Dynamical behaviour of a two-predator model with prey refuge

Abstract

A three-component model consisting on one-prey and two-predator populations is considered with a Holling type II response function incorporating a constant proportion of prey refuge. We also consider the competition among predators for their food (prey) and shelter. The essential mathematical features of the model have been analyzed thoroughly in terms of stability and bifurcations arising in some selected situations. Threshold values for some parameters indicating the feasibility and stability conditions of some equilibria are determined. The range of significant parameters under which the system admits different types of bifurcations is investigated. Numerical illustrations are performed in order to validate the applicability of the model under consideration.

Appendices

Appendix A

The definition of the second additive compound matrix can be found in the paper of Li and Muldowney [33]. Let A = (a _ij ) be an n ×n matrix. The second additive compound A ^[2] is the $\left( {_2^n } \right)\times \left( {_2^n } \right)$ matrix defined as follows:

For any integer $i = 1, {\ldots}, \left( {_2^n } \right)$, let (i) = (i ₁, i ₂) be the ith member in the lexicographic ordering of integer pairs (i ₁, i ₂), such that, 1 ≤ i ₁ < i ₂ ≤ n.

Then the element in the ith row and jth column of A ^[2] is

• $a_{i_{1}}{_{i_{1}}} +a_{i_{2}}{_{i_{2}}},$   if (i) = (j )

• $\left( {-1} \right)^{r+s}a_{i_{r}j_{\,s}},$   if exactly one entry i _r of (i) doesn’t occur in (j ) and j _s doesn’t occur in (j )

• 0,   if neither entry from (i) occurs in (j )

For n = 3

$$ \label{eq16} A=\left( {\alpha_{ij} } \right)_{3\times 3}=\left[ {{\begin{array}{*{20}c} {\alpha_{11} } & {\alpha_{12} } & {\alpha_{13} } \\ {\alpha_{21} } & {\alpha_{22} } & {\alpha_{23} } \\ {\alpha_{31} } & {\alpha_{32} } & {\alpha_{33} } \\ \end{array} }} \right], $$

(19)

its second additive compound matrix is

$$ \label{eq17} A_{\,3\times 3}^{\,\left[ 2 \right]} =\left[ {{\begin{array}{*{20}c} {\alpha_{11} +\alpha_{22} } \hfill & {\alpha_{23} } \hfill & {-\alpha_{13} } \hfill \\ {\alpha_{32} } \hfill & {\alpha_{11} +\alpha_{33} } \hfill & {\alpha_{12} } \hfill \\ {-\alpha_{31} } \hfill & {\alpha_{21} } \hfill & {\alpha_{22} +\alpha_{23} } \hfill \\ \end{array} }} \right]. $$

(20)

In this case, (1) = (1, 2), (2) = (1, 3), (3) = (2, 3).

Appendix B

Theorem 11

Bendixson’s criterion in R ⁿ (cf. Arino et al. [40]): A simple closed rectifiable curve that is invariant with respect to (11) cannot exist if any one of the following conditions is satisfied in R ⁿ :

• (i) ${\sup \left\{ {\dfrac{\partial F_r }{\partial x_r }+\dfrac{\partial F_s }{\partial x_s }+\displaystyle\sum\limits_{j\,\ne\, r,s} {\left( {\left| {\dfrac{\partial F_j }{\partial x_r }} \right|+\left| {\dfrac{\partial F_j }{\partial x_s }} \right|} \right):1\le r<s\le n} } \right\}<0,}$

• (ii) ${\sup \left\{ {\dfrac{\partial F_r }{\partial x_r }+\dfrac{\partial F_s }{\partial x_s }+\displaystyle\sum\limits_{j\,\ne\, r,s} {\left( {\left| {\dfrac{\partial F_r }{\partial x_j }} \right|+\left| {\dfrac{\partial F_s }{\partial x_j }} \right|} \right):1\le r<s\le n} } \right\}<0,}$

• (iii) λ ₁ + λ ₂ < 0,

• (iv) ${\inf \left\{ {\dfrac{\partial F_r }{\partial x_r }+\dfrac{\partial F_s }{\partial x_s }+\displaystyle\sum\limits_{j\,\ne\, r,s} {\left( {\left| {\dfrac{\partial F_j }{\partial x_r }} \right|+\left| {\dfrac{\partial F_j }{\partial x_s }} \right|} \right):1\le r<s\le n} } \right\}<0,}$

• (v) ${\inf \left\{ {\dfrac{\partial F_r }{\partial x_r }+\dfrac{\partial F_s }{\partial x_s }+\displaystyle\sum\limits_{j\,\ne\, r,s} {\left( {\left| {\dfrac{\partial F_r }{\partial x_j }} \right|+\left| {\dfrac{\partial F_s }{\partial x_j }} \right|} \right):1\le r<s\le n} } \right\}<0,}$

• (vi) λ _n − 1 + λ _n  < 0,

where λ ₁ ≥ λ ₂ ≥ ⋯ ≥ λ _n are eigenvalues of $\frac{1}{2}\left( {\left( {{\partial F} \mathord{\left/ {\vphantom {{\partial F} {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)^\ast +\left( {{\partial F} \mathord{\left/ {\vphantom {{\partial F} {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)} \right)$ . ${{\partial F} \mathord{\left/ {\vphantom {{\partial F} {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}$ is the Jacobian matrix of F and the asterisk denotes the transposition.