On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation

Kimun Ryu; Wonlyul Ko

doi:10.3934/dcdsb.2022015

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2022, Volume 27, Issue 11: 6679-6709. Doi: 10.3934/dcdsb.2022015 open access

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On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation

Kimun Ryu and
Wonlyul Ko^,

Department of Mathematics Education, Cheongju University, Cheongju, Chungbuk 28503, Republic of Korea

^* Corresponding author: kowl@cju.ac.kr (Wonlyul Ko)
^* Corresponding author: kowl@cju.ac.kr (Wonlyul Ko)

Received: July 31, 2021

Revised: September 30, 2021

Early access: February 2022

Published: October 2022

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Abstract

The current paper provides a qualitative study of a diffusive predator-prey system with a hunting-cooperation functional response under homogeneous Neumann boundary conditions. The considered functional response explains a behavioral mechanism in which predators search for and hunt their prey cooperatively. We investigate the global attractor for nonnegative time-dependent solutions to the system and the nonpersistence leading to the extinction of the predator species in the system. Moreover, we study the local stability at all feasible nonnegative constant steady states and the occurrence of Hopf bifurcation (i.e., the existence of time-periodic orbits) in the system. Finally, we study the existence and nonexistence of nonconstant positive steady states in the system, and the limiting behavior of the positive steady states according to the diffusion rate. Through this study, we observe interesting results, such as bistability, extinction of the predator species induced by a large consumption rate, and stationary patterns in the system with hunting cooperation in predators.

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Mathematics Subject Classification: 35B36, 35K57, 92D25.

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