Qualitative structure of a discrete predator-prey model with nonmonotonic functional response

Yanlin Zhang; Qi Cheng; Shengfu Deng

doi:10.3934/dcdss.2022065

Article Contents

2023, Volume 16, Issue 3&4: 773-786. Doi: 10.3934/dcdss.2022065

This issue Previous Article Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise Next Article Some remarks on lattice and nonlocal dispersal evolution systems

Qualitative structure of a discrete predator-prey model with nonmonotonic functional response

1.
Minnan Science and Technology University, Quanzhou, Fujian 362332, China
2.
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

^* Corresponding author: Shengfu Deng
^* Corresponding author: Shengfu Deng

This paper is dedicated to Professor Jibin Li for his 80th birthday

Received: December 2021

Revised: February 2022

Early access: March 18, 2022

Published online: March 18, 2022

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Abstract

In this paper, we study the qualitative structure of a discrete predator-prey model with nonmonotonic functional response near a degenerate fixed point whose eigenvalues are $ \pm1 $. Firstly, the model is transformed into an ordinary differential system by using the normal form theory and the Takens's theorem. Then, the qualitative properties of this ordinary differential system near the degenerate equilibrium are analyzed with the blowing-up method. Finally, according to the conjugacy between the discrete model and the time-one mapping of the vector field, the qualitative structure of this discrete model is obtained. Numerical simulations are also given.

Keywords:

Mathematics Subject Classification: Primary: 37C05, 37G05, 39A30; Secondary: 37C75.

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Figure 1. Phase portraits of system (20) in the rigion $ U_{+} $

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Figure 2. The phase portraits for $ 0<D<\frac{8}{3} $

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Figure 3. The phase portraits for $ D = \frac{8}{3} $

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Figure 4. The phase portraits for $ D>\frac{8}{3} $

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Figure 5. Phase portraits of system (1)

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