A systematic study of autonomous and nonautonomous predator–prey models with combined effects of fear, migration and switching

Abstract

In this paper, we study the dynamics of a predator–prey system under the combined effects of fear, migration and switching phenomena. This study is new in the sense that the dynamics of such system was studied either through fear or migration, or switching, but the combined effects of such three factors are yet to be explored. We observe oscillatory behavior of the system in the absence of fear, migration and switching, whereas the system shows stable dynamics if anyone of these three factors is introduced. After analyzing the behavior of system with fear, migration and switching, we find that the system does not possess periodic solution whenever the predator population experiences intraspecies competition, but in the absence of intraspecies competition among predator population, fear of predators destabilizes the system, whereas on increasing the migration rates, the system first undergoes subcritical Hopf-bifurcation and then supercritical Hopf-bifurcation settling the system to stable coexistence. Existence of multiple limit cycles is also observed. Our results show that switching behavior of predator population supports the survival of prey and predator populations. We extend our model by assuming the fear parameters as time dependent. We find that the nonautonomous system exhibits periodic solutions, whereas the corresponding autonomous system shows stable focus. Moreover, we observe that if the autonomous system undergoes a Hopf-bifurcation through limit cycle oscillations, the corresponding nonautonomous system shows higher periodic solutions. Almost periodic behavior of the system is also observed by setting the fear parameters as almost periodic functions of time.

Appendices

Appendix A

We define a new variable $U=x+y+z$. For an arbitrary $\sigma >0$, by summing up the equations in system (1), we obtain

$$\begin{aligned}&\frac{\mathrm{d}U}{\mathrm{d}t}+\sigma U\le (r_1+\sigma )x\\&\quad +(r_2+\sigma )y-b_1x^2-b_2y^2-(d_1-\sigma )z. \end{aligned}$$

Since $0<p_{12},p_{21},\delta _1,\delta _2\le 1$, choosing $d_1\ge \sigma$, from the above equation, we get

$$\begin{aligned}&\frac{\mathrm{d}U}{\mathrm{d}t}+\sigma U\le (r_1+\sigma )x\\&\quad +(r_2+\sigma )y-b_1x^2-b_2y^2. \end{aligned}$$

Thus, we obtain the following upper bound,

$$\begin{aligned}&\frac{\mathrm{d}U}{\mathrm{d}t}+\sigma U\le \frac{(r_1+\sigma )^2}{4b_1}\\&\quad +\frac{(r_2+\sigma )^2}{4b_2}=H \ \text {(say)}. \end{aligned}$$

Applying standard results on differential inequalities [52], we have

$$\begin{aligned}&U(t)\le \frac{H}{\sigma }+e^{-\sigma t}\\&\quad \left( U(0)-\frac{H}{\sigma }\right) \\&\quad \le \max \left\{ \frac{H}{\sigma },U(0)\right\} =M \ \text {(say)}. \end{aligned}$$

Thus, there exists an $M>0$, depending only on the parameters of system (1), such that $0\le U(t)\le M$ for all sufficiently large values of t. Hence, the solutions of system (1) and consequently all the densities appearing in the system are ultimately bounded above [51].

Fig. 20

Extinction of a z population, b x and z populations and c y and z populations in system (24). Parameters are at the same values as in Table 1 except in a $r_1=0.07$, $\beta _1=0.06$, $r_2=0.06$, $\beta _2=0.049$, $\delta _1=0.01$, $d_1=0.38$, $d_2=0.1$, $a_{11}=0.05$, $a_{22}=0.25$, b $r_1=0.1$, $e_1=0.99$, $p_{21}=0.0002$, $r_2=0.08$, $e_2=0.0005$, $\beta _2=0.0049$, $\delta _2=0.02$, $d_2=0.01$, $a_{11}=0.05$, $a_{22}=0.25$ and c $r_1=0.1$, $e_1=0.00001$, $\beta _1=0.0006$, $r_2=0.08$, $e_2=0.3$, $p_{12}=0.00075$, $\delta _1=0.05$, $d_2=0.01$, $a_{11}=0.05$, $a_{22}=0.25$

Fig. 21

Time series of system (24). Parameters are at the same values as in Table 1 except $a_1=0.1$, $d_2=0$, $a_{11}=0.05$ and $a_{22}=0.25$

Fig. 22

Global stability of positive periodic solution for the system (24). Parameters are at the same values as in Table 1 except $a_1=0.1$, $d_2=0$, $a_{11}=0.05$ and $a_{22}=0.25$. Figure shows that solution trajectories starting from three different initial points, (0.4557, 0.3224, 1.8601), (0.4557, 0.3224, 1.9601) and (0.4557, 0.3224, 1.5601), ultimately converge to a unique positive periodic solution

Fig. 23

Time series of system (24). Parameters are at the same values as in Table 1 except $a_1=0.5$, $d_2=0$, $a_{11}=0.05$ and $a_{22}=0.25$

Fig. 24

System (24) exhibits almost periodic solution for the parameter values as in Table 1 except $r_1=0.9$, $a_1=0.1$, $r_2=0.9$, $\delta _1=0.8$, $\delta _2=0.5$, $d_2=0.5$, $a_{11}=0.05$ and $a_{22}=0.25$. Here, the forms of time-dependent parameters are chosen as $a_1(t)=a_1+a_{11}[\sin (\omega t)+\sin (\sqrt{5}\omega t)]$ and $a_2(t)=a_2+a_{22}[\sin (\omega t)+\sin (\sqrt{5}\omega t)]$

Since $\displaystyle \lim _{t\rightarrow \infty }\sup [x(t)+y(t)+z(t)]\le M,$ there exist $T_1,T_2,T_3>0$ such that $x(t)\le x_M$ $\forall$ $t\ge T_1$, $y(t)\le y_M$ $\forall$ $t\ge T_2$ and $z(t)\le z_M$ $\forall$ $t\ge T_3$, where $x_M$, $y_M$ and $z_M$ are finite positive constants with $x_M+y_M+z_M\le M$. Hence, for all $t\ge \max \{T_1,T_2,T_3\}=T$, $x(t)\le x_M$, $y(t)\le y_M$ and $z(t)\le z_M$. Let us define $M_1=\max \{x_M,y_M,z_M\}$.

Now, from the first equation of system (1), we have

$$\begin{aligned}&\frac{\mathrm{d}x}{\mathrm{d}t}\ge \frac{rx}{1+a_1z_M}\\&\quad -r_{01}x-b_1x^2-e_1x-\beta _1xz_M. \end{aligned}$$

Hence, it follows that for some $x_m$,

$$\begin{aligned}&\lim _{t\rightarrow \infty }\inf x(t)\\&\quad \ge \frac{\frac{r_1}{1+a_1z_M}-r_{01}-e_1-\beta _1z_M}{b_1}=x_m. \end{aligned}$$

Also, from the second equation of system (1), we have

$$\begin{aligned} \frac{\mathrm{d}y}{\mathrm{d}t}\ge \frac{r_2y}{1+a_2z_M}-r_{02}y-b_2y^2-e_2y-\beta _2yz_M. \end{aligned}$$

Thus, for some $y_m$, we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\inf y(t)\ge \frac{\frac{r_2}{1+a_2z_M}-r_{02}-e_2-\beta _2z_M}{b_2}=y_m. \end{aligned}$$

Now, from the last equation of system (1), we obtain

$$\begin{aligned}&\frac{\mathrm{d}z}{\mathrm{d}t}\ge \frac{\delta _1\beta _1x^2_mz}{x_M+y_M}\\&\quad +\frac{\delta _2\beta _2y^2_mz}{x_M+y_M}-d_1z-d_2z^2. \end{aligned}$$

Therefore, for some $z_m$, we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\inf z(t)\ge \frac{\frac{\delta _1\beta _1x^2_m+\delta _2\beta _2y^2_m}{x_M+y_M}-d_1}{d_2}=z_m. \end{aligned}$$

Let $M_2=\min \{x_m,y_m,z_m\}$. For $M_2$ to be positive, the conditions in (4) must hold. Hence, the theorem follows.

Appendix B

Jacobian of the system (1) is obtained as,

$$\begin{aligned} J=\left( \begin{array}{ccc} J_{11} &{}\quad J_{12} &{}\quad J_{13} \\ J_{21} &{}\quad J_{22} &{}\quad J_{23} \\ J_{31} &{}\quad J_{32} &{}\quad J_{33} \end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} J_{11}= & {} \frac{r_1}{1+a_1z}-r_{01}-2b_1x-e_1-\frac{\beta _1xz(x+2y)}{(x+y)^2}, \\ J_{12}= & {} e_2p_{21}+\frac{\beta _1x^2z}{(x+y)^2}, \\ J_{13}= & {} -\frac{r_1a_1x}{(1+a_1z)^2}-\frac{\beta _1x^2}{x+y},\\ J_{21}= & {} e_1p_{12}+\frac{\beta _2y^2z}{(x+y)^2}, \\ J_{22}= & {} \frac{r_2}{1+a_2z}-r_{02}-2b_2y-e_2-\frac{\beta _2yz(2x+y)}{(x+y)^2}, \\ J_{23}= & {} -\frac{r_2a_2y}{(1+a_2z)^2}-\frac{\beta _2y^2}{x+y},\\ J_{31}= & {} \frac{\delta _1\beta _1xz(x+2y)}{(x+y)^2}-\frac{\delta _2\beta _2y^2z}{(x+y)^2}, \\ J_{32}= & {} -\frac{\delta _1\beta _1x^2z}{(x+y)^2}+\frac{\delta _2\beta _2yz(2x+y)}{(x+y)^2},\\ J_{33}= & {} \frac{\delta _1\beta _1x^2}{x+y}+\frac{\delta _2\beta _2y^2}{x+y}-d_1-2d_2z. \end{aligned}$$

• 1. Evaluating the Jacobian at the equilibrium $E_1$, one eigenvalue is obtained as $-d_1$, while the remaining two are given by roots of the quadratic:

  $$\begin{aligned} \lambda ^2+{\overline{A}}_1\lambda +{\overline{A}}_2=0, \end{aligned}$$

  (57)

  where

  $$\begin{aligned} {\overline{A}}_1= & {} -\{r_1+r_2-(e_1+e_2+r_{01}+r_{02})\}, \\ {\overline{A}}_2= & {} \{r_1-(r_{01}+e_1)\}\{r_2-(r_{02}+e_2)\}\\&\quad -e_1e_2p_{12}p_{21}. \end{aligned}$$

  Clearly, under conditions in (16), both roots of equation (57) are either negative or have negative real parts.

• 2. At the equilibrium $E_2$, one root of Jacobian is found to be $\displaystyle \frac{\delta _1\beta _1\{r_1-(r_{01}+e_1)\}}{b_1}-d_1,$ and the others two can be obtained as roots of the quadratic:

  $$\begin{aligned} \lambda ^2+{\widetilde{A}}_1\lambda +{\widetilde{A}}_2=0, \end{aligned}$$

  (58)

  where

  $$\begin{aligned} {\widetilde{A}}_1= & {} r_1-(r_{01}+e_1)-\{r_2-(r_{02}+e_2)\}, \\ {\widetilde{A}}_2= & {} -[\{r_1-(r_{01}+e_1)\}\{r_2-(r_{02}+e_2)\}\\&\quad +e_1e_2p_{12}p_{21}]. \end{aligned}$$

  For the existence of equilibrium $E_2$, $r_1-(r_{01}+e_1)$ is positive, while the positive value of $r_2-(r_{02}+e_2)$ leads to existence of equilibrium $E_3$, and in that case ${\widetilde{A}}_2<0$. Hence, Eq. (58) has one positive and one negative roots, and thus the equilibrium $E_2$ is unstable. Moreover, if conditions in (17) hold, then both roots of Eq. (58) are either negative or have negative real parts, and hence then the equilibrium $E_2$ is stable.

• 3. At the equilibrium $E_3$, the Jacobian gives one eigenvalue as $\displaystyle \frac{\delta _2\beta _2\{r_2-(r_{02}+e_2)\}}{b_2}-d_1,$  while others two as roots of the quadratic:

  $$\begin{aligned} \lambda ^2+{\widehat{A}}_1\lambda +{\widehat{A}}_2=0, \end{aligned}$$

  (59)

  where

  $$\begin{aligned} {\widehat{A}}_1= & {} r_2-(r_{02}+e_2)-\{r_1-(r_{01}+e_1)\}, \\ {\widehat{A}}_2= & {} -[\{r_1-(r_{01}+e_1)\}\{r_2-(r_{02}+e_2)\}+e_1e_2p_{12}p_{21}]. \end{aligned}$$

  For the existence of equilibrium $E_3$, $r_2-(r_{02}+e_2)$ is positive, while the positive value of $r_1-(r_{01}+e_1)$ leads to existence of equilibrium $E_2$, and in that case ${\widehat{A}}_2<0$. Hence, Eq. (59) has one positive and one negative roots, and thus the equilibrium $E_3$ is unstable. Moreover, if conditions in (18) hold, then both roots of Eq. (59) are either negative or have negative real parts, and hence the equilibrium $E_3$ is stable.

• 4. Characteristic equation of the Jacobian evaluated at the equilibrium $E_4$ is obtained as,

  $$\begin{aligned} \lambda ^3+\check{A}_1\lambda ^2+\check{A}_2\lambda +\check{A}_3=0, \end{aligned}$$

  (60)

  where

  $$\begin{aligned} \check{A}_1= & {} b_1x_4+d_2z_4-\left( \frac{r_2}{1+a_2z_4}-r_{02}-e_2\right) ,\\ \check{A}_2= & {} x_4z_4\left\{ b_1d_2+\delta _1\beta _1\right. \\&\quad \left. \left( \beta _1+\frac{r_1a_1}{(1+a_1z_4)^2}\right) \right\} \\&\quad -(b_1x_4+d_2z_4)\left( \frac{r_2}{1+a_2z_4}-r_{02}-e_2\right) \\&\quad -e_1p_{12}(e_2p_{21}+\beta _1z_4),\\ \check{A}_3= & {} -x_4z_4\left( \frac{r_2}{1+a_2z_4}-r_{02}-e_2\right) \\&\quad \left\{ b_1d_2+\delta _1\beta _1\left( \beta _1+\frac{r_1a_1}{(1+a_1z_4)^2}\right) \right\} \\&\quad -e_1p_{12}z_4\left\{ d_2(e_2p_{21}+\beta _1z_4)\right. \\&\quad \left. +\delta _1\beta _1x_4\left( \beta _1+\frac{r_1a_1}{(1+a_1z_4)^2}\right) \right\} . \end{aligned}$$

  Note that if $\displaystyle \frac{r_2}{1+a_2z_4}-r_{02}-e_2>0,$ then $\check{A}_3<0$, and hence Eq. (60) has at least one positive root. Thus, the equilibrium $E_4$ is unstable. On the other hand, if $\displaystyle \frac{r_2}{1+a_2z_4}-r_{02}-e_2<0,$ then $\check{A}_1$ is positive. Further, if $\check{A}_3$ and $\check{A}_1\check{A}_2-\check{A}_3$ are positive, then the roots of Eq. (60) are either negative or have negative real parts, and hence the equilibrium $E_4$ is stable.

• 5. Characteristic equation of the Jacobian evaluated at the equilibrium $E_5$ is obtained as,

  $$\begin{aligned} \lambda ^3+\acute{A}_1\lambda ^2+\acute{A}_2\lambda +\acute{A}_3=0, \end{aligned}$$

  (61)

  where

  $$\begin{aligned} \acute{A}_1= & {} b_2y_5+d_2z_5-\left( \frac{r_1}{1+a_1z_5}-r_{01}-e_1\right) ,\\ \acute{A}_2= & {} y_5z_5\left\{ b_2d_2+\delta _2\beta _2\right. \\&\quad \left. \left( \beta _2+\frac{r_2a_2}{(1+a_2z_5)^2}\right) \right\} \\&\quad -(b_2y_5+d_2z_5)\left( \frac{r_1}{1+a_1z_5}-r_{01}-e_1\right) \\&\quad -e_2p_{21}(e_1p_{12}+\beta _2z_5),\\ \acute{A}_3= & {} -y_5z_5\left( \frac{r_1}{1+a_1z_5}-r_{01}-e_1\right) \\&\quad \left\{ b_2d_2+\delta _2\beta _2\left( \beta _2+\frac{r_2a_2}{(1+a_2z_5)^2}\right) \right\} \\&\quad -e_2p_{21}z_5\left\{ d_2(e_1p_{12}+\beta _2z_5)\right. \\&\quad \left. +\delta _2\beta _2y_5\left( \beta _2+\frac{r_2a_2}{(1+a_2z_5)^2}\right) \right\} . \end{aligned}$$

  Note that if $\displaystyle \frac{r_1}{1+a_1z_5}-r_{01}-e_1>0,$ then $\acute{A}_3<0$, and hence, Eq. (61) has at least one positive root. Thus, the equilibrium $E_5$ is unstable. On the other hand, if $\displaystyle \frac{r_1}{1+a_1z_5}-r_{01}-e_1<0,$ then $\acute{A}_1$ is positive. Further, if $\acute{A}_3$ and $\acute{A}_1\acute{A}_2-\acute{A}_3$ are positive, then the roots of Eq. (61) are either negative or have negative real parts, and hence the equilibrium $E_5$ is stable.

• 6. One eigenvalue of the Jacobian at the equilibrium $E_6$ is given by

  $$\begin{aligned} \frac{\delta _1\beta _1x^2_6}{x_6+y_6}+\frac{\delta _2\beta _2y^2_6}{x_6+y_6}-d_1, \end{aligned}$$

  which is negative if the first condition in (21) holds; the remaining two eigenvalues are given by roots of the quadratic,

  $$\begin{aligned} \lambda ^2+\grave{A}_1\lambda +\grave{A}_2=0, \end{aligned}$$

  (62)

  where

  $$\begin{aligned} \grave{A}_1= & {} -[r_1+r_2-\{r_{01}+r_{02}\\&\quad +e_1+e_2+2(b_1x_6+b_2y_6)\}],\\ \grave{A}_2= & {} \{r_1-(r_{01}+e_1+2b_1x_6)\}\\&\quad \{r_2-(r_{02}+e_2+2b_2y_6)\}-e_1e_2p_{12}p_{21}. \end{aligned}$$

  Note that roots of Eq. (62) are either negative or have negative real parts if the last two conditions in (21) hold.

• 7. At the equilibrium $E^*$, the Jacobian takes the form,

  $$\begin{aligned} J_{E^*}=\left( \begin{array}{ccc} a_{11} &{}\quad a_{12} &{}\quad -a_{13} \\ a_{21} &{}\quad a_{22} &{}\quad -a_{23} \\ a_{31} &{}\quad a_{32} &{}\quad -a_{33} \end{array}\right) , \end{aligned}$$

  where

  $$\begin{aligned} a_{11}= & {} \frac{r_1}{1+a_1z^*}-2b_1x^*-r_{01}\\&\quad -e_1-\frac{\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2}, \\ a_{12}= & {} e_2p_{21}+\frac{\beta _1{x^*}^2z^*}{(x^*+y^*)^2}, \\ a_{13}= & {} \frac{r_1a_1x^*}{(1+a_1z^*)^2}+\frac{\beta _1{x^*}^2}{x^*+y^*},\\ a_{21}= & {} e_1p_{12}+\frac{\beta _2{y^*}^2z^*}{(x^*+y^*)^2}, \\ a_{22}= & {} \frac{r_2}{1+a_2z^*}-2b_2y^*-r_{02}\\&\quad -e_2-\frac{\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}, \\ a_{23}= & {} \frac{r_2a_2y^*}{(1+a_2z^*)^2}+\frac{\beta _2{y^*}^2}{x^*+y^*},\\ a_{31}= & {} \frac{\delta _1\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2}\\&\quad -\frac{\delta _2\beta _2{y^*}^2z^*}{(x^*+y^*)^2},\\ a_{32}= & {} -\frac{\delta _1\beta _1{x^*}^2z^*}{(x^*+y^*)^2}\\&\quad +\frac{\delta _2\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2},\\ a_{33}= & {} -d_2z^* \end{aligned}$$

  The associated characteristic equation is given by

  $$\begin{aligned} \lambda ^3+\breve{A}_1\lambda ^2+\breve{A}_2\lambda +\breve{A}_3=0, \end{aligned}$$

  (63)

  where

  $$\begin{aligned} \breve{A}_1= & {} -\frac{r_1}{1+a_1z^*}+2b_1x^*+r_{01}+e_1\\&\quad +\frac{\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2}-\frac{r_2}{1+a_2z^*}\\&\quad +2b_2y^*+r_{02}+e_2+\frac{\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}+d_2z^*,\\ \breve{A}_2= & {} \left\{ \frac{r_1}{1+a_1z^*}-2b_1x^*-r_{01}\right. \\&\quad \left. -e_1-\frac{\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2}\right\} \\&\quad \left\{ \frac{r_2}{1+a_2z^*}-2b_2y^*-r_{02}\right. \\&\quad \left. -e_2 -\frac{\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}-d_2z^*\right\} \\&\quad +\left( \frac{r_2a_2y^*}{(1+a_2z^*)^2}+\frac{\beta _2{y^*}^2}{x^*+y^*}\right) \\&\quad \left( \frac{\delta _2\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}\right. \\&\quad \left. -\frac{\delta _1\beta _1{x^*}^2z^*}{(x^*+y^*)^2}\right) \\&\quad +\left( \frac{r_1a_1x^*}{(1+a_1z^*)^2}+\frac{\beta _1{x^*}^2}{x^*+y^*}\right) \\&\quad \left( \frac{\delta _1\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2}\right. \\&\quad \left. -\frac{\delta _2\beta _2{y^*}^2z^*}{(x^*+y^*)^2}\right) \\&\quad -\left( e_2p_{21}+\frac{\beta _1{x^*}^2z^*}{(x^*+y^*)^2}\right) \\&\quad \left( e_1p_{12}+\frac{\beta _2{y^*}^2z^*}{(x^*+y^*)^2}\right) \\&\quad -d_2z^*\left\{ \frac{r_2}{1+a_2z^*}-2b_2y^*-r_{02}-e_2\right. \\&\quad \left. -\frac{\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}\right\} ,\\ \breve{A}_3= & {} \left\{ \frac{r_1}{1+a_1z^*}-2b_1x^*-r_{01}\right. \\&\quad \left. -e_1-\frac{\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2}\right\} \\&\quad \left[ d_2z^*\left\{ \frac{r_2}{1+a_2z^*}-2b_2y^*-r_{02}-e_2\right. \right. \\&\quad \left. \left. -\frac{\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}\right\} \right. \\&\quad -\left( \frac{r_2a_2y^*}{(1+a_2z^*)^2}+\frac{\beta _2{y^*}^2}{x^*+y^*}\right) \\&\quad \left( \frac{\delta _2\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}\right. \\&\quad \left. \left. -\frac{\delta _1\beta _1{x^*}^2z^*}{(x^*+y^*)^2}\right) \right] \\&\quad +\left( e_2p_{21}+\frac{\beta _1{x^*}^2z^*}{(x^*+y^*)^2}\right) \\&\quad \left[ \left( \frac{\delta _1\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2}\right. \right. \\&\quad \left. -\frac{\delta _2\beta _2{y^*}^2z^*}{(x^*+y^*)^2}\right) \\&\quad \left. \left( \frac{r_2a_2y^*}{(1+a_2z^*)^2}+\frac{\beta _2{y^*}^2}{x^*+y^*}\right) \right. \\&\quad \left. -d_2z^*\left( e_1p_{12}+\frac{\beta _2{y^*}^2z^*}{(x^*+y^*)^2}\right) \right] \\&\quad +\left( \frac{r_1a_1x^*}{(1+a_1z^*)^2}+\frac{\beta _1{x^*}^2}{x^*+y^*}\right) \\&\quad \left[ \left( e_1p_{12}+\frac{\beta _2{y^*}^2z^*}{(x^*+y^*)^2}\right) \right. \\&\quad \left( \frac{\delta _2\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}\right. \\&\quad \left. \left. -\frac{\delta _1\beta _1{x^*}^2z^*}{(x^*+y^*)^2}\right) \right. \\&\quad -\left( \frac{\delta _1\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2}\right. \\&\quad \left. -\frac{\delta _2\beta _2{y^*}^2z^*}{(x^*+y^*)^2}\right) \\&\quad \left\{ \frac{r_2}{1+a_2z^*}-2b_2y^*-r_{02}\right. \\&\quad \left. \left. -e_2-\frac{\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}\right\} \right] . \end{aligned}$$

  Employing Routh–Hurwitz criterion together with equation (63), we get that roots of Eq. (63) are either negative or with negative real parts if and only if the conditions in (22) hold; thus, the equilibrium $E^*$ is locally asymptotically stable.

Appendix C

The second additive compound matrix of the Jacobian of system (1) at the coexistence equilibrium $E^*$ is given by,

$$\begin{aligned} J^{[2]}=\left( \begin{array}{ccc} a_{11}+a_{22} &{}\quad a_{23} &{}\quad -a_{13} \\ a_{32} &{}\quad a_{11}+a_{33} &{}\quad a_{12} \\ -a_{31} &{}\quad a_{21} &{}\quad a_{22}+a_{33} \end{array}\right) . \end{aligned}$$

Let $\displaystyle |X|_\infty =\sup _i|X_i|.$ The logarithmic norm $\mu _\infty (J^{[2]})$ of $J^{[2]}$ endowed with the vector norm $|X|_\infty$ is the supremum of $a_{11}+a_{22}+|a_{23}|+|a_{13}|$, $a_{11}+a_{33}+|a_{32}|+|a_{12}|$ and $a_{22}+a_{33}+|a_{31}|+|a_{21}|$.

Now, $a_{11}+a_{22}+|a_{23}|+|a_{13}|<0$ if

$$\begin{aligned}&\frac{r_1\{1+a_1(x^*+z^*)\}}{(1+a_1z^*)^2}+\frac{r_2\{1+a_2(y^*+z^*)\}}{(1+a_2z^*)^2}\\&\quad +\frac{\beta _1{x^*}^2+\beta _2{y^*}^2}{x^*+y^*}\\&\quad <r_{01}+r_{02}+e_1+e_2+2(b_1x^*+b_2y^*)\\&\quad +\frac{z^*\{\beta _1{x^*}^2+\beta _2{y^*}^2+2x^*y^*(\beta _1+\beta _2)\}}{(x^*+y^*)^2}; \end{aligned}$$

$a_{11}+a_{33}+|a_{32}|+|a_{12}|<0$ if

$$\begin{aligned}&\frac{r_1}{1+a_1z^*}+e_2p_{21}+\frac{(1-\delta _1)\beta _1{x^*}^2z^*+\delta _2\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}\\&\quad <r_{01}+e_1+2b_1x^*+d_2z^*+\frac{\beta _1x^*z^*(x^*+2y^*)}{(x^*+y^*)^2} \end{aligned}$$

and $a_{22}+a_{33}+|a_{31}|+|a_{21}|<0$ if

$$\begin{aligned}&\frac{r_2}{1+a_2z^*}+e_1p_{12}+\frac{\delta _1\beta _1x^*z^*(x^*+2y^*)+(1-\delta _2)\beta _2{y^*}^2z^*}{(x^*+y^*)^2}\\&\quad <r_{02}+e_2+2b_2y^*+d_2z^*+\frac{\beta _2y^*z^*(2x^*+y^*)}{(x^*+y^*)^2}. \end{aligned}$$

Following [53], system (1) has no periodic solution around the coexistence equilibrium $E^*$ provided condition (23) is satisfied.

Appendix D

Let the critical value of $a_1$, say $a^*_1$, be defined by

$$\begin{aligned} \breve{A}_1(a^*_1)\breve{A}_2(a^*_1)-\breve{A}_3(a^*_1)=0. \end{aligned}$$

(64)

Thus, at $a_1=a^*_1$, characteristic Eq. (63) can be rewritten as $(\lambda +\breve{A}_1)(\lambda ^2+\breve{A}_2)=0$. This equation has three roots, a pair of purely imaginary roots $\lambda _{1,2}=\pm i\sqrt{\breve{A}_2}$ and a negative one $\lambda _3=-\breve{A}_1$. This ensures the presence of Hopf-bifurcation.

To show the transversality condition, let us consider a point $a_1$ in an $\epsilon -$ neighborhood of $a^*_1$; the above roots become function of $a_1$, namely $\lambda _{1,2}=\kappa (a_1)\pm i\rho (a_1)$. Substituting them into Eq. (63) and separating real and imaginary parts, we have

$$\begin{aligned}&\kappa ^3-3\kappa \rho ^2+\breve{A}_1(\kappa ^2-\rho ^2)+\breve{A}_2\kappa +\breve{A}_3=0, \end{aligned}$$

(65)

$$\begin{aligned}&3\kappa ^2\rho -\rho ^3+2\breve{A}_1\kappa \rho +\breve{A}_2\rho =0. \end{aligned}$$

(66)

As $\rho (a_1)\ne 0$, from Eq. (66), it follows that

$$\begin{aligned} \rho ^2=3\kappa ^2+2\breve{A}_1\kappa +\breve{A}_2. \end{aligned}$$

Substituting this in Eq. (65), we find

$$\begin{aligned} 8\kappa ^3+8\breve{A}_1\kappa ^2+2\kappa (\breve{A}^2_1+\breve{A}_2)+\breve{A}_1\breve{A}_2-\breve{A}_3=0.\nonumber \\ \end{aligned}$$

(67)

From the above equation, recalling that $\kappa (a^*_1)=0$, we get

$$\begin{aligned} \left[ \frac{\mathrm{d}\kappa }{da_1}\right] _{a_1=a^*_1}=-\left[ \frac{1}{2(\breve{A}^2_1+\breve{A}_2)}\frac{d}{\mathrm{d}a_1}(\breve{A}_1\breve{A}_2-\breve{A}_3)\right] _{a_1=a^*_1} \end{aligned}$$

and the latter does not vanish provided that

$$\begin{aligned} \left[ \frac{d}{\mathrm{d}a_1}(\breve{A}_1\breve{A}_2-\breve{A}_3)\right] _{a_1=a^*_1}\ne 0. \end{aligned}$$

(68)

To better understand the nature of the instability, we determine the initial period and the amplitude of the oscillatory solutions. Set $\breve{A}_3=\psi \breve{A}_1\breve{A}_2$ in Eq. (63). Assuming that $\lambda$ depends continuously on $\psi$, we rewrite Eq. (63) as

$$\begin{aligned} \lambda ^3+\breve{A}_1\lambda ^2+\breve{A}_2\lambda +\psi \breve{A}_1\breve{A}_2=0. \end{aligned}$$

(69)

At $\psi =\psi ^*=1$, because $\breve{A}_3=\breve{A}_1\breve{A}_2$, Eq. (69), as seen above, factorizes into $(\lambda +\breve{A}_1)(\lambda ^2+\breve{A}_2)$, which has a pair of purely imaginary roots, $\lambda (\psi ^*)=\pm i\sqrt{\breve{A}_2}$, while the other one is $\lambda (\psi ^*)=-\breve{A}_1$.

Further, $\breve{A}_1\breve{A}_2-\breve{A}_3=(1-\psi )\breve{A}_1\breve{A}_2$. Thus, if $\psi \in (0,1)$, then $\breve{A}_1\breve{A}_2-\breve{A}_3>0$ and this ensures stability, conversely, we have instability for $\psi >1$.

If we set $\psi =\psi ^*+\epsilon ^2\xi$, where $|\epsilon |\ll 1$ and $\xi =\pm 1$, then $\lambda (\psi )=\lambda (\psi ^*+\epsilon ^2\xi )$, so that the linear portion in $\epsilon ^2\xi$ of the Taylor series expansion of $\lambda$ about $\psi ^*$ is

$$\begin{aligned} \lambda (\psi )=\lambda (\psi ^*)+\lambda '(\psi ^*)\epsilon ^2\xi +O(\epsilon ^4), \end{aligned}$$

(70)

where prime denotes differentiation with respect to $\psi$.

Differentiating and simplifying Eq. (69) yields

$$\begin{aligned} \lambda '(\psi )\equiv \frac{\breve{A}_1\breve{A}_2}{2(\breve{A}^2_1+\breve{A}_2)}\pm i\frac{\breve{A}^2_1\sqrt{\breve{A}_2}}{2(\breve{A}^2_1+\breve{A}_2)}. \end{aligned}$$

(71)

Using the fact that $\mathfrak {R}(\lambda (\psi ^*))=0$ and $\displaystyle \mathfrak {R}(\lambda '(\psi ^*))=\frac{\breve{A}_1\breve{A}_2}{2(\breve{A}^2_1+\breve{A}_2)}>0,$ and substituting $\lambda (\psi ^*)$ and $\lambda '(\psi )$ into Eq. (70), we obtain the approximation

$$\begin{aligned} \lambda (\psi )= & {} \lambda (\psi ^*)+\lambda '(\psi ^*)\epsilon ^2\xi \nonumber \\&\quad =\frac{\breve{A}_1\breve{A}_2\epsilon ^2\xi }{2(\breve{A}^2_1+\breve{A}_2)} \nonumber \\&\quad \pm i\sqrt{\breve{A}_2}\left( 1+\frac{\breve{A}^2_1\epsilon ^2\xi }{2(\breve{A}^2_1+\breve{A}_2)}\right) +O(\epsilon ^4). \end{aligned}$$

(72)

Thus, the initial period and amplitude of the oscillations associated with the loss of stability when $\psi >\psi ^*$, respectively, are

$$\begin{aligned}&\frac{2\pi }{\sqrt{\breve{A}_2}\left( 1+\frac{\breve{A}^2_1\epsilon ^2\xi }{2(\breve{A}^2_1+\breve{A}_2)}\right) } \quad \text {and} \\&\quad \exp \left( \frac{\breve{A}_1\breve{A}_2\epsilon ^2\xi }{2(\breve{A}^2_1+\breve{A}_2)}\right) ,\\&\quad \epsilon =\sqrt{\frac{|\psi -\psi ^*|}{|\xi |}}. \end{aligned}$$