# American Institute of Mathematical Sciences

May  2020, 25(5): 1631-1647. doi: 10.3934/dcdsb.2019244

## Stability and bifurcation analysis of Filippov food chain system with food chain control strategy

 1 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China 2 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

* Corresponding author: Soliman A. A. Hamdallah

Received  March 2019 Revised  June 2019 Published  May 2020 Early access  November 2019

In the present work, we introduce a control model to describe three species food chain interaction model composed of prey, middle predator, and top predator. The middle predator preys on prey and the top predator preys on middle predator. The control techniques of the exploited natural resources are used to modulate the harvesting effort to avoid high risks of extinction of the middle predator and keep stability of the food chain, by prohibiting fishing when the population density drops below a prescribed threshold. The behavior of the system stability of the regular, virtual, pseudo-equilibrium and tangent points are discussed. The complicated non-smooth dynamic behaviors (sliding and crossing segment and their domains) are analyzed. The bifurcation set of pseudo-equilibrium and the sliding crossing bifurcation have been investigated. Our analytical findings are verified through numerical investigations.

Citation: Soliman A. A. Hamdallah, Sanyi Tang. Stability and bifurcation analysis of Filippov food chain system with food chain control strategy. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1631-1647. doi: 10.3934/dcdsb.2019244
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##### References:
Illustration of existence of sliding segment of Filippov system (3): where $A\;c_2< a_3,\;c2<a_3+E,B\;c_2> a_3,\;c2>a_3+E,$ and $C\; c_2> a_3,\;c2<a_3+E$
Bifurcation of pseudo-equilibrium
Bifurcation of pseudo-equilibrium
Limit cycle without sliding segment
Limit cycle with sliding segment $\Sigma^S_1$
Stable crossing sliding cycle
The phase portrait of Filippov system (3). We choose $a_3 = 0.0135,\;E = 0.02$
The phase portrait of Filippov system (3). We choose $a_3 = 0.0124,\;E = 0.02$
The phase portrait of Filippov system (3). We choose $a_3 = 0.0122,\;E = 0.0002$
The phase portrait of Filippov system (3). We choose $a_3 = 0.0123,\;E = 0.0002$
The phase portrait of Filippov system (3). We choose $a_3 = 0.01,\;E = 0.02$
The phase portrait of Filippov system (3). We choose $a_3 = 0.01,E = 0.5,a_1 = 1, a_2 = 0.5, b_1 = 0.09, b_2 = 1.117283951, c_1 = 1.1, c_2 = 1.25, d = 10, and\; ET = 0.9.$
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