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Hopf Bifurcation for an Age-Structured Predator–Prey Model with Crowley–Martin Functional Response and Two Delays

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Abstract

In this paper, we investigate an age-structured predator–prey model with predator-dependent functional response and two delays. The main feature of this article is to study the Crowley–Martin functional response, age structure and delays at the same time in a predator–prey model. We consider two time delays in this work: one is predator–prey reaction delay \(\tau _{1}\) and the other one is predator maturation delay \(\tau _{2}\) that appears in the fertility rate function. After transforming the original model into an abstract non-densely defined Cauchy problem and taking advantage of the integrated semigroup theory, we obtain the global existence and uniqueness of solutions, the existence of equilibria of the model. Then the global asymptotic stability of the boundary equilibrium is investigated under an appropriate condition. Furthermore, the existence of Hopf bifurcation at the positive equilibrium is established when \(\tau _{1}\), \(\tau _{2}\) are changed independently and simultaneously, as well as when \(\tau _{1}=\tau _{2}\). At last, numerical simulations are carried out to support the main theoretical results.

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Yajing Li & Zhihua Liu

  2. School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China

    Ziheng Zhang

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All authors were involved in the design of the research. YL wrote the manuscript with support from ZL and ZZ. ZL and ZZ contributed to the critical revision of the manuscript for important content. All authors have read and approved the final manuscript.

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Correspondence to Yajing Li.

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This work is supported by the National Key R &D Program of China (No. 2020YFA0712900), NSFC (Grant Nos. 11871007, 11811530272 and 12071297) and the Fundamental Research Funds for the Central Universities.

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Li, Y., Liu, Z. & Zhang, Z. Hopf Bifurcation for an Age-Structured Predator–Prey Model with Crowley–Martin Functional Response and Two Delays. Qual. Theory Dyn. Syst. 22, 66 (2023). https://doi.org/10.1007/s12346-023-00765-4

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