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Dynamical Analysis of a Predator-Prey Model with Additive Allee Effect and Prey Group Defense

Resmawan Resmawan, Agus Suryanto, Isnani Darti, Hasan S Panigoro

Abstract

In this article, we develop a predator-prey model with Allee effect and prey group defense. The model has three equilibrium points i.e. the trivial point, the predator extinction point, and the coexistence point. All equilibrium points are locally asymptotically stable under certain conditions. The Allee effect in this model influences the stability of the equilibrium point. If the Allee effect is weak, then the trivial equilibrium point is unstable. Meanwhile, if the Allee effect is strong, then the trivial equilibrium point is locally asymptotically stable. Those mean that a strong Allee effect can lead to the extinction of both populations. Moreover, under weak Allee condition, forward bifurcation and Hopf bifurcation occur at the predator extinction equilibrium point. Meanwhile, a strong Allee effect may induce bistability at both the trivial equilibrium point and the predator extinction equilibrium point. Those mean that prey can survive without the presence of predators, but a strong Allee effect can lead to prey extinction if the population size is very small. To support our analytical findings, we perform some numerical simulations in the final section.

Dynamical Analysis of a Predator-Prey Model with Additive Allee Effect and Prey Group Defense

Abstract

In this article, we develop a predator-prey model with Allee effect and prey group defense. The model has three equilibrium points i.e. the trivial point, the predator extinction point, and the coexistence point. All equilibrium points are locally asymptotically stable under certain conditions. The Allee effect in this model influences the stability of the equilibrium point. If the Allee effect is weak, then the trivial equilibrium point is unstable. Meanwhile, if the Allee effect is strong, then the trivial equilibrium point is locally asymptotically stable. Those mean that a strong Allee effect can lead to the extinction of both populations. Moreover, under weak Allee condition, forward bifurcation and Hopf bifurcation occur at the predator extinction equilibrium point. Meanwhile, a strong Allee effect may induce bistability at both the trivial equilibrium point and the predator extinction equilibrium point. Those mean that prey can survive without the presence of predators, but a strong Allee effect can lead to prey extinction if the population size is very small. To support our analytical findings, we perform some numerical simulations in the final section.

Paper Structure

This paper contains 9 sections, 7 theorems, 14 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. MODEL
  3. Existence and Local Stability of Equilibrium Points
  4. The Existence of Equilibrium Points
  5. Local Stability
  6. Numerical Simulation
  7. The Impact of Predation Conversion Rate
  8. The Impact of Environmental Protection Rate
  9. Conclusion

Key Result

Theorem 1

If the system GrindEQ__2_ has a weak Allee effect $(h<w)$, then the equilibrium point $E_1=\left(N_1,0\right)$ exists and is unique.

Figures (6)

  • Figure 1: Bifurcation diagram of the system \ref{['GrindEQ__2_']} with a weak Allee effect $(h=0.2)$ and parameter values as in Table \ref{['tab1']}: (a) $N$ state and (b) $P$ state
  • Figure 2: Phase portraits of the system \ref{['GrindEQ__2_']} with a weak Allee effect $(h=0.2)$ and parameter values as in Table \ref{['tab1']}: (a)$c=0.1$, (b) $c=0.3$, (c) $c=0.4$
  • Figure 3: Phase portraits of the system \ref{['GrindEQ__2_']} with a strong Allee effect $(h=0.4)$ and parameter values as in Table \ref{['tab1']}: (a)$\ c=0.1$, (b) $c=0.3$, (c) $c=0.4$
  • Figure 4: Bifurcation diagram of the system \ref{['GrindEQ__2_']} with a weak Allee effect $(h=2)$ and parameter values as in Table \ref{['tab2']}: (a) $N$ state and (b) $P$ state
  • Figure 5: Phase portraits of the system \ref{['GrindEQ__2_']} with a weak Allee effect $(h=0.2)$ and parameter values as in Table \ref{['tab2']}: (a)$\ b=0.3$, (b) $b=0.7$, (c) $b=1.1$
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7