Dynamics of Partial Differential Equations

Volume 19 (2022)

Number 4

Dynamic analysis of a diffusive eco-epidemiological system with fear effect and prey refuge

Pages: 247 – 271

DOI: https://dx.doi.org/10.4310/DPDE.2022.v19.n4.a1

Authors

Tingting Ma (College of Mathematics and Systems Science, Shandong University of Science and Technology Qingdao, China)

Xinzhu Meng (College of Mathematics and Systems Science, Shandong University of Science and Technology Qingdao, China)

Abstract

In the evolutionary development of species, the prey can produce the fear effect in the face of predation behaviors. This fear effect may affect the own reproduction growth of the prey. In order to reduce the risk of predation, the prey has the instinct to protect themselves. At the same time, the population is easy vulnerable by disease in the ecosystem. Driven by these biological facts, we propose an eco-epidemiological system with two-predator-one-prey that considers fear effect and prey refuge. At the same time, we consider the impact of spatial diffusion on the stability of the system. We discuss the conditions for the existence of all equilibrium points with biological meanings in non-spatial system. We also obtain local stability conditions for all equilibrium points. We show that the deterministic system is bistable. Kolmogorov analysis is used to analyze the appearance of limit cycles and chaos in the non-spatial system. We consider $k$ as a bifurcation parameter and study the existence of Hopf bifurcation. In the spacial diffusion system, we deduce the local stability conditions of the diffusion system and obtain the occurrence conditions of Turing instability. Numerical simulations are given to explain the phenomena beyond the scope of analytical methods and better understand the complex predator-prey interactions.

Keywords

eco-epidemiological system, fear effect, bi-stability, Hopf bifurcation, turing instability

2010 Mathematics Subject Classification

Primary 37G15, 92B05. Secondary 18F25, 92D30.

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This work was supported by the Research Fund for the Taishan Scholar Project of Shandong Province of China, Shandong Provincial Natural Science Foundation of China (ZR2019MA003), the SDUST Innovation Fund for Graduate Students (YC20210217).

Received 7 May 2022

Published 14 December 2022