A cannibalistic eco-epidemiological model with disease in predator population

Abstract

In this present article, we propose and analyze a cannibalistic predator–prey model with disease in the predator population. We consider two important factors for the dynamics of predator population. The first one is governed through cannibalistic interaction, and the second one is governed through the disease in the predator population via cannibalism. The local stability analysis of the model system around the biologically feasible equilibria are investigated. We perform global dynamics of the model using Lyapunov functions. We analyze and compare the community structure of the system in terms of ecological and disease basic reproduction numbers. The existence of Hopf bifurcation around the interior steady state is investigated. We also derive the sufficient conditions for the permanence and impermanence of the system. The study reveals that the cannibalism acts as a self-regulatory mechanism and controls the disease transmission among the predators by stabilizing the predator–prey oscillations.

Appendices

Appendix 1

• (a) The Jacobian matrix around the interior equilibrium point $B_{*}(x_{*}, y_{*}, z_{*})$ is given by

  $$\begin{aligned} J^{*}= \left[ \begin{array}{ccc} V_{11} &{} V_{12} &{} V_{13} \\ V_{21} &{} V_{22} &{} V_{23} \\ V_{31} &{} V_{32} &{} V_{33} \\ \end{array} \right] , \end{aligned}$$

  where $V_{11}=-\frac{rx_{*}}{k}+\frac{(\alpha _{1}y_{*}+\alpha _{2} z_{*})x_{*}}{(\gamma +x_{*})^{2}}, V_{12}=-\frac{\alpha _{1}x_{*}}{\gamma +x_{*}}, V_{13}=-\frac{\alpha _{2}x_{*}}{\gamma +x_{*}}, V_{21}=\frac{\alpha (\alpha _{1}y_{*}+\alpha _{2} z_{*})}{\gamma +x_{*}} -\frac{\alpha (\alpha _{1}y_{*}+\alpha _{2} z_{*})x_{*}}{(\gamma +x_{*})^{2}}, V_{22}=\frac{\alpha \alpha _{1}x_{*}}{\gamma +x_{*}}\,+\,c_{1}\sigma (2\beta y_{*}+z_{*})\,+\,c_{2}\sigma \beta z_{*} -\sigma (2\beta y_{*}+z_{*})-\sigma l f z_{*} - \lambda z_{*}-d, V_{23}=\frac{\alpha \alpha _{2}x_{*}}{\gamma +x_{*}}+c_{1}\sigma y_{*}+c_{2}\sigma (\beta y_{*}+2z_{*})-\sigma y_{*}-\sigma l f y_{*} - \lambda y_{*}, V_{31}=0, V_{32}=\lambda z_{*}+ \sigma l f z_{*}-\sigma \beta z_{*}, V_{33}=-\sigma z_{*}$. The characteristic equation for $J^{*}$ is

  $$\begin{aligned} \mu ^{3}+\varTheta _{1} \mu ^{2}+\varTheta _{2} \mu + \varTheta _{3}=0, \end{aligned}$$

  (7.1)

  where $\varTheta _{1}=-(V_{11} +V_{22}+V_{33}), \varTheta _{2}=V_{11}V_{22}+V_{11}V_{33}+ V_{22} V_{33}- V_{12} V_{21}- V_{23} V_{32}, \varTheta _{3}=-det(J^{*})=V_{11} V_{23} V_{32} + V_{12} V_{21} V_{33} - V_{13} V_{21} V_{32} - V_{11} V_{22} V_{33}$. According to the Routh–Hurwitz criteria the interior equilibrium point $B_{*}$ is stable if the conditions stated in the theorem hold.

• (b) Let $\varSigma$ be a positive definite Lyapunov function about $B_{*}(x_{*}, y_{*}, z_{*})$, given by

  $$\begin{aligned} \varSigma =\varSigma _{x}+\varSigma _{y}+\varSigma _{z}, \end{aligned}$$

where $\varSigma _{x}=x-x_{*}-x_{*} ln\frac{x}{x_{*}}, \varSigma _{y}=y-y_{*}-y_{*} ln\frac{y}{y_{*}}, \varSigma _{z}=z-z_{*}-z_{*} ln\frac{z}{z_{*}}$.

Now, computing the time derivative of $\varSigma _{x}$ along the solution of (2.4), we obtain

$$\begin{aligned} {\dot{\varSigma }}_{x}= & {} (x-x_{*})\left[ r\left( 1-\frac{x}{k}\right) -\frac{\alpha _{1}y+\alpha _{2} z}{\gamma +x}\right] \\= & {} \left. (x-x_{*})\left[ -\frac{r(x-x_{*})}{k}\right) -\alpha _{1}\left( \frac{y}{\gamma +x}-\frac{y_{*}}{\gamma +x_{*}}\right) -\alpha _{2}\left( \frac{z}{\gamma +x}-\frac{z_{*}}{\gamma +x_{*}}\right) \right] \\= & {} -\frac{r}{k}(x-x_{*})^{2}-\frac{\alpha _{1}}{(\gamma +x)}(x-x_{*})(y-y_{*})-\frac{\alpha _{2}}{(\gamma +x)}(x-x_{*})(z-z_{*})\\&+\,\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{(\gamma +x)(\gamma +x_{*})}(x-x_{*})^{2}. \end{aligned}$$

Similarly,

$$\begin{aligned} {\dot{\varSigma }}_{y}= & {} (y-y_{*})\left[ \frac{\alpha (\alpha _{1}y+\alpha _{2} z)x}{(\gamma +x)y} +c_{1}\sigma (\beta y+z)\right. \\&\left. +\,c_{2}\sigma \left( \beta +\frac{z}{y}\right) z-\sigma (\beta y+z)-\sigma l f z - \lambda z-d\right] \\= & {} (y-y_{*})\left[ \alpha \alpha _{1} \left( \frac{x}{\gamma +x}-\frac{x_{*}}{\gamma +x_{*}}\right) + \alpha \alpha _{2} \left\{ \frac{xz}{(\gamma +x)y}-\frac{x_{*}z_{*}}{(\gamma +x_{*})y_{*}}\right\} \right. \\&-\,\sigma (1-c_{1})\beta (y-y_{*})- \{\sigma (1-c_{1})-c_{2} \sigma \beta + \sigma lf +\lambda \} (z-z_{*})\\&\left. +\,c_{2} \sigma \left( \frac{z^{2}}{y}- \frac{z^{*^{2}}}{y_{*}}\right) \right] \\= & {} \left\{ \frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}+ \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \right\} (x-x_{*})(y-y_{*})\\&+\,\left[ c_{2} \sigma \frac{z}{y}+c_{2} \sigma \frac{z_{*}}{y}-\{\sigma (1-c_{1})-c_{2} \sigma \beta + \sigma lf +\lambda \}\right. \\&\left. +\,\frac{\alpha \alpha _{2} x}{(\gamma +x)y}\right] (y-y_{*})(z-z_{*})\\&-\,\left\{ \sigma (1-c_{1})\beta +\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y} \right\} (y-y_{*})^{2},\\ {\dot{\varSigma }}_{z}= & {} (z-z_{*})[\lambda y+ \sigma l f y-\sigma (\beta y+z) -(d+e) ]\\= & {} (\lambda + \sigma l f -\sigma \beta )(y-y_{*})(z-z_{*})-\sigma (z-z_{*})^{2}.\\ \end{aligned}$$

Adding these quantities and after some algebraic calculations, we can obtain

$$\begin{aligned} {\dot{\varSigma }}= & {} -\left[ \{\frac{r}{k}-\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{(\gamma +x)(\gamma +x_{*})} \} (x-x_{*})^{2}\right. \\&+\,\left\{ \sigma (1-c_{1})\beta +\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y} \right\} (y-y_{*})^{2}+\sigma (z-z_{*})^{2} \\&+\,\left\{ \frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}- \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \right\} \\&\times (x-x_{*})(y-y_{*})\\&+\,\left\{ \sigma (1-c_{1})+\sigma \beta (1-c_{2})-c_{2} \sigma \frac{z}{y}-c_{2} \sigma \frac{z_{*}}{y}- \frac{\alpha \alpha _{2} x}{(\gamma +x)y}\right\} (y-y_{*})(z-z_{*})\\&+\,\left. \frac{\alpha _{2}}{(\gamma +x)}(x-x_{*})(z-z_{*}) \right] \end{aligned}$$

The above expression can be written in the form $-\mathbf {w}^{T}\varOmega \mathbf {w}$, where $\mathbf {w}=(x-x_{*}, y-y_{*}, z-z_{*})$ and $\varOmega$ is the symmetric matrix given by

$$\begin{aligned} \varOmega =\left( \begin{array}{ccc} \varOmega _{11}&{} \varOmega _{12} &{} \varOmega _{13} \\ \varOmega _{12} &{} \varOmega _{22} &{}\varOmega _{23} \\ \varOmega _{13} &{} \varOmega _{23} &{} \varOmega _{33} \\ \end{array} \right) , \end{aligned}$$

with $\varOmega _{11}= \frac{r}{k}-\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{(\gamma +x)(\gamma +x_{*})}, \varOmega _{12}= \frac{1}{2} \{ \frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}- \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \}, \varOmega _{13}= \frac{1}{2} \frac{\alpha _{2}}{(\gamma +x)}, \varOmega _{22}=\sigma (1-c_{1})\beta +\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y}, \varOmega _{23}=\frac{1}{2} \{ \sigma (1-c_{1})+\sigma \beta (1-c_{2})-c_{2} \sigma \frac{z}{y}-c_{2} \sigma \frac{z_{*}}{y}- \frac{\alpha \alpha _{2} x}{(\gamma +x)y}\}, \varOmega _{33}=\sigma$.

Thus, ${\dot{\varSigma }}$ is negative definite if the symmetric matrix $\varOmega$ is positive definite. The matrix $\varOmega$ is positive definite, if all the principal minors ${\mathbb {P}}_{1}= \varOmega _{11}, {\mathbb {P}}_{2}= \varOmega _{11} \varOmega _{22}-\varOmega _{12}^{2}, {\mathbb {P}}_{3}=\varOmega _{11}\varOmega _{22}\varOmega _{33}+2\varOmega _{12}\varOmega _{13}\varOmega _{23}-\varOmega _{11}\varOmega _{23}^{2}-\varOmega _{22}\varOmega _{13}^{2}-\varOmega _{33}\varOmega _{12}^{2}$ of $\varOmega$ are positive, i.e.,

(i):

${\mathbb {P}}_{1}=\frac{r}{k}-\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{(\gamma +x)(\gamma +x_{*})} >0$,

(ii):

${\mathbb {P}}_{2}=[\frac{r}{k}-\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{(\gamma +x)(\gamma +x_{*})}] [ \sigma (1-c_{1})\beta +\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y}]-[\frac{1}{2} \{\frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}- \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \}]^{2} >0$,

(iii):

${\mathbb {P}}_{3}=\varOmega _{33}(\varOmega _{11}\varOmega _{22}-\varOmega _{12}^{2})+\varOmega _{23}(\varOmega _{12}\varOmega _{13}-\varOmega _{11}\varOmega _{23})+\varOmega _{13}(\varOmega _{12}\varOmega _{23}-\varOmega _{22}\varOmega _{13})>0$.

For ${\mathbb {P}}_{1}$, we have by (4.3) ${\mathbb {P}}_{1}(x)=\frac{r}{k}-\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{(\gamma +x)(\gamma +x_{*})}>{\mathbb {P}}_{1}(0)=\frac{r}{k}-\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{\gamma (\gamma +x_{*})}>0$ since $\dot{{\mathbb {P}}_{1}}(x)=\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{(\gamma +x)^{2}(\gamma +x_{*})}>0$.

For ${\mathbb {P}}_{2}$, using (4.3) and (4.4), we have

$$\begin{aligned} {\mathbb {P}}_{2}= & {} \left[ \frac{r}{k}-\frac{\alpha _{1} y_{*}+ \alpha _{2}z_{*}}{(\gamma +x)(\gamma +x_{*})}\right] \left[ \sigma (1-c_{1})\beta +\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y}\right] \\&-\,\left[ \frac{1}{2} \left\{ \frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}- \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \right\} \right] ^{2} \\= & {} \Bigl [\Bigl \{\Bigl (\frac{r}{k}-\frac{\alpha _{1} y_{*}+ \alpha _{2}z_{*}}{(\gamma +x)(\gamma +x_{*})} \Bigl ) \sigma (1-c_{1})\beta -\frac{1}{4}\left( \frac{\alpha _{1}}{\gamma +x}\right) ^{2} \Bigr \}\\&+\,\Bigl \{\frac{r}{k}-\frac{\alpha _{1} y_{*}+ \alpha _{2}z_{*}}{(\gamma +x)(\gamma +x_{*})} \Bigr \} \Bigl \{\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y} \Bigr \} \\&+\,\frac{1}{2} \frac{\alpha \alpha _{1} \alpha _{2} \gamma z_{*}}{(\gamma +x)^{2}(\gamma +x_{*})y}\Bigl (1-\frac{\alpha \gamma }{\gamma +x_{*}}\Bigr )+\frac{1}{4} \frac{\alpha \alpha _{1}^{2} \gamma }{(\gamma +x)^{2}(\gamma +x_{*})} \Bigl (1-\frac{\alpha \gamma }{\gamma +x_{*}} \Bigr )\\&+\,\frac{1}{4} \frac{\alpha \gamma }{(\gamma +x)^{2}(\gamma +x_{*})} \Bigl \{\alpha _{1}^{2}-\frac{\alpha \alpha _{2}^{2}\gamma z^{*^{2}} }{(\gamma +x_{*})y^{2}} \Bigr \} \Bigr ]\\> & {} \Bigl [\Bigl \{\Bigl (\frac{r}{k}-\frac{\alpha _{1} y_{*}+ \alpha _{2}z_{*}}{\gamma (\gamma +x_{*})} \Bigl ) \sigma (1-c_{1})\beta \\&-\,\frac{1}{4}\Bigl (\frac{\alpha _{1}}{\gamma }\Bigr )^{2} \Bigr \}+\Bigl \{\frac{r}{k}-\frac{\alpha _{1} y_{*}+ \alpha _{2}z_{*}}{\gamma (\gamma +x_{*})} \Bigr \} \Bigl \{\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y} \Bigr \} \\&+\,\frac{1}{2} \frac{\alpha \alpha _{1} \alpha _{2} \gamma z_{*}}{(\gamma +x)^{2}(\gamma +x_{*})y}\Bigl (1-\frac{\alpha \gamma }{\gamma +x_{*}}\Bigr )+\frac{1}{4} \frac{\alpha \alpha _{1}^{2} \gamma }{(\gamma +x)^{2}(\gamma +x_{*})} \Bigl (1-\frac{\alpha \gamma }{\gamma +x_{*}} \Bigl )\\&+\,\frac{1}{4} \frac{\alpha \gamma }{(\gamma +x)^{2}(\gamma +x_{*})} \Bigl \{\alpha _{1}^{2}-\frac{\alpha \alpha _{2}^{2}\gamma z^{*^{2}} }{(\gamma +x_{*})y^{2}} \Bigr \} \Bigr ] >0. \end{aligned}$$

Now for ${\mathbb {P}}_{3}$, we have

$$\begin{aligned}&\varOmega _{12}\varOmega _{13}-\varOmega _{11}\varOmega _{23}\\&\quad =\frac{1}{4}\Bigl [ \Bigl \{ \frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}- \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \Bigr \}\\&\qquad \times \frac{\alpha _{2}}{(\gamma +x)}-2 \Bigl \{\frac{r}{k}-\frac{(\alpha _{1} y_{*}+ \alpha _{2}z_{*})}{(\gamma +x)(\gamma +x_{*})}\Bigr \} \\&\qquad \times \Bigl \{ \sigma (1-c_{1})+\sigma \beta (1-c_{2})-c_{2} \sigma \frac{z}{y}-c_{2} \sigma \frac{z_{*}}{y}- \frac{\alpha \alpha _{2} x}{(\gamma +x)y}\Bigr \} \Bigr ] \\&\quad =\frac{1}{4} \Bigl [ \Bigl \{ \frac{\alpha _{2}}{(\gamma +x)} \Bigl ( \frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}\\&\quad \quad -\,\frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \Bigr ) -\,\frac{2r}{k} \Bigl ( \sigma (1-c_{1})+\sigma \beta (1-c_{2})\Bigl ) \Bigr \} \\&\quad \quad +\,2 \{\sigma (1-c_{1})+\sigma \beta (1-c_{2}) \}\frac{\alpha _{1} y_{*}+ \alpha _{2}z_{*}}{(\gamma +x)(\gamma +x_{*})}+2 \Bigl \{\frac{r}{k}-\frac{\alpha _{1} y_{*}+ \alpha _{2}z_{*}}{(\gamma +x)(\gamma +x_{*})}\Bigr \}\\&\quad \quad \times \Bigl \{ c_{2} \sigma \frac{z}{y}+c_{2} \sigma \frac{z_{*}}{y}+ \frac{\alpha \alpha _{2} x}{(\gamma +x)y} \Bigr \} \Bigr ] \\&\quad =\frac{1}{4} \Bigl [ \Bigl \{ \frac{\alpha _{2}}{(\gamma +x)} \Bigl ( \frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}- \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \Bigr ) \\&\quad \quad -\,\frac{2r}{k} \Bigl ( \sigma (1-c_{1})+\sigma \beta (1-c_{2})\Bigl ) \Bigr \} +2 \{\sigma (1-c_{1})\\&\quad \quad +\,\sigma \beta (1-c_{2}) \}\frac{\alpha _{1} y_{*}+ \alpha _{2}z_{*}}{(\gamma +x)(\gamma +x_{*})}+2 \Bigl \{\frac{r}{k}-\frac{\alpha _{1} y_{*} +\alpha _{2}z_{*}}{\gamma (\gamma +x_{*})}\Bigr \}\\&\quad \quad \times \Bigl \{ c_{2} \sigma \frac{z}{y}+c_{2} \sigma \frac{z_{*}}{y}+ \frac{\alpha \alpha _{2} x}{(\gamma +x)y} \Bigr \} \Bigr ]>0 \,(\hbox {Using } (4.3)\hbox { and }(4.4)). \end{aligned}$$

From (4.5), we have $\varOmega _{23}=\frac{1}{2} \{ \sigma (1-c_{1})+\sigma \beta (1-c_{2})-c_{2} \sigma \frac{z}{y}-c_{2} \sigma \frac{z_{*}}{y}- \frac{\alpha \alpha _{2} x}{(\gamma +x)y}\}>0$.

Using (4.4) and (4.6), for ${\mathbb {P}}_{3}$, we get

$$\begin{aligned} \varOmega _{12}\varOmega _{23}-\varOmega _{22}\varOmega _{13}= & {} \frac{1}{4}\Bigl [ \Bigl \{ \frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}- \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \Bigr \} \\&\times \Bigl \{ \sigma (1-c_{1})+\sigma \beta (1-c_{2})-c_{2} \sigma \frac{z}{y}-c_{2} \sigma \frac{z_{*}}{y}- \frac{\alpha \alpha _{2} x}{(\gamma +x)y} \Bigr \}\\&-\,\Bigl \{ \sigma (1-c_{1})\beta +\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y} \Bigr \} \frac{2\alpha _{2}}{(\gamma +x)} \Bigr ] \\= & {} \frac{1}{4}\Bigl [ \Bigl \{ \Bigl ( \sigma (1-c_{1})+\sigma \beta (1-c_{2}) \Bigr ) \\&\times \Bigl ( \frac{\alpha _{1}}{(\gamma +x)}-\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}- \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y} \Bigr )\\&-\,\Bigl ( \sigma (1-c_{1})\beta +\frac{c_{2} \sigma z^{*^{2}}}{y_{*}y}+ \frac{\alpha \alpha _{2} x_{*}z_{*}}{(\gamma +x_{*})y_{*}y}\Bigr ) \frac{2\alpha _{2}}{(\gamma +x)}\Bigr \}\\&+\,\Bigl \{ c_{2} \sigma \frac{z}{y}+c_{2} \sigma \frac{z_{*}}{y} \Bigr \} \times \Bigl \{ \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y}-\frac{\alpha _{1}}{(\gamma +x)} \Bigr \} \\&+\,\frac{\alpha \alpha _{2} x}{(\gamma +x)y} \Bigl \{ \frac{\alpha \alpha _{2} \gamma z_{*}}{(\gamma +x)(\gamma +x_{*})y}-\frac{\alpha _{1}}{(\gamma +x)} \Bigr \} \\&+\,\frac{\alpha \alpha _{1} \gamma }{(\gamma +x)(\gamma +x_{*})}\Bigl \{ c_{2} \sigma \frac{z}{y}+c_{2} \sigma \frac{z_{*}}{y}+\frac{\alpha \alpha _{2} x}{(\gamma +x)y} \Bigr \} \Bigr ] >0 \end{aligned}$$

Above results suggest that ${\mathbb {P}}_{3}>0$, so the conditions (i), (ii) and (iii) imply that ${\dot{\varSigma }}<0$ along the trajectories. Therefore, system (2.4) is globally stable around the interior equilibrium point $B_{*}$ according as the conditions stated in the theorem hold.

Appendix 2

Let $\mu (\sigma )=\varrho (\sigma )+ i \varsigma (\sigma )$ for all $\sigma \in R^{+}$, the characteristic value of the characteristic Eq. (7.1). Substituting this value in Eq. (7.1), and separate real and imaginary parts as

$$\begin{aligned}&\varrho ^{3}-3\varrho \varsigma ^{2}+ (\varrho ^{2}-\varsigma ^{2})\varTheta _{1}+ \varrho \varTheta _{2}+\varTheta _{3} = 0, \end{aligned}$$

(7.2)

$$\begin{aligned}&3\varrho ^{2}\varsigma -\varsigma ^{3}+2\varrho \varsigma \varTheta _{1} +\varsigma \varTheta _{2} = 0. \end{aligned}$$

(7.3)

A necessary condition for a Hopf bifurcation of $B_{*}$ is that the characteristic equation (7.1) should have purely imaginary solutions. For the Hopf bifurcation to occur at $\sigma =\sigma ^{*}$, substituting $\varTheta _{1}(\sigma ^{*})\varTheta _{2}(\sigma ^{*})=\varTheta _{3}(\sigma ^{*})$ into Eq. (7.1), the characteristic equation must be of the form

$$\begin{aligned}{}[\mu ^{2}+\varTheta _{2}(\sigma ^{*})] [\mu +\varTheta _{1}(\sigma ^{*}) ]=0. \end{aligned}$$

(7.4)

Thus, the characteristic values of the Eq. (7.4) are $\mu _{1,2}(\sigma ^{*})=\pm i\sqrt{\varTheta _{2}(\sigma ^{*})}=\pm i\varsigma ^{*}$ and $\mu _{3}(\sigma ^{*})=-\varTheta _{1}(\sigma ^{*})$. By using the condition (i) of the Theorem 4, we have $\varsigma ^{*}>0$ and $\mu _{3}(\sigma ^{*})<0$.

Now, we can verify the transversality condition

$$\begin{aligned} \left[ \frac{d}{d\sigma }[Re\{\mu (\sigma )\}]\right] _{\sigma =\sigma ^{*}}\ne 0. \end{aligned}$$

(7.5)

Calculating the derivative of (7.2) and (7.3) w.r.t. $\sigma$ and substituting $\varrho =0, \varsigma =\varsigma ^{*}$ and $\sigma =\sigma ^{*}$, we get

$$\begin{aligned} \begin{aligned}&\mathcal {P}\left[ \frac{d[Re\{\mu (\sigma \}]}{d\sigma }\right] _{\sigma =\sigma ^{*}}-\mathcal {Q}\left[ \frac{d[Im\{\mu (\sigma )\}]}{d\sigma }\right] _{\sigma =\sigma ^{*}}= \mathcal {L},\\&\mathcal {Q}\left[ \frac{d[Re\{\mu (\sigma )\}]}{d\sigma }\right] _{\sigma =\sigma ^{*}}+\mathcal {P}\left[ \frac{d[Im\{\mu (\sigma )\}]}{d\sigma }\right] _{\sigma =\sigma ^{*}}= \mathcal {M},\\ \end{aligned} \end{aligned}$$

(7.6)

where $\mathcal {P}=-2\varTheta _{2}(\sigma ^{*}), \mathcal {Q}=2\varTheta _{1}(\sigma ^{*})\sqrt{\varTheta _{2}(\sigma ^{*})}, \mathcal {L}=\acute{\varTheta _{1}}(\sigma ^{*})\varTheta _{2}(\sigma ^{*})-\acute{\varTheta _{3}}(\sigma ^{*}), \mathcal {M}=-\acute{\varTheta _{2}}(\sigma ^{*})\sqrt{\varTheta _{2}(\sigma ^{*})}$.

Solving for $\acute{\mu }(\sigma ^{*})$ from system (7.6) we obtained

$$\begin{aligned} \left[ \frac{d}{d\sigma }[Re\{\mu (\sigma )\}]\right] _{\sigma =\sigma ^{*}}= & {} \acute{\mu }(\sigma ^{*}) =\frac{\mathcal {L}\mathcal {P}+\mathcal {M}\mathcal {Q}}{\mathcal {P}^{2}+\mathcal {Q}^{2}}\\= & {} \frac{2\varTheta _{2}(\sigma ^{*})\{\acute{\varTheta _{3}}(\sigma ^{*})-\varTheta _{1}(\sigma ^{*}) \acute{\varTheta _{2}}(\sigma ^{*})-\acute{\varTheta _{1}}(\sigma ^{*})\varTheta _{2}(\sigma ^{*}) \}}{\mathcal {P}^{2}+\mathcal {Q}^{2}}. \end{aligned}$$

Hence $\acute{\mu }(\sigma ^{*})>0$, if $\varTheta _{2}(\sigma ^{*}) > 0$ and $\acute{\varTheta _{3}}(\sigma ^{*})>[\varTheta _{1}(\sigma ^{*})\varTheta _{2}(\sigma ^{*})]^{'}$. Therefore, the transversality condition is satisfied and a Hopf bifurcation occurs when $\sigma$ passes through the critical value $\sigma ^{*}$.

Appendix 3

Let $\tilde{p}$ be a point in the positive quadrant and $o(\tilde{p})$ be orbit through $\tilde{p}$ and $\tilde{\varOmega }(\tilde{p})$ be the bounded omega limit set of the orbit through $\tilde{p}$. $W^{s}(B_{r})$ denotes the stable manifold of $B_{r}, r=1,2$.

Clearly $B_{0}\notin \tilde{\varOmega }(\tilde{p})$. If possible, let $B_{0}\in \tilde{\varOmega }(\tilde{p})$ then by the Butler-McGehee lemma there exists a point m in $\tilde{\varOmega }(\tilde{p})\cap W^{s}(B_{0})$. But, o(m) lies in $\tilde{\varOmega }$ and $W^{s}(B_{0})$ is the yz-plane, which shows that o(m) is unbounded, a contradiction.

Next we show that $B_{1}\notin \tilde{\varOmega }(\tilde{p})$. If $B_{1}\in \tilde{\varOmega }(\tilde{p})$, the condition $R_{01}>1$ implies that $B_{1}$ is a saddle point, then applying the Butler–McGehee lemma there exists a point m in $\tilde{\varOmega }(\tilde{p})\cap W^{s}(B_{1})$. Now $\tilde{\varOmega }(\tilde{p})\cap W^{s}(B_{1})$ is the xz-space and hence orbit in this plane emanate from either $B_{1}$ or an unbounded orbit lies in $\tilde{\varOmega }(\tilde{p})$, which is a contradiction.

Finally, we show that no periodic orbit solution in the xy-space or $B_{2}\in \tilde{\varOmega }(\tilde{p})$. If $q_{j}, j=1, 2,\ldots , n$ denote the closed orbit of the periodic solution $(\varphi _{q}(t), \psi _{q}(t))$ in xy-space such that $q_{j}$ lies inside $q_{j-1}$. Let, the Jacobian matrix J of the system (2.4) corresponding to $q_{j}$ be denoted by $J_{q}(\varphi _{q}(t), \psi _{q}(t), 0)$. The fundamental matrix of the linear periodic system is given by

$$\begin{aligned} {\dot{Z}}=J_{q}(t), \quad Z(0)=Z_{0}. \end{aligned}$$

(7.7)

We find that, $e^{\zeta _{q}\mathcal {T}}$ is the Floquet multiplier of (7.7) in the direction of z. Then applying the process as proposed by Kumar and Freedman (1989), we conclude that no $q_{j}$ lies on $\tilde{\varOmega }$. Therefore, $\tilde{\varOmega }$ lies in the positive quadrant and system (2.4) is persistent. Since only the closed orbits and the equilibria from the omega limit set of the solutions are on the boundary of $R^{3}_{+}$ and system (2.4) is dissipative. Thus, the system (2.4) is uniformly persistent by the main theorem of Butler et al. (1986).