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Discontinuous harvesting policy in a Filippov system involving prey refuge

Rajesh Ranjan Patra, Sarit Maitra

Abstract

In this article, a non-smooth predator-prey dynamical system is considered. Here, we discuss about sustainable harvesting in a Filippov predator-prey system, which can produce yield and at the same time prevent over-exploitation of bioresources. The local and global stability analysis of the two subsystems, with and without harvesting, are studied. Furthermore, for the Filippov system, we have performed bifurcation analysis for several key parameters like predation rate, threshold quantity and prey refuge. Some local sliding bifurcations are also observed for the system. Numerical simulations are presented to illustrate the dynamical behaviour of the system.

Discontinuous harvesting policy in a Filippov system involving prey refuge

Abstract

In this article, a non-smooth predator-prey dynamical system is considered. Here, we discuss about sustainable harvesting in a Filippov predator-prey system, which can produce yield and at the same time prevent over-exploitation of bioresources. The local and global stability analysis of the two subsystems, with and without harvesting, are studied. Furthermore, for the Filippov system, we have performed bifurcation analysis for several key parameters like predation rate, threshold quantity and prey refuge. Some local sliding bifurcations are also observed for the system. Numerical simulations are presented to illustrate the dynamical behaviour of the system.
Paper Structure (17 sections, 5 theorems, 35 equations, 5 figures, 1 table)

This paper contains 17 sections, 5 theorems, 35 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Model
  3. Basic results: boundedness, existence & stability equilibria, and subsystems
  4. Boundedness
  5. Existence of interior equilibrium point
  6. Analysis of subsystem-I
  7. Analysis of subsystem-II
  8. Analysis of Filippov system \ref{['sys']}
  9. Preliminaries
  10. Sliding segment & sliding mode dynamics
  11. Equilibria and some special points
  12. Stability of the equilibrium points
  13. Numerical analysis
  14. Bifurcation set of equilibria
  15. Impact of prey refuge
  16. ...and 2 more sections

Key Result

Lemma 3.1

Let $\eta(t)$ be an absolute continuous function with real numbers $k_1$ and $k_2$ with $k_1 \neq 0$ s.t. $d\eta/dt+k_1 \eta(t) \le k_2,~ \forall t\ge 0$, then for all $t\ge \tilde{T} \ge 0$,

Figures (5)

  • Figure 1: Bifurcation diagram of system \ref{['sys']} with respect to the predation rate $p$ and the threshold $S$ showing existence of different type of equilibria with their stability properties like asymptotically stable (AS) or unstable (US): (a) For parameters in set A$_1$ with varying $S$ & $p$(b) For parameters in set A$_2$ with varying $S$ & $p$.
  • Figure 2: Dynamics of \ref{['sys']} for parameters in set A$_2$ with $S=4$(a) Time series starting from (0.2,4) reaches $E_R^1$(b) Basins of attraction of the two regular equilibria (c) Time series starting from (0.8,5) reaches $E_R^2$.
  • Figure 3: Bifurcation diagrams in $Sp$-plane for different $m$-values with other parameters from set A$_1$
  • Figure 4: System \ref{['sys']} undergoes boundary focus bifurcation at $S=0.1416$ when $r_1=0.9$, $k_1=2$, $p=0.6$, $m=0.2$, $b=0.4$, $q_1=0.2$, $E=1$, $r_2=0.8$, $k_2=1.5$, $e=0.6$ & $q_2=0.1$
  • Figure 5: System \ref{['sys']} undergoes boundary node bifurcation at $S=0.4005$ when $r_1=0.9$, $k_1=2$, $p=0.6$, $m=0.2$, $b=0.4$, $q_1=0.2$, $E=1$, $r_2=0.8$, $k_2=1.5$, $e=0.6$ & $q_2=0.1$

Theorems & Definitions (13)

  • Lemma 3.1: aziz2002
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • ...and 3 more