.3934/mbe.2023782

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 10: 17608-17624. doi: 10.3934/mbe.2023782

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Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response

College of Science, Donghua University, Shanghai 201620, China

Academic Editor: Yang Kuang

Received: 05 July 2023 Revised: 01 September 2023 Accepted: 08 September 2023 Published: 14 September 2023

In this paper, we consider the dynamics of a slow-fast Bazykin's model with piecewise-smooth Holling type Ⅰ functional response. We show that the model has Saddle-node bifurcation and Boundary equilibrium bifurcation. Furthermore, it is also proven that the model has a homoclinic cycle, a heteroclinic cycle or two relaxation oscillation cycles for different parameters conditions. These results imply the dynamical behavior of the model is sensitive to the predator competition rate and the initial densities of prey and predators. In order to support the theoretical analysis, we present some phase portraits corresponding to different values of parameters by numerical simulation. These phase portraits include two relaxation oscillation cycles, an unstable relaxation oscillation cycle surrounded by a stable homoclinic cycle; the coexistence of a heteroclinic cycle and an unstable relaxation oscillation cycle. These results reveal far richer and much more complex dynamics compared to the model without different time scale or with smooth Holling type Ⅰ functional response.
Citation: Xiao Wu, Shuying Lu, Feng Xie. Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 17608-17624. doi: 10.3934/mbe.2023782

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Abstract

In this paper, we consider the dynamics of a slow-fast Bazykin's model with piecewise-smooth Holling type Ⅰ functional response. We show that the model has Saddle-node bifurcation and Boundary equilibrium bifurcation. Furthermore, it is also proven that the model has a homoclinic cycle, a heteroclinic cycle or two relaxation oscillation cycles for different parameters conditions. These results imply the dynamical behavior of the model is sensitive to the predator competition rate and the initial densities of prey and predators. In order to support the theoretical analysis, we present some phase portraits corresponding to different values of parameters by numerical simulation. These phase portraits include two relaxation oscillation cycles, an unstable relaxation oscillation cycle surrounded by a stable homoclinic cycle; the coexistence of a heteroclinic cycle and an unstable relaxation oscillation cycle. These results reveal far richer and much more complex dynamics compared to the model without different time scale or with smooth Holling type Ⅰ functional response.

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