Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect

Abstract

The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.

Appendices

Appendix

Positivity and Boundedness of the Model Solutions

From the model system (1), we have

$$\begin{aligned} x(t)=\left\{ \begin{array}{ll} x(0) exp \left[ \displaystyle \int _{0}^{t}\left( r \left( 1- \frac{x(s)}{K}\right) (x(s)-\theta )\right) ds\right] &{} \text{ if } \ \ \ \ \ 0 \le x \le m \\ x(0) exp \left[ \displaystyle \int _{0}^{t}\left( r \left( 1- \frac{x(s)}{K}\right) (x(s)-\theta ) - \frac{a (x(s)-m)y(s)}{(x(s)-m+y(s))x(s)}\right) ds\right] &{} \text{ if } \ \ \ \ \ x > m, \end{array} \right. \nonumber \\ \end{aligned}$$

(25)

and

$$\begin{aligned} y(t)=\left\{ \begin{array}{ll} y(0) exp(-dt) &{} \text{ if } \ \ \ \ \ 0 \le x \le m \\ y(0) exp \left[ \displaystyle \int _{0}^{t}\left( \frac{e a (x(s)-m)}{x(s)-m+y(s)}-d \right) ds \right] &{} \text{ if } \ \ \ \ \ x > m. \end{array} \right. \end{aligned}$$

(26)

From the above, we can see that $x(t) \ge 0$ and $y(t) \ge 0$ whenever $x(0) > 0$ and $y(0) >0$. Thus, all the solutions starting in the interior of the positive quadrant remain in it for all $t \ge 0$.

Now, we show the boundedness of the model solutions. For this, consider Eq. (25), which can be written as

$$\begin{aligned} x(t)= & {} x(0) exp \left[ \int _{0}^{t}F (x(s), y(s))ds\right] , \end{aligned}$$

where,

$$\begin{aligned} F(x(s), y(s))=\left\{ \begin{array}{ll} r \left( 1- \frac{x(s)}{K}\right) (x(s)-\theta ) &{} \text{ if } \ \ \ \ \ 0 \le x \le m \\ r \left( 1- \frac{x(s)}{K}\right) (x(s)-\theta ) - \frac{a (x(s)-m)y(s)}{(x(s)-m+y(s))x(s)} &{} \text{ if } \ \ \ \ \ x > m. \end{array} \right. \end{aligned}$$

Now, two cases arise:

Case I: $x(0)\in (0,K)$

In this case, we claim that $x(t) \le K$ for all $t \ge 0$. Otherwise, there exists two positive real numbers $t_1$ and $t_2$ such that $x(t_1)=K$ and $x(t)> K$ for all $t \in (t_1, t_2)$. Then, for all $t \in (t_1, t_2)$,

$$\begin{aligned} x(t)= & {} x(0) exp \left[ \int _{0}^{t}F (x(s), y(s))ds\right] \nonumber \\= & {} x(0) exp \left[ \int _{0}^{t_1}F (x(s), y(s))ds\right] exp \left[ \int _{t_1}^{t}F (x(s), y(s))ds\right] \nonumber \\= & {} x(t_1) exp \left[ \int _{t_1}^{t}F (x(s), y(s))ds\right] < x(t_1)=K, \end{aligned}$$

(27)

as $F (x(t), y(t))<0$ for all $t \in (t_1, t_2)$. This contradicts our hypothesis. Hence $x(t) \le K$ for all future time.

Case II: When $x(0)>K$.

In this case, as long as $x(t)\ge K$

$$\begin{aligned} x(t)= & {} x(0) exp \left[ \int _{0}^{t}F (x(s), y(s))ds\right] \le x(0), \end{aligned}$$

as $F (x(t), y(t))\le 0$ for $x(t)\ge K$.

Combining both the above cases, it can be concluded that for any positive solution

$$\begin{aligned} x(t) \le max\{x(0),K\}. \end{aligned}$$

Now, from model system (1), we have

$$\begin{aligned} \frac{d x(t)}{dt}+ \frac{1}{e}\frac{d y(t)}{dt}\le & {} r x \left( 1- \frac{x}{K}\right) (x-\theta ) -\frac{d y}{e}. \end{aligned}$$

Let $\xi = max_{t\ge 0}r x \left( 1- \frac{x}{K}\right) (x-\theta ) + d x$, then

$$\begin{aligned} \frac{d x(t)}{dt}+ \frac{1}{e}\frac{d y(t)}{dt}\le \xi - d \left( x+ \frac{y}{e}\right) . \end{aligned}$$

(28)

Using Gronwall’s inequality, we have

$$\begin{aligned} x(t)+ \frac{y(t)}{e} \le \left( x(0)+ \frac{y(0)}{e} \right) e^{-d t}+\frac{\xi }{d}(1-e^{-d t }). \end{aligned}$$

(29)

For large enough t, we can write

$$\begin{aligned} x(t)+ \frac{y(t)}{e} \le \frac{\xi }{d}+\epsilon , \end{aligned}$$

(30)

for arbitrary $\epsilon >0$. Since x(t) is bounded, (30) implies that y(t) is also bounded.

Stability of Bifurcating Periodic Solutions

Applying the transformation

$$\begin{aligned} z_1= x-x_2^*, \ \ z_2=y-y_2^*, \end{aligned}$$

(31)

the model system (3) reduces to the following system

$$\begin{aligned} \dot{z_1}= & {} a_{10} z_1 +a_{01} z_2 + a_{20} z_1^2 + a_{11} z_1 z_2 +a_{02}z_2^2+a_{30} z_1^3\nonumber \\&+ a_{21} z_1^2 z_2 + a_{12}z_1 z_2^2+a_{03}z_2^3+0(\mid z\mid ^4),\nonumber \\ \dot{z_2}= & {} b_{10} z_1 + b_{01} z_2+b_{20} z_1^2 + b_{11} z_1 z_2 +b_{02}z_2^2+ b_{30} z_1^3\nonumber \\&+ b_{21} z_1^2 z_2+ b_{12}z_1 z_2^2+b_{03}z_2^3+0(\mid z\mid ^4), \end{aligned}$$

(32)

where

$$\begin{aligned}&a_{10}=\frac{d}{e}f'(x_2^*)-\frac{a y_2^*}{x_2^*-m+y_2^*}+\frac{ a (x_2^*-m)y_2^*}{(x_2^*-m+y_2^*)^2}, a_{01}=-\frac{ a (x_2^*-m)^2}{(x_2^*-m+y_2^*)^2},\\&a_{20}=\frac{r}{K}(K+\theta -3 x_2^*)+\frac{a {y_2^*}^2}{(x_2^*-m+y_2^*)^3},\\&a_{11}=\frac{-2 a (x_2^*-m)y_2^*}{(x_2^*-m+y_2^*)^3}, \ a_{02}=\frac{a (x_2^*-m)^2}{(x_2^*-m+y_2^*)^3}, \ a_{30}= -\frac{r}{K}-\frac{a {y_2^*}^2}{(x_2^*-m+y_2^*)^4},\\&a_{21}=-\frac{a y_2^*}{(x_2^*-m+y_2^*)^3}+\frac{3 a (x_2^*-m)y_2^*}{(x_2^*-m+y_2^*)^4},\\&\quad a_{12}=-\frac{a (x_2^*-m)}{(x_2^*-m+y_2^*)^3}+\frac{3 a (x_2^*-m)y_2^*}{(x_2^*-m+y_2^*)^4},\\&a_{03}=-\frac{a (x_2^*-m)^2}{(x_2^*-m+y_2^*)^4}, b_{10}=\frac{e a {y_2^*}^2}{(x_2^*-m+y_2^*)^2}, b_{01}=\frac{-e a (x_2^*-m)y_2^*}{(x_2^*-m+y_2^*)^2},\\&b_{20}=-\frac{e a {y_2^*}^2}{(x_2^*-m+y_2^*)^3}, b_{11}=\frac{2 e a (x_2^*-m)y_2^*}{(x_2^*-m+y_2^*)^3}, \ b_{02}=-\frac{e a (x_2^*-m)^2}{(x_2^*-m+y_2^*)^3}, \\&b_{30}= \frac{e a {y_2^*}^2}{(x_2^*-m+y_2^*)^4},\ b_{21}=\frac{e a y_2^*}{(x_2^*-m+y_2^*)^3}-\frac{3 e a (x_2^*-m)y_2^*}{(x_2^*-m+y_2^*)^4},\\&b_{12}=\frac{e a (x_2^*-m)}{(x_2^*-m+y_2^*)^3}-\frac{3 e a (x_2^*-m)y_2^*}{(x_2^*-m+y_2^*)^4},\ b_{03}=\frac{e a (x_2^*-m)^2}{(x_2^*-m+y_2^*)^4}. \end{aligned}$$

The system (32) can be written as

$$\begin{aligned} \dot{Z}= J_{E_2^*} Z+ Q (Z), \end{aligned}$$

(33)

where,

$$\begin{aligned} Z= & {} \left( \begin{array}{c} z_1\\ z_2 \end{array} \right) \ and \ Q= \left( \begin{array}{c} Q^1\\ Q^2 \end{array} \right) \\= & {} \left( \begin{array}{c} a_{20} z_1^2 + a_{11} z_1 z_2 +a_{02}z_2^2+a_{30} z_1^3+ a_{21} z_1^2 z_2+a_{12}z_1 z_2^2+a_{03}z_2^3 \\ b_{20} z_1^2 + b_{11} z_1 z_2 +b_{02}z_2^2+ b_{30} z_1^3+ b_{21} z_1^2 z_2+b_{12}z_1 z_2^2+b_{03}z_2^3 \end{array} \right) .\end{aligned}$$

The eigenvector v of Jacobian matrix $J_{E_2^*}$ corresponding to the eigenvalue $i \omega _0$ at $a=a_c$ is found to be $v=( a_{01}, i \omega _0-a_{10})^T$. Now define

$$\begin{aligned} A= & {} (Re(v), -\,Im(v))\\= & {} \left( \begin{array}{cc} a_{01}\ \ \ &{}0\\ -\,a_{10}\ \ \ &{} -\,\omega _0 \end{array}\right) . \end{aligned}$$

Let $Z=A Y$ or $Y=A^{-1} Z$, where $Y= (y_1, y_2)^T$. Under this linear transformation, system (33) becomes

$$\begin{aligned} \dot{Y}= (A^{-1} J_{E_2^*} A) Y + A^{-1} Q(A Y), \end{aligned}$$

(34)

This can be written as,

$$\begin{aligned} \left( \begin{array}{c} \dot{y_1}\\ \dot{y_2} \end{array} \right) =\left( \begin{array}{cc} 0 &{} -\omega _0\\ \omega _0 &{} 0 \end{array} \right) \left( \begin{array}{c} y_1\\ y_2 \end{array} \right) + \left( \begin{array}{c} F^1(y_1,y_2; a_c)\\ F^2(y_1,y_2;a_c) \end{array} \right) \end{aligned}$$

(35)

where

$$\begin{aligned}&F^1(y_1,y_2; a_c)= \frac{1}{a_{01}}Q^1,\\&\quad F^2(y_1,y_2; a_c)= -\frac{1}{\omega _0 a_{01}}(a_{10} Q^1+a_{01} Q^2).\\ Q^1&= (a_{20} {a_{01}}^2-a_{11}a_{01} a_{10}+ a_{02}{a_{10}}^2)y_1^2+(2 a_{02} a_{10}-a_{11}a_{01})\omega _0 y_1 y_2\\&\quad +\, a_{02}\omega _0^2 y_2^2+(a_{30}{a_{01}}^3+a_{12} a_{01} {a_{10}}^2-a_{21}{a_{01}}^2 a_{10}-a_{03} {a_{10}}^3)y_1^3-a_{03}\omega _0^3 y_2^3\\&\quad +\, (2 a_{12}a_{10}a_{01}-a_{21}{a_{01}}^2-3 a_{03}{a_{10}}^2)\omega _0y_1^2y_2+(a_{12}a_{01}-3a_{03}a_{10})\omega _0^2 y_1 y_2^2,\\ Q^2&= (b_{20} {a_{01}}^2-b_{11}a_{01} a_{10}+ b_{02}{a_{10}}^2)y_1^2+(2 b_{02} a_{10}-b_{11}a_{01})\omega _0 y_1 y_2\\&\quad +\, b_{02}\omega _0^2 y_2^2+(b_{30}{a_{01}}^3+b_{12} a_{01} {a_{10}}^2-b_{21}{a_{01}}^2 a_{10}-b_{03} {a_{10}}^3)y_1^3-b_{03}\omega _0^3 y_2^3\\&\quad +\, (2 b_{12}a_{10}a_{01}-b_{21}{a_{01}}^2-3 b_{03}{a_{10}}^2)\omega _0y_1^2y_2+(b_{12}a_{01}-3b_{03}a_{10})\omega _0^2 y_1 y_2^2. \end{aligned}$$

In order to determine the stability and direction of the periodic solution, we need to calculate the following quantity called Lyapunov coefficient (Wiggins 1990).

$$\begin{aligned} l_1= & {} \frac{1}{16}\left( F^1_{111}+ F^2_{112}+F^1_{122}+ F^2_{222}\right) +\frac{1}{16 \omega _0}\left[ F^1_{12}\left( F^1_{11}+F^1_{22}\right) \right. \\&\left. -F^2_{12}\left( F^2_{11}+F^2_{22}\right) -F^1_{11}F^2_{11}+F^1_{22}F^2_{22}\right] . \end{aligned}$$

where $\displaystyle F^k_{ij}=\left[ \frac{\partial F^k}{\partial y_i \partial y_j}\right] _{(0,0;a_c)}$ and $\displaystyle F^k_{ijl}=\left[ \frac{\partial F^k}{\partial y_i \partial y_j \partial y_l}\right] _{(0,0; a_c)}$.

The bifurcating periodic solutions are orbitally stable or unstable according as $l_1<0$ or $l_1>0$.

Conditions for the Non-degeneracy of BT Bifurcation

In the following, we will show that the conditions for non-degeneracy of BT bifurcation are satisfied using the algorithm given in (Kuznetsov 1998). Consider the system

$$\begin{aligned} \frac{dx}{dt}= & {} r x \left( 1- \frac{x}{K}\right) (x-\theta ) - \frac{(a_{\mathrm{SN}}+\nu _1) (x-m)y}{x-m+y}= f_1(x,y;\nu ), \nonumber \\ \frac{dy}{dt}= & {} \frac{(1+\nu _2)(a_{\mathrm{SN}}+\nu _1) (x-m)y}{x-m+y}- d y=f_2(x,y;\nu ), \end{aligned}$$

(36)

where $\nu _1$ and $\nu _2$ are small. When $\nu _1=0$ and $\nu _2=0$, system (36) has one positive equilibrium $(x^*, y^*)$, which is a cusp of codimension 2.

Let $u_1 = x - x^*$ and $u_2 = y - y^*$. Then, system (36) becomes

$$\begin{aligned} \dot{u_1}= & {} \left( c_{10}-\frac{\nu _1 {y^*}^2}{(x^*-m+y^*)^2}\right) u_1 + \left( c_{01}-\frac{\nu _1 (x^*-m)^2}{(x^*-m+y^*)^2}\right) u_2\nonumber \\&+ c_{20} u_1^2 + c_{11} u_1 u_2 +c_{02}u_2^2+0(\mid u\mid ^3),\nonumber \\ \dot{u_2}= & {} \left( d_{10}+\frac{(\nu _2 a_{\mathrm{SN}}+\nu _1+\nu _1 \nu _2) {y^*}^2}{(x^*-m+y^*)^2}\right) u_1\nonumber \\&+ \left( d_{01}+\frac{(\nu _2 a_{\mathrm{SN}}+\nu _1+\nu _1 \nu _2) (x^*-m)^2}{(x^*-m+y^*)^2}\right) u_2 \nonumber \\&+ d_{20} u_1^2 + d_{11} u_1 u_2 +d_{02}u_2^2+0(\mid u\mid ^3), \end{aligned}$$

(37)

where

$$\begin{aligned}&c_{10}=\frac{ a_{\mathrm{SN}} (x^*-m)y^*}{(x^*-m+y^*)^2}, c_{01}=-\frac{ a_{\mathrm{SN}} (x^*-m)^2}{(x^*-m+y^*)^2},\ c_{20}=\frac{r}{K}(K+\theta -3 x^*)\\&\qquad +\frac{(a_{\mathrm{SN}}+\nu _1) {y^*}^2}{(x^*-m+y^*)^3},\\&c_{11}=\frac{-2 (a_{\mathrm{SN}}+\nu _1) (x^*-m)y^*}{(x_2^*-m+y^*)^3}, \ c_{02}=\frac{(a_{\mathrm{SN}}+\nu _1) (x^*-m)^2}{(x^*-m+y^*)^3},\\&d_{10}=\frac{a_{\mathrm{SN}} {y^*}^2}{(x^*\!-\!m+y^*)^2}, d_{01}\!=\!\frac{- a_{\mathrm{SN}} (x^*\!-\!m)y^*}{(x^*-m+y^*)^2},\ d_{20}=-\frac{(1+\nu _2) (a_{\mathrm{SN}}+\nu _1) {y_2^*}^2}{(x_2^*-m+y_2^*)^3},\\&d_{11}=\frac{2 (1+\nu _2) (a_{\mathrm{SN}}+\nu _1) (x^*-m)y^*}{(x^*-m+y^*)^3}, \ d_{02}=-\frac{(1+\nu _2) (a_{\mathrm{SN}}+\nu _1) (x^*-m)^2}{(x^*-m+y^*)^3}. \end{aligned}$$

Now, using the affine transformation $w_1=u_1$ and $w_2=c_{10}u_1+c_{01} u_2$, the system (37) reduces to

$$\begin{aligned} \dot{w_1}= & {} w_2+\xi _{00}(\nu )+\xi _{10}(\nu ) w_1+\xi _{01}(\nu ) w_2+\frac{1}{2} \xi _{20}(\nu ) w_1^2 +\xi _{11}(\nu ) w_1 w_2\nonumber \\&+ \frac{1}{2}\xi _{02}(\nu ) w_2^2 + 0(\mid w\mid ^3),\nonumber \\ \dot{w_2}= & {} \eta _{00}(\nu )+\eta _{10}(\nu ) w_1+\eta _{01}(\nu ) w_2+\frac{1}{2} \eta _{20}(\nu ) w_1^2 +\eta _{11}(\nu ) w_1 w_2\nonumber \\&+ \frac{1}{2}\eta _{02}(\nu ) w_2^2 + 0(\mid w\mid ^3), \end{aligned}$$

(38)

where

$$\begin{aligned}&\xi _{00}(\nu )= f_1(x^*,y^*,\nu ), \ \xi _{10}(\nu )=\nu _1\frac{ c_{10} (x^*-m)^2-c_{01} {y^*}^2}{(x^*-m+y^*)^2 c_{01}},\\&\quad \xi _{01}(\nu )=-\frac{\nu _1 (x^*-m)^2}{(x^*-m+y^*)^2c_{01}}\\&\xi _{20}(\nu )= 2\left( c_{20}-\frac{c_{11} c_{10}}{c_{01}}+\frac{c_{02} c_{10}^2 }{c_{01}^2}\right) , \ \xi _{11}(\nu )= \left( \frac{c_{11}}{c_{01}}-\frac{2 c_{10} c_{02} }{c_{01}^2}\right) , \\&\quad \xi _{02}(\nu )= \frac{2 c_{02}}{c_{01}^2},\\&\eta _{00}(\nu )= c_{10} f_1(x^*,y^*,\nu )+ c_{01} f_2(x^*,y^*,\nu ), \\&\eta _{10}(\nu )= \frac{(c_{10}\nu _1-c_{01} (\nu _2 a_{\mathrm{SN}}+\nu _1+\nu _1 \nu _2))(c_{10}(x^*-m)^2-c_{01} {y^*}^2)}{(x^*-m+y^*)^2 c_{01}},\\&\eta _{01}(\nu )= -\frac{(c_{10}\nu _1-c_{01} (\nu _2 a_{\mathrm{SN}}+\nu _1+\nu _1 \nu _2))(x^*-m)^2}{(x^*-m+y^*)^2 c_{01}},\\&\eta _{20}(\nu )= 2 c_{10}\left( c_{20}-\frac{c_{11} c_{10}}{c_{01}}+\frac{c_{02} c_{10}^2 }{c_{01}^2}\right) + 2 c_{01}\left( d_{20}-\frac{d_{11} c_{10}}{c_{01}}+\frac{d_{02} c_{10}^2 }{c_{01}^2}\right) ,\\&\eta _{11}(\nu )= c_{10}\left( \frac{c_{11}}{c_{01}}-\frac{2 c_{10} c_{02} }{c_{01}^2}\right) +c_{01}\left( \frac{d_{11}}{c_{01}}-\frac{2 c_{10} d_{02} }{c_{01}^2}\right) ,\\&\eta _{02}(\nu )= \frac{2 c_{10} c_{02}}{c_{01}^2}+\frac{2 c_{01} d_{02}}{c_{01}^2}. \end{aligned}$$

The degeneracy conditions of BT bifurcation are

$$\begin{aligned} \left( \begin{array}{cc} c_{10}\ \ \ &{}c_{01}\\ d_{10}\ \ \ &{} d_{01} \end{array}\right) \ne \Theta _{2x2}, \ \ \ \xi _{20}(0)+\eta _{11}(0) \ne 0, \ \ \ \eta _{20}(0) \ne 0. \end{aligned}$$

We find that

$$\begin{aligned} \xi _{20}(0)+\eta _{11}(0)= \frac{2 r}{K}(K+\theta -3 x^*) \ne 0 \end{aligned}$$

and

$$\begin{aligned} \eta _{20}(0) = \frac{2 a_{\mathrm{SN}} (x^*-m) y^* r}{(x^*-m+y^*)^2 K}(K+\theta -3 x^*) \ne 0. \end{aligned}$$

Thus, the degeneracy conditions of BT bifurcation are satisfied. The bifurcation structure of the BT point is given by the following quantity

$$\begin{aligned} \sigma= & {} sign(\eta _{20}(0)(\xi _{20}(0)+\eta _{11}(0)))\\= & {} sign\left( \frac{4 a_{\mathrm{SN}} (x^*-m) y^* r^2}{(x^*-m+y^*)^2 K^2}(K+\theta -3 x^*)^2 \right) =1. \end{aligned}$$