Diffusive patterns in a predator–prey system with fear and hunting cooperation

Abstract

Cooperation among species is a ubiquitous behavior to better understand the system dynamics from an ecological perspective. Hunting cooperation among predators can impose fear effects on the prey population, thereby decreasing the prey’s birth rate. Considering this fact, we propose a model that incorporates hunting cooperation among predators and the fear-induced birth reduction in the prey population. We have done the complete dynamical analysis, including boundedness of solutions, persistence of the system, existence of all equilibria and their local and global stability, existence of Hopf bifurcation and its direction and stability, and existence of saddle-node bifurcation. We analyze Hopf bifurcation with respect to the hunting cooperation parameter and saddle-node bifurcation by varying the predation rate. Moreover, we analyze the multi-stability of the system and observe that bi-stability occurs in two different scenarios. In the spatially extended system, we provide a detailed stability analysis and obtain the conditions for Turing instability. Various Turing patterns such as spots, holes, and stripes are obtained and discussed the biological significance of these patterns for the two-dimensional spatial model. We performed numerical simulations to validate our analytical results for both spatial and non-spatial models.

Appendix

1.1 Proof of Theorem 8

Proof

To find the stability and direction of Hopf bifurcation, we calculate the first Lyapunov coefficient. Let $u = x -x^*$ and $v = y-y^*$, then system (2) becomes

$$\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}t}= & {} \frac{r(u+x^*)}{1+k\alpha _1(v+y^*)} - r_0(u+x^*) - r_1(u+x^*)^2 - (\alpha _0+\alpha _1(v+y^*))(u+x^*)(v+y^*) := f(u,v).\\ \frac{\mathrm{d}v}{\mathrm{d}t}= & {} c(\alpha _0+\alpha _1(v+y^*))(u+x^*)(v+y^*)-[\frac{\gamma +\delta (v+y^*)}{1+(v+y^*)}](v+y^*) :=g(u,v). \end{aligned}$$

Now, expanding the above system in Taylor’s series at $(u,v)=(0,0)$ up to third order, we get

$$\begin{aligned} \begin{array}{ccc} \frac{\mathrm{d}u}{\mathrm{d}t} &{}=&{} J_{11}u + J_{12}v + f_1(u,v), \\ \frac{\mathrm{d}v}{\mathrm{d}t} &{}=&{} J_{21}u + J_{22}v + g_1(u,v), \end{array} \end{aligned}$$

(18)

$f_1(u,v)$ and $g_1(u,v)$ are the higher-order terms of u and v, given by

$$\begin{aligned} \begin{array}{ccc} f_1(u,v) &{}=&{} f_{uu}u^2 + f_{uv}uv + f_{vv}v^2 + f_{uuu}u^3 + f_{uuv}u^2v + f_{uvv}uv^2 + f_{vvv}v^3, \\ g_1(u,v) &{}=&{} g_{uu}u^2 + g_{uv}uv + g_{vv}v^2 + g_{uuu}u^3 + g_{uuv}u^2v + g_{uvv}uv^2 + g_{vvv}v^3, \end{array} \end{aligned}$$

where

$$\begin{aligned} f_u&= J_{11} = \frac{r}{1+k\alpha _1 y^*} - r_0 - 2r_1x^* - (\alpha _0+\alpha _1 y^*)y^*, \\ f_v&= J_{12} = -\frac{rk\alpha _1 x^*}{(1+k\alpha _1 y^*)^2} -x^*(\alpha _0+2\alpha _1y^*), \\ f_{uu}&= -2r_1, \\ f_{uv}&= [\frac{-rk\alpha _1}{(1+k\alpha _1y^*)^2}-(\alpha _0+2\alpha _1y^*)],\\ f_{vv}&= 2x^*\alpha _1(\frac{rk^2\alpha _1}{(1+k\alpha _1y^*)^3}-1), \\ f_{uuu}&= 0, \\ f_{uuv}&= 0, \\ f_{uvv}&= 2(\frac{rk^2\alpha _1^2}{(1+k\alpha _1y^*)^3}-\alpha _1),\\ f_{vvv}&= -\frac{6rk^3\alpha _1^3x^*}{(1+k\alpha _1 y^*)^4}, \end{aligned}$$

and

$$\begin{aligned} g_u&= J_{21} = c(\alpha _0+\alpha _1y^*)y^*, \\ g_v&= J_{22} = cx^*(2\alpha _1y^*+\alpha _0)-\frac{\gamma +\delta y^*}{1+y^*}-y^*\frac{\delta -\gamma }{(1+y^*)^2}, \\ g_{uu}&= 0, \\ g_{uv}&= c(\alpha _0+2\alpha _1y^*), \\ g_{vv}&= 2(c\alpha _1x^*-\frac{\delta -\gamma }{(1+y^*)^3}),\\ g_{uuu}&= 0,\\ g_{uuv}&= 0,\\ g_{uvv}&= 2c\alpha _1,\\ g_{vvv}&= \frac{6(\delta -\gamma )}{(1+y^*)^4}. \end{aligned}$$

System (18) can be written as

$$\begin{aligned} \begin{array}{ccc} {\dot{U}} = J_{E_1^*}U + H(U), \end{array} \end{aligned}$$

where

$$\begin{aligned} U= & {} \Big (u,v\Big )^T, \\ H= & {} \Big (f_1(u,v),g_1(u,v)\Big )^T, \\= & {} \Big (f_{uu}u^2 + f_{uv}uv + f_{vv}v^2 + f_{uvv}uv^2 + f_{vvv}v^3, g_{uv}uv + g_{vv}u^2 + g_{uvv}uv^2 + g_{vvv}v^3\Big )^T. \end{aligned}$$

Now, Hopf bifurcation occurs when $Tr(J_{E_1^*})=0$ and $det(J_{E_1^*})>0$, i.e., at the Hopf bifurcation point, the eigenvalue will be purely imaginary, which is given by $i\sqrt{f_ug_v-f_vg_u}$. Eigenvector corresponding to this eigenvalue $i\sqrt{f_ug_v-f_vg_u}$ is given by ${\overline{v}} = \Big (f_v, i\sqrt{f_ug_v-f_vg_u}-f_u\Big )^T$. Now, we define $S = \Big (Re({\overline{v}}), - Im({\overline{v}})\Big ) = \left[ \begin{array}{cc} f_v &{} 0 \\ -f_u &{} -\sqrt{f_ug_v-f_vg_u} \end{array} \right] .$ Now, let $U = SZ$ or $Z = S^{-1}U$, where $Z = \Big (z_1,z_2\Big )^T$. Therefore, under this transformation, the system is reduced to

$$\begin{aligned} {\dot{Z}} = \Big (S^{-1}J_{E_1^*}S\Big )Z + S^{-1}H\Big (SZ\Big ). \end{aligned}$$

This can be written as

$$\begin{aligned} \left[ \begin{array}{cc} \dot{z_1} \\ \dot{z_2} \end{array} \right] = \left[ \begin{array}{cc} 0 &{} -\sqrt{f_ug_v-f_vg_u} \\ \sqrt{f_ug_v-f_vg_u} &{} 0 \end{array} \right] \left[ \begin{array}{cc} z_1 \\ z_2 \end{array} \right] + \left[ \begin{array}{cc} H_1(z_1,z_2) \\ H_2(z_1,z_2) \end{array} \right] , \end{aligned}$$

where $H_1(z_1,z_2)$ and $H_2(z_1,z_2)$ are given by

$$\begin{aligned} H_1(z_1,z_2)&= \frac{1}{f_v}\Big [f_{uu}f_v^2 z_1^2 - f_{uv}f_vz_1(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)+f_{vv}(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^2 \\&\quad + f_{uvv}f_vz_1(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^2 - f_{vvv}(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^3\Big ] ,\\ H_2(z_1,z_2)&= -\frac{1}{f_v\sqrt{f_ug_v-f_vg_u}}\Big [f_u(f_{uu}f_v^2z_1^2-f_{uv}f_vz_1(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)+f_{vv}(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^2\\&\quad + f_{uvv}f_vz_1(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^2-f_{vvv}(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^3)\\&\quad +f_v(-g_{uv}f_vz_1(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)+g_{vv}(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^2\\&\quad +g_{uvv}f_vz_1(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^2 -g_{vvv}(f_uz_1+\sqrt{f_ug_v-f_vg_u}z_2)^3) \Big ]. \end{aligned}$$

The direction of Hopf bifurcation is determined by the sign of the first Lyapunov coefficient, which is given by

$$\begin{aligned} L:= & {} \frac{1}{16}\left[ \frac{\partial ^3 H_1}{\partial z_1^3} + \frac{\partial ^3 H_1}{\partial z_1\partial z_2^2} + \frac{\partial ^3 H_2}{\partial z_1^2\partial z_2} + \frac{\partial ^3 H_2}{\partial z_2^3}\right] \\&+ \frac{1}{16\sqrt{f_ug_v-f_vg_u}}\left[ \frac{\partial ^2 H_1}{\partial z_1\partial z_2}\left( \frac{\partial ^2 H_1}{\partial z_1^1} + \frac{\partial ^2 H_1}{\partial z_2^2}\right) - \frac{\partial ^2 H_2}{\partial z_1\partial z_2}\left( \frac{\partial ^2 H_2}{\partial z_1^2} + \frac{\partial ^2 H_2}{\partial z_2^2}\right) - \frac{\partial ^2 H_1}{\partial z_1^2}\frac{\partial ^2 H_2}{\partial z_1^2} + \frac{\partial ^2 H_1}{\partial z_2^2}\frac{\partial ^2 H_2}{\partial z_2^2}\right] . \end{aligned}$$

$\square$