The dynamics of a delayed predator-prey model with square root functional response and stage structure

Miao Peng; Rui Lin; Zhengdi Zhang; Lei Huang; Miao Peng; Rui Lin; Zhengdi Zhang; Lei Huang

doi:10.3934/era.2024150

Electronic Research Archive

2024, Volume 32, Issue 5: 3275-3298. doi: 10.3934/era.2024150

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Research article

The dynamics of a delayed predator-prey model with square root functional response and stage structure

Miao Peng ^{1
,
,},
Rui Lin ¹,
Zhengdi Zhang ¹,
Lei Huang ^{2
,
,}

1.
School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
2.
School of Zhenjiang, Jiangsu Union Technical Institute, Zhenjiang 212016, China

Received: 10 March 2024 Revised: 30 April 2024 Accepted: 11 May 2024 Published: 15 May 2024

In recent years, one of the most prevalent matters in population ecology has been the study of predator-prey relationships. In this context, this paper investigated the dynamic behavior of a delayed predator-prey model considering square root type functional response and stage structure for predators. First, we obtained positivity and boundedness of the solutions and existence of equilibrium points. Second, by applying the stability theory of delay differential equations and the Hopf bifurcation theorem, we discussed the system's local stability and the existence of a Hopf bifurcation at the positive equilibrium point. Moreover, the properties of the Hopf bifurcation were deduced by using the central manifold theorem and normal form method. Analytical results showed that when the time delay was less than the critical value, the two populations will coexist, otherwise the ecological balance will be disrupted. Finally, some numerical simulations were also included to verify the theoretical results.
- predator-prey model,
- square root functional response,
- stage structure,
- Hopf bifurcation,
- stability
Citation: Miao Peng, Rui Lin, Zhengdi Zhang, Lei Huang. The dynamics of a delayed predator-prey model with square root functional response and stage structure[J]. Electronic Research Archive, 2024, 32(5): 3275-3298. doi: 10.3934/era.2024150

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Abstract

In recent years, one of the most prevalent matters in population ecology has been the study of predator-prey relationships. In this context, this paper investigated the dynamic behavior of a delayed predator-prey model considering square root type functional response and stage structure for predators. First, we obtained positivity and boundedness of the solutions and existence of equilibrium points. Second, by applying the stability theory of delay differential equations and the Hopf bifurcation theorem, we discussed the system's local stability and the existence of a Hopf bifurcation at the positive equilibrium point. Moreover, the properties of the Hopf bifurcation were deduced by using the central manifold theorem and normal form method. Analytical results showed that when the time delay was less than the critical value, the two populations will coexist, otherwise the ecological balance will be disrupted. Finally, some numerical simulations were also included to verify the theoretical results.

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