Hopf bifurcation in an age-structured prey-predator model with Holling Ⅲ response function

Lijun Wang; Chuanjun Dai; Min Zhao; Lijun Wang; Chuanjun Dai; Min Zhao

doi:10.3934/mbe.2021156

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 3144-3159. doi: 10.3934/mbe.2021156

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Hopf bifurcation in an age-structured prey-predator model with Holling Ⅲ response function

1.
Zhejiang Provincial Key Laboratory for Water Environment and Marine Biological Resources Protection, Wenzhou University, Wenzhou, Zhejiang, 325035, China
2.
School of Mathematics and Physics, Wenzhou University, Wenzhou, Zhejiang, 325035, China
3.
School of Life and Environmental Science, Wenzhou University, Wenzhou, Zhejiang, 325035, China

Academic Editor: Moxun Tang

Received: 11 January 2021 Accepted: 30 March 2021 Published: 02 April 2021

In this paper, we propose a prey-predator model with age structure which is described by the mature period. The aim of this paper is to study how mature period affect the dynamics of interaction between prey and predator. The sufficient condition of the existence of non-negative steady state is derived. By using integrated semigroup theory, we obtain the characteristic equation, by which we find that the non-negative steady state will lose its stability via Hopf bifurcation induced by mature period, and the corresponding periodic solutions emerge. Additionally, some numerical simulations are provided to illustrate the results predicted by linear analysis. Especially, the numerical results indicate that both mature period and age can affect the amplitude and period of periodic solutions.
- age-structure,
- Holling Ⅲ functional response,
- Hopf bifurcation,
- prey-predator model,
- non-densely defined
Citation: Lijun Wang, Chuanjun Dai, Min Zhao. Hopf bifurcation in an age-structured prey-predator model with Holling Ⅲ response function[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 3144-3159. doi: 10.3934/mbe.2021156

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Abstract

In this paper, we propose a prey-predator model with age structure which is described by the mature period. The aim of this paper is to study how mature period affect the dynamics of interaction between prey and predator. The sufficient condition of the existence of non-negative steady state is derived. By using integrated semigroup theory, we obtain the characteristic equation, by which we find that the non-negative steady state will lose its stability via Hopf bifurcation induced by mature period, and the corresponding periodic solutions emerge. Additionally, some numerical simulations are provided to illustrate the results predicted by linear analysis. Especially, the numerical results indicate that both mature period and age can affect the amplitude and period of periodic solutions.

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