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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Stationary Pattern in a Predator-Prey Model with Predator-Harvesting Policy

Discontinuity, Nonlinearity, and Complexity 12(2) (2023) 411--436 | DOI:10.5890/DNC.2023.06.013

$^1$ School of Mathematical Sciences, Anhui University, Hefei 230601, China

$^2$ College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

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Abstract

Stationary patterns in a predator-prey model with Holling-III functional response and harvesting policy are investigated in this work. For the proposed model, nonnegativity, uniformly boundedness, permanence, stability, the existence and direction of Hopf bifurcation are analyzed. For the spatial system, the existence conditions for the Turing instability are also established. Then using weakly nonlinear analysis, amplitude equations near critical values of the Turing instability are derived. Different kinds of solutions can be shown analytically by analyzing amplitude equations. Based on numerical analysis, the stationary patterns can be found, such as hexagonal patterns, stripe patterns and mixed states of hexagonal pattern and stripe pattern. Numerical simulations are well consistent with theoretical results. It is further noted that the behavior of harvesting is a factor for existence and stability of equilibria, the occurrence of the transcritical bifurcation, pattern formation and the permanence of the system.

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