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Stability and Hopf bifurcation for a ratio-dependent predator-prey system with stage structure and time delay

Advances in Difference Equations20152015:255

https://doi.org/10.1186/s13662-015-0548-x

  • Received: 7 April 2015
  • Accepted: 17 June 2015
  • Published:

Abstract

A ratio-dependent predator-prey system with time delay due to the gestation of the predator and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of the predator-extinction equilibrium and the coexistence equilibrium of the system are discussed, respectively. Further, the existence of Hopf bifurcation at the coexistence equilibrium is also studied. By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the analytical results.

Keywords

  • predator-prey system
  • stage structure
  • time delay
  • stability
  • Hopf bifurcation

MSC

  • 34K18
  • 34K20
  • 34K60
  • 92D25

1 Introduction

Since the pioneering works of Arditi et al. [1, 2], ratio-dependent predator-prey models have received much attention from scientists (see, for example, [37]). In [4], Kuang and Beretta studied the following ratio-dependent predator-prey model with the Michaelis-Menten type functional response
$$ \left \{ \textstyle\begin{array}{l} \dot{x}(t)= {rx(t) (1-\frac{x(t)}{K} )-\frac {a_{1}x(t)y(t)}{x(t)+my(t)}}, \\ \dot{y}(t)= {\frac{a_{2}x(t)y(t)}{x(t)+my(t)}} -dy(t), \end{array}\displaystyle \right . $$
(1.1)
where \(x(t)\) and \(y(t)\) are the densities of the prey and the predator population at time t, respectively. The parameters r, K, \(a_{1}\), \(a_{2}\), d and m are positive constants representing the prey intrinsic growth rate, carrying capacity, capturing rate, conversion rate, the predator death rate and half capturing saturation constant, respectively. System (1.1) was systematically studied by Kuang and Beretta [4], and the global stability of boundary equilibria, the positive equilibrium, and permanence of the system were discussed. In [3], Beretta and Kuang incorporated a time delay due to the gestation of the predator into system (1.1). Sufficient conditions were derived for the global stability of positive equilibrium of the delayed system. In [6], Xu and Ma incorporated stage structure for predator and time delay due to the gestation of the predator into system (1.1). Sufficient conditions were obtained for the global stability of the boundary equilibrium and the global attractivity of the positive equilibrium of the stage-structured system. Stage-structured models have received great attention in recent years (for example, [610]). Most of the researchers consider models with stage structure only for one species. However, it is of importance to discuss the effects of stage structure for both the predator and the prey species.
Based on the above discussions, in this paper, we incorporate stage structure for both the predator and the prey into system (1.1) and discuss the effects of time delay due to the gestation of the predator on the global dynamics of the model. To this end, we study the following delay differential system:
$$ \left \{ \textstyle\begin{array}{l} \dot{x}_{1}(t)=rx_{2}(t)-(r_{1}+d_{1})x_{1}(t)-ax_{1}^{2}(t)- {\frac {a_{1}x_{1}(t)y_{2}(t)}{x_{1}(t)+my_{2}(t)}}, \\ \dot{x}_{2}(t)=r_{1}x_{1}(t)-d_{2}x_{2}(t), \\ \dot{y}_{1}(t)= {\frac{a_{2}x_{1}(t-\tau)y_{2}(t-\tau)}{x_{1}(t-\tau)+my_{2}(t-\tau )}} -(r_{2}+d_{3})y_{1}(t), \\ \dot{y}_{2}(t)= r_{2}y_{1}(t)-d_{4}y_{2}(t), \end{array}\displaystyle \right . $$
(1.2)
where \(x_{1}(t)\), \(x_{2}(t)\), \(y_{1}(t)\) and \(y_{2}(t)\) represent the densities of the immature and the mature prey, the immature and the mature predator at time t, respectively. The parameters a, \(a_{1}\), \(a_{2}\), \(d_{1}\), \(d_{2}\), \(d_{3}\), \(d_{4}\), r, \(r_{1}\), \(r_{2}\) and m are positive constants, in which r is the birth rate of the prey; a is the intra-specific competition rate of the immature prey; \(d_{1}\) and \(d_{2}\) are the death rates of the immature prey and the mature prey, respectively; \(r_{1}\) and \(r_{2}\) are the transformation rates from the immature individuals to mature individuals for the prey and the predator, respectively; \(a_{1}\) is the capturing rate of the predator, \(a_{2}/a_{1}\) is the conversion rate of nutrients into the reproduction of the predator; \(d_{3}\) and \(d_{4}\) are the death rates of the immature predator and the mature predator, respectively; m is the half capturing saturation constant. \(\tau\geq0\) is a constant delay due to the gestation of the predator. We note that it is assumed in (1.2) that only mature predators capture immature prey and have the ability to reproduce.
The initial conditions for system (1.2) take the form
$$ \begin{aligned} &x_{1}(\theta)= \phi_{1}(\theta)\geq0,\qquad x_{2}(\theta)= \phi_{2}(\theta)\geq0, \\ &y_{1}(\theta)= \varphi_{1}(\theta)\geq0,\qquad y_{2}(\theta)= \varphi_{2}(\theta)\geq0,\quad \theta\in[-\tau,0), \\ &\phi_{1}(0)>0, \qquad \phi_{2}(0)>0,\qquad \varphi_{1}(0)>0, \\ &\varphi_{2}(0)>0, \quad \bigl( \phi_{1}(\theta ),\phi_{2}(\theta),\varphi_{1}( \theta),\varphi_{2}(\theta)\bigr)\in{C\bigl([-\tau,0], R_{+0}^{4}\bigr)}, \end{aligned} $$
(1.3)
where \(R_{+0}^{4}=\{(x_{1}, x_{2}, x_{3}, x_{4}): {x_{i}\geq{0}},i=1,2,3,4\}\).

It is well known by the fundamental theory of functional differential equations [11] that system (1.2) has a unique solution \((x_{1}(t), x_{2}(t), y_{1}(t), y_{2}(t))\) satisfying initial conditions (1.3).

The organization of this paper is as follows. In the next section, we investigate the local stability of the predator-extinction equilibrium and the coexistence equilibrium of system (1.2). Further, we study the existence of a Hopf bifurcation for system (1.2) at the coexistence equilibrium. In Section 3, by means of an iterative technique, sufficient conditions are derived for the global stability of the coexistence equilibrium of system (1.2). By comparison arguments, we discuss the global stability of the predator-extinction of system (1.2). Numerical simulations are carried out to illustrate the main results. A brief discussion is given in Section 4 to conclude this work.

2 Local stability and Hopf bifurcation

In this section, we discuss the local stability of equilibria and the existence of a Hopf bifurcation at the coexistence equilibrium of system (1.2).

It is easy to show that if \(rr_{1}>d_{2}(r_{1}+d_{1})\), system (1.2) admits a predator-extinction equilibrium \(E_{1}(x_{1}^{+}, x_{2}^{+}, 0, 0)\), where
$${x_{1}^{+}=\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}, \qquad x_{2}^{+}= \frac {r_{1}[rr_{1}-d_{2}(r_{1}+d_{1})]}{ad_{2}^{2}}}. $$
Further, if the following condition holds
  1. (H1)

    \({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m}>0}\),

     
then system (1.2) has a unique coexistence equilibrium \(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*},y_{2}^{*})\), where
$$\begin{aligned}& {x_{1}^{*}=\frac{a_{1}}{a} \biggl[\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}-\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m} \biggr], \qquad x_{2}^{*}=\frac{r_{1}}{d_{2}}x_{1}^{*}}, \\& {y_{1}^{*}=\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{mr_{2}(r_{2}+d_{3})}x_{1}^{*},\qquad y_{2}^{*}=\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})}x_{1}^{*}}. \end{aligned}$$
The characteristic equation of (1.2) at the equilibrium \(E_{1}(x_{1}^{+}, x_{2}^{+}, 0, 0)\) is of the form
$$\begin{aligned}& \bigl[\lambda^{2}+(r_{2}+d_{3}+d_{4}) \lambda+d_{4}(r_{2}+d_{3})-a_{2}r_{2}e^{-\lambda\tau } \bigr] \\& \quad {}\times\bigl[\lambda^{2}+\bigl(r_{1}+d_{1}+2ax_{1}^{+}+d_{2} \bigr)\lambda+rr_{1}-d_{2}(r_{1}+d_{1}) \bigr]=0. \end{aligned}$$
(2.1)
Note that \(rr_{1}>d_{2}(r_{1}+d_{1})\), it is easy to show that the equation
$$\lambda^{2}+\bigl(r_{1}+d_{1}+2ax_{1}^{+}+d_{2} \bigr)\lambda+rr_{1}-d_{2}(r_{1}+d_{1})=0 $$
always has two negative real roots. All other roots are given by the roots of equation
$$\lambda^{2}+(r_{2}+d_{3}+d_{4}) \lambda+d_{4}(r_{2}+d_{3})-a_{2}r_{2}e^{-\lambda\tau}=0. $$
Let \(f(\lambda)=\lambda^{2}+(r_{2}+d_{3}+d_{4})\lambda +d_{4}(r_{2}+d_{3})-a_{2}r_{2}e^{-\lambda\tau}\). If (H1) holds and λ is a real number, then we have
$$\begin{aligned}& f(0)=d_{4}(r_{2}+d_{3})-a_{2}r_{2}< 0, \\& \lim_{x\rightarrow+\infty}f(x)=+\infty. \end{aligned}$$
Hence, \(f(\lambda)=0\) has at least one positive real root. Therefore, the equilibrium \(E_{1}\) is unstable. If \(a_{2}r_{2}< d_{4}(r_{2}+d_{3})\), we have \(f(0)>0\) and \(f'(\lambda)=2\lambda+r_{2}+d_{3}+d_{4}+a_{2}r_{2}\tau{e^{-\lambda\tau}}\), then it is easily seen that all roots of \(f(\lambda)=0\) have only negative real parts, that is, the equilibrium \(E_{1}\) is stable. Therefore, if \(a_{2}r_{2}< d_{4}(r_{2}+d_{3})\), by Kuang and So ([12], Lemma B), we see that the equilibrium \(E_{1}\) is locally stable for all \(\tau>0\).
The characteristic equation of system (1.2) at the equilibrium \(E^{*}\) takes the form
$$ \lambda^{4}+p_{3}\lambda^{3}+p_{2} \lambda^{2}+p_{1}\lambda+p_{0}+ \bigl(q_{2}\lambda ^{2}+q_{1}\lambda+q_{0} \bigr)e^{-\lambda\tau}=0, $$
(2.2)
where
$$\begin{aligned}& p_{3}=\gamma+d_{2}+r_{2}+d_{3}+d_{4}, \\& p_{2}=d_{4}(\gamma+d_{2}+r_{2}+d_{3})+( \gamma+d_{2}) (r_{2}+d_{3})+\gamma{d_{2}} -rr_{1}, \\& p_{1}=d_{4}(\gamma+d_{2}) (r_{2}+d_{3})+(r_{2}+d_{3}+d_{4}) (\gamma{d_{2}} -rr_{1}), \\& p_{0}=d_{4}(r_{2}+d_{3}) ( \gamma{d_{2}}-rr_{1}), \\& q_{2}=-a_{2}r_{2}\beta^{2},\qquad q_{1}=-a_{2}r_{2}\beta^{2} \bigl(2ax_{1}^{*}+r_{1}+d_{1}+d_{2} \bigr), \\& q_{0}=a_{2}r_{2}\beta^{2} \bigl(rr_{1}+d_{2}a_{1}m\alpha^{2}- \gamma{d_{2}}\bigr), \\& \alpha=\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m}, \qquad \beta=\frac {d_{4}(r_{2}+d_{3})}{a_{2}r_{2}}, \qquad \gamma=r_{1}+d_{1}+2ax_{1}^{*}+a_{1}m \alpha^{2}. \end{aligned}$$
When \(\tau=0\), equation (2.2) becomes
$$ \lambda^{4}+p_{3}\lambda^{3}+(p_{2}+q_{2}) \lambda+p_{0}+q_{0}=0. $$
(2.3)
Clearly \(p_{3}>0\). If the following condition holds
  1. (H2)

    \({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m} (1+\frac{d_{4}(r_{2}+d_{3})}{a_{2}r_{2}} )>0}\),

     
then we have
$$\begin{aligned}& p_{3}(p_{2}+q_{2})-(p_{1}+q_{1})>0, \\& (p_{1}+q_{1})\bigl[p_{3}(p_{2}+q_{2})-(p_{1}+q_{1}) \bigr]>p_{3}^{2}(p_{0}+q_{0}). \end{aligned}$$
Hence, the coexistence equilibrium \(E^{*}\) of system (1.2) is locally asymptotically stable.
If (\(\omega>0\)) is a solution of (2.2), separating real and imaginary parts, we have the following:
$$ \begin{aligned} &\bigl(q_{2} \omega^{2}-q_{0}\bigr)\sin\omega\tau+q_{1}\omega \cos\omega\tau=p_{3}\omega ^{3}-p_{1}\omega, \\ &\bigl(q_{2}\omega^{2}-q_{0}\bigr)\cos\omega \tau-q_{1}\omega\sin\omega\tau=\omega ^{4}-p_{2} \omega^{2}+p_{0}. \end{aligned} $$
(2.4)
Squaring and adding the two equations of (2.4), it follows that
$$ \omega^{8}+h_{3}\omega^{6}+h_{2} \omega^{4}+h_{1}\omega^{2}+h_{0}=0, $$
(2.5)
where
$$\begin{aligned}& h_{0}=p_{0}^{2}-q_{0}^{2},\qquad h_{1}=p_{1}^{2}-2p_{0}p_{2}+2q_{0}q_{2}-q_{1}^{2}, \\& h_{2}=p_{2}^{2}+2p_{0}-2p_{1}p_{3}-q_{2}^{2},\qquad h_{3}=p_{3}^{2}-2p_{2}. \end{aligned}$$
Let (H1) hold, it is easy to show that
$$\begin{aligned}& h_{3}=\gamma^{2}+d_{2}^{2}+(r_{2}+d_{3})^{2}+d_{4}^{2}+2rr_{1}>0, \\& h_{2}=\bigl(d_{2}^{2}+\gamma^{2}+2rr_{1} \bigr)\bigl[(r_{2}+d_{3})^{2}+d_{4}^{2} \bigr]+(\gamma {d_{2}}-rr_{1})^{2}+d_{4}^{2}(r_{2}+d_{3})^{2}- \bigl(a_{2}r_{2}\beta^{2}\bigr)^{2}>0, \\& h_{1}=(\gamma {d_{2}}-rr_{1})^{2} \bigl[d_{4}^{2}+(r_{2}+d_{3})^{2} \bigr] \\& \hphantom{h_{1}=}{}+\bigl[d_{4}^{2}(r_{2}+d_{3})^{2}- \bigl(a_{2}r_{2}\beta ^{2}\bigr)^{2}\bigr] \bigl(d_{2}^{2}+\gamma^{2}+2rr_{1}\bigr) \\& \hphantom{h_{1}=}{}+2\bigl(a_{2}r_{2}\beta^{2} \bigr)^{2}a_{1}m\alpha^{2}\bigl(r_{1}+d_{1}+2ax_{1}^{*} \bigr)>0. \end{aligned}$$
If the following condition holds
  1. (H3)

    \({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m} (1+2\frac {d_{4}(r_{2}+d_{3})}{a_{2}r_{2}+d_{4}(r_{2}+d_{3})} )>0}\),

     
then \(h_{0}>0\). Hence, the coexistence equilibrium \(E^{*}\) of system (1.2) is locally asymptotically stable for all \(\tau>0\). If the following condition holds
  1. (H4)

    \({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}<\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m} (1+2\frac {d_{4}(r_{2}+d_{3})}{a_{2}r_{2}+d_{4}(r_{2}+d_{3})} )}\),

     
then \(h_{0}<0\). Hence, equation (2.5) has a unique positive root \(\omega_{0}\), that is, the characteristic equation (2.2) admits a pair of purely imaginary roots of the form \(\pm i\omega_{0}\). From (2.4), we see that \(\tau_{k}\) corresponding to \(\omega_{0}\) is
$$\begin{aligned} \tau_{k} =&\frac{1}{\omega_{0}}\arccos\frac{q_{1}\omega_{0}^{2}(p_{3}\omega _{0}^{2}-p_{1})+(q_{2}\omega_{0}^{2}-q_{0}) (\omega_{0}^{4}-p_{2}\omega_{0}^{2}+p_{0})}{(q_{1}\omega_{0})^{2}+(q_{2}\omega _{0}^{2}-q_{0})^{2}} \\ &{}+ \frac{2k\pi}{\omega_{0}},\quad k=0,1,2,\ldots. \end{aligned}$$
(2.6)
Hence, by the general theory on characteristic equation of delay differential equation from [13], if (H2) and (H4) hold, \(E^{*}\) remains stable for \(\tau<\tau_{0}\).
We now claim that
$${\biggl.\frac{d(\operatorname{Re}(\lambda))}{d\tau}\biggr|_{\tau=\tau_{0}}}>0. $$
This will signify that there exists at least one eigenvalue with positive real part for \(\tau>\tau_{0}\). Moreover, the conditions for the existence of a Hopf bifurcation [14] are then satisfied. To this end, differentiating equation (2.2) with respect to τ, it follows that
$$\begin{aligned}& \bigl[\bigl(4\lambda^{3}+3p_{3}\lambda^{2}+2p_{2} \lambda+p_{1}\bigr) +(2q_{2}\lambda +q_{1})e^{-\lambda\tau} - \tau\bigl(q_{2}\lambda^{2}+q_{1} \lambda+q_{0}\bigr)e^{-\lambda\tau}\bigr]\frac{d\lambda}{d\tau } \\& \quad =\lambda \bigl(q_{2}\lambda^{2}+q_{1}\lambda+q_{0} \bigr)e^{-\lambda\tau}. \end{aligned}$$
From this equation, we can obtain
$$\biggl(\frac{d\lambda}{d\tau} \biggr)^{-1}= {\frac{4\lambda^{3}+3p_{3}\lambda ^{2}+2p_{2}\lambda+p_{1}}{-\lambda(\lambda^{4}+p_{3}\lambda^{3}+p_{2}\lambda ^{2}+p_{1}\lambda+p_{0})}+ \frac{2q_{2}\lambda+q_{1}}{\lambda(q_{2}\lambda^{2}+q_{1}\lambda+q_{0})}-\frac{\tau }{\lambda}}. $$
Hence, we derive that
$$\begin{aligned}& \operatorname{sign} \biggl\{ \frac{d(\operatorname{Re}\lambda)}{d\tau} \biggr\} _{\lambda=i\omega _{0}} \\& \quad = \operatorname{sign} \biggl\{ \operatorname{Re} \biggl(\frac{d\lambda}{d\tau} \biggr)^{-1} \biggr\} _{\lambda=i\omega_{0}} \\& \quad = \operatorname{sign} \biggl\{ \operatorname{Re} \biggl[\frac{4\lambda^{3}+3p_{3}\lambda^{2}+2p_{2}\lambda +p_{1}}{-\lambda(\lambda^{4}+p_{3}\lambda^{3}+p_{2}\lambda^{2}+p_{1}\lambda+p_{0})}+ \frac{2q_{2}\lambda+q_{1}}{\lambda(q_{2}\lambda^{2}+q_{1}\lambda+q_{0})} \biggr]_{\lambda=i\omega_{0}} \biggr\} \\& \quad = \operatorname{sign}\biggl\{ \frac{2(2\omega_{0}^{2}-p_{2})(\omega_{0}^{4}-p_{2}\omega _{0}^{4}+p_{0})+(3p_{3}\omega_{0}^{2}-p_{1})(p_{3}\omega_{0}^{2}-p_{1})}{(\omega_{0}^{4}-p_{2}\omega _{0}^{2}+p_{0})^{2}+\omega_{0}^{2}(p_{3}\omega_{0}^{2}-p_{1})^{2}} \\& \qquad {}+\frac{-q_{1}^{2}-2q_{2}(q_{2}\omega_{0}^{2}-q_{0})}{(q_{2}\omega _{0}^{2}-q_{0})^{2}+(q_{1}\omega_{0})^{2}}\biggr\} . \end{aligned}$$
Note that \((\omega_{0}^{4}-p_{2}\omega_{0}^{2}+p_{0})^{2}+\omega_{0}^{2}(p_{3}\omega _{0}^{2}-p_{1})^{2}=(q_{2}\omega_{0}^{2}-q_{0})^{2}+(q_{1}\omega_{0})^{2}\), then
$$ \operatorname{sign} \biggl\{ \frac{d(\operatorname{Re}\lambda)}{d\tau} \biggr\} _{\lambda=i\omega _{0}}= \operatorname{sign} \biggl\{ \frac{4\omega_{0}^{6}+3h_{3}\omega_{0}^{4}+2h_{2}\omega _{0}^{2}+h_{1}}{(q_{2}\omega_{0}^{2}-q_{0})^{2}+(q_{1}\omega_{0})^{2}} \biggr\} . $$
Accordingly, if (H1) holds, then we have that
$$ \operatorname{sign} \biggl\{ \frac{d(\operatorname{Re}\lambda)}{d\tau} \biggr\} _{\lambda=i\omega_{0}}>0. $$
Therefore, the transversal condition holds and a Hopf bifurcation occurs at \(\tau=\tau_{0}\), \(\omega=\omega_{0}\).

We therefore obtain the following results.

Theorem 2.1

For system (1.2), we have the following:
  1. (i)

    Let \(rr_{1}>d_{2}(r_{1}+d_{1})\), if \(a_{2}r_{2}< d_{4}(r_{2}+d_{3})\), then the predator-extinction equilibrium \(E_{1}(x_{1}^{+}, x_{2}^{+}, 0, 0)\) is locally asymptotically stable; if \(a_{2}r_{2}>d_{4}(r_{2}+d_{3})\), then the equilibrium \(E_{1}\) is unstable.

     
  2. (ii)

    If (H3) holds, then the positive equilibrium \(E^{*}\) is locally asymptotically stable for all \(\tau\geq0\).

     
  3. (iii)

    If (H2) and (H4) hold, then there exists a positive number \(\tau_{0}\) such that the coexistence equilibrium \(E^{*}\) is locally asymptotically stable if \(0<\tau<\tau_{0}\) and unstable if \(\tau>\tau_{0}\). Further, system (1.2) undergoes a Hopf bifurcation at \(E^{*}\) when \(\tau=\tau_{0}\).

     

3 Global stability

In this section, we are concerned with the global stability of the coexistence equilibrium \(E^{*}\) and the predator-extinction equilibrium \(E_{1}\) of system (1.2), respectively.

Theorem 3.1

Let (H3) hold, then the coexistence equilibrium \(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\) of system (1.2) is globally stable provided that
  1. (H5)

    \(a_{2}r_{2}<2d_{4}(r_{2}+d_{3})\), \({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac{1}{m}}\).

     

Proof

Let \((x_{1}(t), x_{2}(t), y_{1}(t), y_{2}(t))\) be any positive solution of system (1.2) with initial conditions (1.3). Let
$$\begin{aligned}& U_{x_{i}}=\limsup_{t\rightarrow+\infty}{x_{i}(t)},\qquad L_{x_{i}}=\liminf_{t\rightarrow+\infty}{x_{i}(t)}, \\& U_{y_{i}}=\limsup_{t\rightarrow+\infty}{y_{i}(t)},\qquad L_{y_{i}}=\liminf_{t\rightarrow+\infty}{y_{i}(t)}\quad (i=1,2). \end{aligned}$$
We now claim that \(U_{x_{i}}=L_{x_{i}}=x_{i}^{*}\), \(U_{y_{i}}=L_{y_{i}}=y_{i}^{*}\). The strategy of the proof is to use an iteration technique.
We derive from the first and the second equations of system (1.2) that
$$\begin{aligned} \begin{aligned} &\dot{x}_{1}(t)\leq{r}x_{2}(t)-(r_{1}+d_{1})x_{1}(t)-ax_{1}^{2}(t), \\ &\dot{x}_{2}(t)=r_{1}x_{1}(t)-d_{2}x_{2}(t). \end{aligned} \end{aligned}$$
Consider the following auxiliary equations:
$$ \begin{aligned} &\dot{z}_{1}(t)=rz_{2}(t)-(r_{1}+d_{1})z_{1}(t)-az_{1}^{2}(t), \\ &\dot{z}_{2}(t)=r_{1}z_{1}(t)-d_{2}z_{2}(t). \end{aligned} $$
(3.1)
Since \(rr_{1}>d_{2}(r_{1}+d_{1})\) holds, by Lemma 2.2 of [10], it follows from (3.1) that
$$\lim_{t\rightarrow+\infty}{z}_{1}(t)={ {\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}}},\qquad \lim_{t\rightarrow+\infty}{z}_{2}(t)={ {\frac{r_{1}[rr_{1}-d_{2}(r_{1}+d_{1})]}{ad_{2}^{2}}}}. $$
A comparison argument shows that
$$ \begin{aligned} &U_{x_{1}}= {\limsup _{t\rightarrow+\infty}{x_{1}(t)}\leq\frac {rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}}:=M_{1}^{x_{1}}, \\ &U_{x_{2}}= {\limsup_{t\rightarrow+\infty}{x_{2}(t)}\leq \frac {r_{1}[rr_{1}-d_{2}(r_{1}+d_{1})]}{ad_{2}^{2}}}:=M_{1}^{x_{2}}. \end{aligned} $$
(3.2)
Hence, for \(\varepsilon>0\) sufficiently small, there exists \(T_{1}>0\) such that if \(t>T_{1}\), \(x_{1}(t)\leq{M_{1}^{x_{1}}}+\varepsilon\).
For \(\varepsilon>0\) sufficiently small, we derive from the third and the fourth equations of system (1.2) that for \(t>T_{1}+\tau\),
$$\begin{aligned}& \dot{y}_{1}(t) {\leq{\frac{a_{2}(M_{1}^{x_{1}}+\varepsilon)y_{2}(t-\tau )}{M_{1}^{x_{1}}+\varepsilon+my_{2}(t-\tau)}} -(d_{3}+r_{2})y_{1}(t)}, \\& \dot{y}_{2}(t)= r_{2}y_{1}(t)-d_{4}y_{2}(t). \end{aligned}$$
Consider the following auxiliary equations:
$$ \begin{aligned} &\dot{u}_{1}(t) {={ \frac{a_{2}(M_{1}^{x_{1}}+\varepsilon)u_{2}(t-\tau )}{M_{1}^{x_{1}}+\varepsilon+mu_{2}(t-\tau)}} -(d_{3}+r_{2})u_{1}(t)}, \\ &\dot{u}_{2}(t)= r_{2}u_{1}(t)-d_{4}u_{2}(t). \end{aligned} $$
(3.3)
Since (H3) holds, by Lemma 2.4 of [6], it follows from (3.3) that
$$\begin{aligned}& \lim_{t\rightarrow+\infty}{u}_{1}(t)=\frac{(M_{1}^{x_{1}}+\varepsilon )[a_{2}r_{2}-d_{4}(r_{2}+d_{3})]}{mr_{2}(r_{2}+d_{3})}, \\& \lim_{t\rightarrow+\infty}{u}_{2}(t)=\frac{(M_{1}^{x_{1}}+\varepsilon )[a_{2}r_{2}-d_{4}(r_{2}+d_{3})]}{md_{4}(r_{2}+d_{3})}. \end{aligned}$$
By comparison, we derive that
$$\begin{aligned}& U_{y_{1}}= {\limsup_{t\rightarrow+\infty}{y_{1}(t)}\leq \frac {(M_{1}^{x_{1}}+\varepsilon)[a_{2}r_{2}-d_{4}(r_{2}+d_{3})]}{mr_{2}(r_{2}+d_{3})}}, \\& U_{y_{2}}= {\limsup_{t\rightarrow+\infty}{y_{2}(t)}\leq \frac {(M_{1}^{x_{1}}+\varepsilon)[a_{2}r_{2}-d_{4}(r_{2}+d_{3})]}{md_{4}(r_{2}+d_{3})}}. \end{aligned}$$
Since these inequalities are true for arbitrary \(\varepsilon>0\), it follows that \(U_{y_{1}}\leq{M_{1}^{y_{1}}}\), \(U_{y_{2}}\leq{M_{1}^{y_{2}}}\), where
$$ M_{1}^{y_{1}}= {\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{mr_{2}(r_{2}+d_{3})}M_{1}^{x_{1}}}, \qquad M_{1}^{y_{2}}= {\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})}M_{1}^{x_{1}}}. $$
(3.4)
Therefore, for \(\varepsilon>0\) sufficiently small, there is \(T_{2}\geq {T_{1}+\tau}\) such that if \(t>T_{2}\), \(y_{2}(t)\leq{M_{1}^{y_{2}}+\varepsilon}\).
For \(\varepsilon>0\) sufficiently small, it follows from the first and the second equations of system (1.2) that for \(t>T_{2}\),
$$ \begin{aligned} &\dot{x}_{1}(t)\geq rx_{2}(t)-(r_{1}+d_{1})x_{1}(t)-ax_{1}^{2}(t)- \frac {a_{1}}{m}x_{1}(t), \\ &\dot{x}_{2}(t)=r_{1}x_{1}(t)-d_{2}x_{2}(t). \end{aligned} $$
(3.5)
Since (H5) holds, by Lemma 2.2 of [10] and a comparison argument, it follows from (3.5) that
$$\begin{aligned}& V_{x_{1}}= {\liminf_{t\rightarrow+\infty}{x_{1}(t)}\geq \frac {m[rr_{1}-d_{2}(d_{1}+r_{1})]-a_{1}d_{2}}{amd_{2}}}:=N_{1}^{x_{1}}, \\& V_{x_{2}}= {\liminf_{t\rightarrow+\infty}{x_{2}(t)}\geq \frac {mr_{1}[rr_{1}-d_{2}(d_{1}+r_{1})]-a_{1}r_{1}d_{2}}{amd_{2}^{2}}}:=N_{1}^{x_{2}}. \end{aligned}$$
Therefore, for \(\varepsilon>0\) sufficiently small, there is \(T_{3}\geq {T_{2}}\) such that if \(t>T_{3}\), \(x_{1}(t)\geq{N_{1}^{x_{1}}}-\varepsilon\).
For \(\varepsilon>0\) sufficiently small, it follows from the third and the fourth equations of system (1.2) that for \(t>T_{3}+\tau\),
$$ \begin{aligned} &\dot{y}_{1}(t)\geq{ \frac{a_{2}(N_{1}^{x_{1}}-\varepsilon)y_{2}(t-\tau )}{N_{1}^{x_{1}}-\varepsilon+my_{2}(t-\tau)} -(r_{2}+d_{3})y_{1}(t)}, \\ &\dot{y}_{2}(t)= r_{2}y_{1}(t)-d_{4}y_{2}(t). \end{aligned} $$
(3.6)
Since \(a_{2}r_{2}>d_{4}(r_{2}+d_{3})\) holds, by Lemma 2.4 of [6] and a comparison argument, it follows from (3.6) that
$$\begin{aligned}& V_{y_{1}}= {\liminf_{t\rightarrow+\infty}{y_{1}(t)}\geq \frac {[a_{2}r_{2}-d_{4}(d_{3}+r_{2})](N_{1}^{x_{1}}-\varepsilon)}{mr_{2}(d_{3}+r_{2})}}, \\& V_{y_{2}}= {\liminf_{t\rightarrow+\infty}{y_{2}(t)}\geq \frac {[a_{2}r_{2}-d_{4}(d_{3}+r_{2})](N_{1}^{x_{1}}-\varepsilon)}{md_{4}(d_{3}+r_{2})}}. \end{aligned}$$
Since these two inequalities hold for arbitrary \(\varepsilon>0\) sufficiently small, we conclude that \(V_{y_{1}}\geq{N_{1}^{y_{1}}}\), \(V_{y_{2}}\geq{N_{1}^{y_{2}}}\), where
$$N_{1}^{y_{1}}= {\frac{a_{2}r_{2}-d_{4}(d_{3}+r_{2})}{mr_{2}(d_{3}+r_{2})}N_{1}^{x_{1}}}, \qquad N_{1}^{y_{2}}= {\frac{a_{2}r_{2}-d_{4}(d_{3}+r_{2})}{md_{4}(d_{3}+r_{2})}N_{1}^{x_{1}}}. $$
Therefore, for \(\varepsilon>0\) sufficiently small, there is \(T_{4}\geq {T_{3}}+\tau\) such that if \(t>T_{4}\), \(y_{2}(t)\geq{N_{1}^{y_{2}}}-\varepsilon\).
For \(\varepsilon>0\) sufficiently small, it follows from the first and the second equations of system (1.2) that for \(t>T_{4}\),
$$ \begin{aligned} &\dot{x}_{1}(t) \leqslant{rx_{2}(t)-(r_{1}+d_{1})x_{1}(t)-ax_{1}^{2}(t)- \frac {a_{1}(N_{1}^{y_{2}}-\varepsilon)}{M_{1}^{x_{1}}+\varepsilon +m(N_{1}^{y_{2}}-\varepsilon)}x_{1}(t)}, \\ &\dot{x}_{2}(t)={r_{1}}x_{1}(t)-d_{2}x_{2}(t). \end{aligned} $$
(3.7)
Since (H3) holds, by Lemma 2.2 of [10] and a comparison argument, for arbitrary \(\varepsilon>0\) sufficiently small, it follows from (3.7) that
$$\begin{aligned}& U_{x_{1}}= {\limsup_{t\rightarrow+\infty}{x_{1}(t)}\leq \frac {rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}-\frac {a_{1}N_{1}^{y_{2}}}{a(M_{1}^{x_{1}}+mN_{1}^{y_{2}})}:=M_{2}^{x_{1}}}, \\& U_{x_{2}}= {\limsup_{t\rightarrow+\infty}{x_{2}(t)}\leq \frac {r_{1}}{d_{2}}M_{2}^{x_{1}}:=M_{2}^{x_{2}}}. \end{aligned}$$
Hence, for \(\varepsilon>0\) sufficiently small, there is \(T_{5}\geq{T}_{4}\) such that if \(t>T_{5}\), \(x_{1}(t)\leq{M_{2}^{x_{1}}}+\varepsilon\). We therefore obtain from the third and the fourth equations of system (1.2) that for \(t>T_{5}+\tau\),
$$ \begin{aligned} &\dot{y}_{1}(t) {\leq \frac{a_{2}(M_{2}^{x_{1}}+\varepsilon)y_{2}(t-\tau) }{M_{2}^{x_{1}}+\varepsilon+my_{2}(t-\tau)}-(r_{2}+d_{3})y_{1}(t)}, \\ &\dot{y}_{2}(t)= r_{2}y_{1}(t)-d_{4}y_{2}(t). \end{aligned} $$
(3.8)
Since \(a_{2}r_{2}>d_{4}(r_{2}+d_{3})\) holds, by Lemma 2.4 of [6] and a comparison argument, for arbitrary \(\varepsilon>0\) sufficiently small, it follows from (3.8) that
$$\begin{aligned}& U_{y_{1}}= {\limsup_{t\rightarrow+\infty}{y_{1}(t)}\leq \frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{mr_{2}(r_{2}+d_{3})}M_{2}^{x_{1}}:=M_{2}^{y_{1}}}, \\& U_{y_{2}}= {\limsup_{t\rightarrow+\infty}{y_{2}(t)}\leq \frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})}M_{2}^{x_{1}}:=M_{2}^{y_{2}}}. \end{aligned}$$
Hence, for \(\varepsilon>0\) sufficiently small, there is \(T_{6}\geq {T}_{5}+\tau\) such that if \(t>T_{6}\), \(y_{2}(t)\leq{M_{2}^{y_{2}}}+\varepsilon\).
Again, for \(\varepsilon>0\) sufficiently small, it follows from the first and the second equations of system (1.2) that for \(t>T_{6}\),
$$ \begin{aligned} &\dot{x}_{1}(t) \geq rx_{2}(t)-(r_{1}+d_{1})x_{1}(t)-ax_{1}^{2}(t)- \frac {a_{1}(M_{2}^{y_{2}}+\varepsilon)}{N_{1}^{x_{1}}-\varepsilon +m(M_{2}^{y_{2}}+\varepsilon)}x_{1}(t), \\ &\dot{x}_{2}(t)=r_{1}x_{1}(t)-d_{2}x_{2}(t). \end{aligned} $$
(3.9)
Since (H3) holds, by Lemma 2.2 of [10] and a comparison argument, for arbitrary \(\varepsilon>0\) sufficiently small, it follows from (3.9) that
$$\begin{aligned}& V_{x_{1}}= {\liminf_{t\rightarrow+\infty}{x_{1}(t)}\geq \frac {rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}-\frac {a_{1}M_{2}^{y_{2}}}{a(N_{1}^{x_{1}}+mM_{2}^{y_{2}})}:=N_{2}^{x_{1}}}, \\& V_{x_{2}}= {\liminf_{t\rightarrow+\infty}{x_{2}(t)}\geq \frac {r_{1}}{d_{2}}N_{2}^{x_{1}}:=N_{2}^{x_{2}}}. \end{aligned}$$
Therefore, for \(\varepsilon>0\) sufficiently small, there is \(T_{7}\geq {T}_{6}\) such that if \(t>T_{7}\), \(x_{1}(t)\geq{N_{2}^{x_{1}}}-\varepsilon\).
For \(\varepsilon>0\) sufficiently small, we derive from the third and the fourth equations of system (1.2) that for \(t>T_{7}+\tau\),
$$ \begin{aligned} &\dot{y}_{1}(t)\geq{ \frac{a_{2}(N_{2}^{x_{1}}-\varepsilon)y_{2}(t-\tau )}{N_{2}^{x_{2}}-\varepsilon+my_{2}(t-\tau)} -(d_{3}+r_{2})y_{1}(t)}, \\ &\dot{y}_{2}(t)= r_{2}y_{1}(t)-d_{4}y_{2}(t). \end{aligned} $$
(3.10)
Since \(a_{2}r_{2}>d_{4}(d_{3}+r_{2})\) holds, by Lemma 2.4 of [6] and a comparison argument, for arbitrary \(\varepsilon>0\) sufficiently small, it follows from (3.10) that
$$\begin{aligned}& V_{y_{1}}= {\liminf_{t\rightarrow+\infty}{y_{1}(t)}\geq \frac {a_{2}r_{2}-d_{4}(d_{3}+r_{2})}{mr_{2}(d_{3}+r_{2})}N_{2}^{x_{1}}:=N_{2}^{y_{1}}}, \\& V_{y_{2}}= {\liminf_{t\rightarrow+\infty}{y_{2}(t)}\geq \frac {a_{2}r_{2}-d_{4}(d_{3}+r_{2})}{md_{4}(d_{3}+r_{2})}N_{2}^{x_{1}}:=N_{2}^{y_{2}}}. \end{aligned}$$
Continuing this process, we derive eight sequences \(M_{k}^{x_{1}}\), \(M_{k}^{x_{2}}\), \(M_{k}^{y_{1}}\), \(M_{k}^{y_{2}}\), \(N_{k}^{x_{1}}\), \(N_{k}^{x_{2}}\), \(N_{k}^{y_{1}}\), \(N_{k}^{y_{2}}\) (\(k=1,2,\ldots\)) such that for \(k\geq2\),
$$ \begin{aligned} &M_{k}^{x_{1}}= { \frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}-\frac {a_{1}N_{k-1}^{y_{2}}}{a(M_{k-1}^{x_{1}}+mN_{k-1}^{y_{2}})}},\qquad M_{k}^{x_{2}}= { \frac{r_{1}}{d_{2}}M_{k}^{x_{1}}}, \\ &M_{k}^{y_{1}}= {\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{mr_{2}(r_{2}+d_{3})}M_{k}^{x_{1}}}, \qquad M_{k}^{y_{2}}= {\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})}M_{k}^{x_{1}}}, \\ &N_{k}^{x_{1}}= {\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}-\frac {a_{1}M_{k}^{y_{2}}}{a(N_{k-1}^{x_{1}}+mM_{k}^{y_{2}})}},\qquad N_{k}^{x_{2}}= {\frac{r_{1}}{d_{2}}N_{k}^{x_{1}}}, \\ &N_{k}^{y_{1}}= {\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{mr_{2}(r_{2}+d_{3})}N_{k}^{x_{1}}}, \qquad N_{k}^{y_{2}}= {\frac{a_{2}r_{2}-d_{4}(d_{3}+r_{2})}{md_{4}(r_{2}+d_{3})}N_{k}^{x_{1}}}. \end{aligned} $$
(3.11)
It is readily seen that
$$ N_{k}^{x_{i}}\leq{V_{x_{i}}} \leq{U_{x_{i}}}\leq{M_{k}^{x_{i}}},\qquad N_{k}^{y_{i}}\leq {V_{y_{i}}}\leq{U_{y_{i}}} \leq{M_{k}^{y_{i}}}\quad (i=1,2). $$
(3.12)
It is easy to show that the sequences \(M_{k}^{x_{i}}\), \(M_{k}^{y_{i}}\) are nonincreasing and the sequences \(N_{k}^{x_{i}}\), \(N_{k}^{y_{i}}\) are nondecreasing. Hence, the limit of each sequence in \(M_{k}^{x_{i}}\), \(M_{k}^{y_{i}}\), \(N_{k}^{x_{i}}\), \(N_{k}^{y_{i}}\) exists. Denote
$$\bar{x}_{i}= {\lim_{t\rightarrow+\infty}M_{n}^{x_{i}}}, \qquad \underline{x}_{i}= {\lim_{t\rightarrow+\infty}N_{n}^{x_{i}}}, \qquad \bar{y}_{i}= {\lim_{t\rightarrow+\infty}M_{n}^{y_{i}}}, \qquad \underline{y}_{i}= {\lim_{t\rightarrow+\infty}N_{n}^{y_{i}}}, \quad i=1,2. $$
We therefore obtain from (3.11) that
$$ \begin{aligned} &\bar{x}_{1}= \frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}-\frac{a_{1}\underline {y}_{2}}{a(\bar{x}_{1}+m\underline{y}_{2})}, \qquad \bar{x}_{2}= \frac{r_{1}}{d_{2}}\bar{x}_{1}, \\ &\bar{y}_{1}= \frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{mr_{2}(r_{2}+d_{3})}\bar{x}_{1},\qquad \bar{y}_{2}= \frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})}\bar{x}_{1}, \\ &\underline{x}_{1}= \frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}-\frac{a_{1}\bar {y}_{2}}{a(\underline{x}_{1}+m\bar{y}_{2})}, \qquad \underline{x}_{2}= \frac{r_{1}}{d_{2}}\underline{x}_{1}, \\ &\underline{y}_{1}= \frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{mr_{2}(r_{2}+d_{3})}\underline{x}_{1}, \qquad \underline{y}_{2}= \frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})}\underline{x}_{1}. \end{aligned} $$
(3.13)
To complete the proof, it is sufficient to prove that \(\bar {x}_{i}=\underline{x}_{i}\), \(\bar{y}_{i}=\underline{y}_{i}\) (\(i=1,2\)). It follows from (3.13) that
$$\begin{aligned}& ad_{2} \bar{x}_{1}^{2}+ {ad_{2} \frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{d_{4}(r_{2}+d_{3})}\bar {x}_{1}\underline{x}_{1}} \\& \quad =\bigl[rr_{1}-d_{2}(r_{1}+d_{1}) \bigr]\bar{x}_{1} + {\bigl[m\bigl(rr_{1}-d_{2}(r_{1}+d_{1}) \bigr)-a_{1}d_{2}\bigr]\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})} \underline{x}_{1}}, \end{aligned}$$
(3.14)
$$\begin{aligned}& ad_{2} \underline{x}_{1}^{2}+ {ad_{2} \frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{d_{4}(r_{2}+d_{3})}\bar{x}_{1}\underline {x}_{1}} \\& \quad =\bigl[rr_{1}-d_{2}(r_{1}+d_{1}) \bigr]\underline{x}_{1}+ {\bigl[m\bigl(rr_{1}-d_{2}(r_{1}+d_{1}) \bigr)-a_{1}d_{2}\bigr]\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})} \bar{x}_{1}}. \end{aligned}$$
(3.15)
Equation (3.14) minus (3.15),
$$\begin{aligned} ad_{2}\bigl(\bar{x}_{1}^{2}- \underline{x}_{1}^{2}\bigr) =&\bigl[rr_{1}-d_{2}(r_{1}+d_{1}) \bigr](\bar {x}_{1}-\underline{x}_{1}) \\ &{}-\frac {[a_{2}r_{2}-d_{4}(r_{2}+d_{3})][m(rr_{1}-d_{2}(r_{1}+d_{1}))-a_{1}(r_{1}+d_{1})]}{md_{4}(r_{2}+d_{3})} \\ &{}\times(\bar {x}_{1}-\underline{x}_{1}). \end{aligned}$$
(3.16)
Assume that \(\bar{x}_{1}\neq\underline{x}_{1}\), then we derive from (3.16) that
$$\begin{aligned} ad_{2}(\bar{x}_{1}+\underline{x}_{1}) =&rr_{1}-d_{2}(r_{1}+d_{1}) \\ &{}- \bigl[m\bigl(rr_{1}-d_{2}(r_{1}+d_{1}) \bigr)-a_{1}d_{2}\bigr]\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})}. \end{aligned}$$
(3.17)
Equation (3.14) plus (3.15),
$$\begin{aligned}& ad_{2}(\bar{x}_{1}+\underline{x}_{1})^{2}+2ad_{2} {\frac {a_{2}r_{2}-2d_{4}(r_{2}+d_{3})}{d_{4}(r_{2}+d_{3})}\bar{x}_{1}\underline {x}} \\& \quad =\biggl[rr_{1}-d_{2}(r_{1}+d_{1})+ \bigl(m\bigl(rr_{1}-d_{2}(r_{1}+d_{1}) \bigr)-a_{1}d_{2}\bigr)\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{md_{4}(r_{2}+d_{3})}\biggr] \\& \qquad {}\times( \bar{x}_{1}+\underline{x}_{1}). \end{aligned}$$
(3.18)
Substituting (3.17) into (3.18), it follows that
$$\begin{aligned}& ad_{2}\bigl[a_{2}r_{2}-2d_{4}(r_{2}+d_{3}) \bigr]\bar{x}_{1}\underline {x} \\& \quad =\bigl[a_{2}r_{2}-d_{4}(r_{2}+d_{3}) \bigr] \biggl[rr_{1}-d_{2}(r_{1}+d_{1})- \frac{a_{1}d_{2}}{m}\biggr](\bar {x}_{1}+\underline{x}_{1}). \end{aligned}$$
(3.19)
Note that \(\bar{x}_{1}>0\), \(\underline{x}_{1}>0\) and (H5) holds. This is a contradiction. Accordingly, we have \(\bar{x}_{1}=\underline{x}_{1}\). It therefore follows from (3.13) that \(\bar{x}_{2}=\underline{x}_{2}\), \(\bar {y}_{1}=\underline{y}_{1}\) and \(\bar{y}_{2}=\underline{y}_{2}\). Hence, the positive equilibrium \(E^{*}\) is global stability. The proof is complete. □

Theorem 3.2

The predator-extinction equilibrium \(E_{1}(x_{1}^{+}, x_{2}^{+}, 0, 0)\) of system (1.2) is globally stable provided that
  1. (H6)

    \(a_{2}r_{2}< d_{4}(r_{2}+d_{3})\), \(\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac{1}{m}\).

     

Proof

Let \((x_{1}(t), x_{2}(t), y_{1}(t), y_{2}(t))\) be any positive solution of system (1.2) with initial conditions (1.3). It follows from the first and the second equations of (1.2) that
$$\begin{aligned}& \dot{x}_{1}(t)\leqslant{rx_{2}(t)-(r_{1}+d_{1})x_{1}(t)-ax_{1}^{2}(t)}, \\& \dot{x}_{2}(t)=r_{1}x_{1}(t)-d_{2}x_{2}(t). \end{aligned}$$
Consider the following auxiliary equations:
$$ \begin{aligned} &\dot{z}_{1}(t)=rz_{2}(t)-(r_{1}+d_{1})z_{1}(t)-az_{1}^{2}(t), \\ &\dot{z}_{2}(t)=r_{1}z_{1}(t)-d_{2}z_{2}(t). \end{aligned} $$
(3.20)
Since \(rr_{1}>d_{2}(r_{1}+d_{1})\), by Lemma 2.2 of [10], it follows from (3.20) that
$$\lim_{t\rightarrow+\infty}{z}_{1}(t)={ {\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}}},\qquad \lim_{t\rightarrow+\infty}{z}_{2}(t)={ {\frac{r_{1}[rr_{1}-d_{2}(r_{1}+d_{1})]}{ad_{2}^{2}}}}. $$
By comparison, we obtain that
$$ \limsup_{t\rightarrow+\infty}{x}_{1}(t)\leq{ { \frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}}},\qquad \limsup_{t\rightarrow+\infty}{x}_{2}(t) \leq{ {\frac {r_{1}[rr_{1}-d_{2}(r_{1}+d_{1})]}{ad_{2}^{2}}}}. $$
(3.21)
Hence, for \(\varepsilon>0\) sufficiently small, there is \(T_{1}>0\) such that if \(t>T_{1}\), then \(x_{1}(t)\leq{x_{1}^{+}}+\varepsilon\). We therefore derive from the third and the fourth equations of system (1.2) that for \(t>T_{1}+\tau\),
$$\begin{aligned}& \dot{y}_{1}(t)\leq{ {\frac{a_{2}(x_{1}^{+}+\varepsilon)y_{2}(t-\tau )}{x_{1}^{+}+\varepsilon+my_{2}(t-\tau)}}} -(r_{2}+d_{3})y_{1}(t), \\& \dot{y}_{2}(t)=r_{2}y_{1}(t)-d_{4}y_{2}(t). \end{aligned}$$
Consider the following auxiliary equations:
$$ \begin{aligned} &\dot{u}_{1}(t)= { \frac{a_{2}(x_{1}^{+}+\varepsilon)u_{2}(t-\tau )}{x_{1}^{+}+\varepsilon+mu_{2}(t-\tau)}} -(r_{2}+d_{3})u_{1}(t), \\ &\dot{u}_{2}(t)=r_{2}u_{1}(t)-d_{4}u_{2}(t). \end{aligned} $$
(3.22)
Since (H6) holds, by Lemma 2.4 of [6], it follows from (3.22) that
$$\lim_{t\rightarrow+\infty}u_{1}(t)=0, \qquad \lim _{t\rightarrow+\infty}u_{2}(t)=0. $$
A comparison argument shows that
$$\lim_{t\rightarrow+\infty}y_{1}(t)=0,\qquad \lim _{t\rightarrow+\infty}y_{2}(t)=0. $$
Hence, for \(\varepsilon>0\) sufficiently small, there is \(T_{2}>T_{1}\) such that if \(t>T_{2}\), \(y_{2}(t)<\varepsilon\).
It follows from the first and the second equations of system (1.2) that for \(t>T_{2}\),
$$\begin{aligned}& \dot{x}_{1}(t)\geq{rx_{2}(t)}-(r_{1}+d_{1})x_{1}(t)-ax_{1}^{2}(t)- {\frac {a_{1}\varepsilon{x_{1}(t)}}{m\varepsilon+x_{1}(t)}}, \\& \dot{x}_{2}(t)={r_{1}x_{1}(t)}-d_{2}x_{2}(t). \end{aligned}$$
Consider the following auxiliary equations:
$$ \begin{aligned} &\dot{z}_{1}(t)=rz_{2}(t)-(r_{1}+d_{1})z_{1}(t)-az_{1}^{2}(t)- {\frac{a_{1}\varepsilon {z_{1}(t)}}{m\varepsilon+z_{1}(t)}}, \\ &\dot{z}_{2}(t)={r_{1}z_{1}(t)}-d_{2}z_{2}(t). \end{aligned} $$
(3.23)
Since (H6) holds, by Lemma 2.2 of [10], we can obtain
$$\begin{aligned}& \lim_{t\rightarrow+\infty}z_{1}(t)=\sqrt{\biggl(\frac {rr_{1}-d_{2}(r_{1}+d_{1})}{2ad_{2}}- \frac{m}{2}\varepsilon\biggr)^{2}+\varepsilon\biggl[m \frac {rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}-\frac{a_{1}}{a}\biggr]} \\& \hphantom{\lim_{t\rightarrow+\infty}z_{1}(t)=}{}+\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{2ad_{2}}-\frac{m}{2}\varepsilon:=z_{1}^{*}, \\& \lim_{t\rightarrow+\infty}z_{2}(t)=\lim_{t\rightarrow+\infty} \frac {r_{1}}{d_{2}}z_{1}^{*}. \end{aligned}$$
For \(\varepsilon>0\) is arbitrary small, by comparison, we derive that
$$\liminf_{t\rightarrow+\infty}{x_{1}(t)}\geq{\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}}, \qquad \liminf_{t\rightarrow+\infty}{x_{2}(t)}\geq{\frac {r_{1}[rr_{1}-d_{2}(r_{1}+d_{1})]}{ad_{2}^{2}}}, $$
which, together with (3.21), yields
$$\lim_{t\rightarrow+\infty}{x_{1}(t)}= {\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{ad_{2}}},\qquad \lim_{t\rightarrow+\infty}{x_{2}(t)}= {\frac{r_{1}[rr_{1}-d_{2}(r_{1}+d_{1})]}{ad_{2}^{2}}}. $$
This completes the proof. □

We now give some examples to illustrate the main results above.

Example 1

In (1.2), let \(a=1\), \(a_{1}=2\), \(a_{2}=1\), \(d_{1}=d_{2}=1\), \(d_{3}=d_{4}=1/2\), \(r=3\), \(r_{1}=2\), \(r_{2}=1\) and \(m=2\). With above coefficients, system (1.2) has a unique coexistence equilibrium \(E^{*}(2.75, 5.5, 0.2292, 0.4583)\). It is easy to show that \(\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}-\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m}(1+\frac{d_{4}(r_{2}+d_{3})}{a_{2}r_{2}})\approx 1.2813\), \(a_{2}r_{2}-2d_{4}(r_{2}+d_{3})=-0.5\), \(\frac {rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}-\frac{1}{m}=1\). By Theorem 3.1, \(E^{*}\) is globally asymptotically stable. Numerical simulation illustrates this fact (see Figure 1).
Figure 1
Figure 1

The temporal solution found by numerical integration of system ( 1.2 ) with \(\pmb{\tau=3}\) ; \(\pmb{(\phi_{1}, \phi_{2}, \varphi_{1}, \varphi_{2})=(1, 1, 1, 1)}\) .

Example 2

In (1.2), we let \(a=1\), \(a_{1}=2\), \(a_{2}=2\), \(d_{1}=d_{2}=2\), \(d_{3}=d_{4}=1\), \(r=1\), \(r_{1}=1\), \(r_{2}=1/3\) and \(m=2\). System (1.2) with the above coefficients has a unique coexistence equilibrium \(E^{*}(0.8889, 2.6667, 2.0741, 6.2222)\). Clearly, \(\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}-\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m}(1+\frac{d_{4}(r_{2}+d_{3})}{a_{2}r_{2}})\approx 0.0988\), \(\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}-\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m}(1+2\frac {d_{4}(r_{2}+d_{3})}{a_{2}r_{2}+d_{4}(r_{2}+d_{3})})\approx-0.1212\). By Theorem 2.1, there is \(\tau_{0}>0\) such that for \(\tau<\tau_{0}\), the coexistence equilibrium \(E^{*}\) is locally stable; for \(\tau>\tau_{0}\), the coexistence equilibrium \(E^{*}\) is unstable. Numerical simulation illustrates this result (see Figure 2).
Figure 2
Figure 2

The temporal solution found by numerical integration of system ( 1.2 ) with \(\pmb{\tau=128}\) ; \(\pmb{(\phi_{1}, \phi_{2}, \varphi _{1}, \varphi_{2})=(1, 1, 1, 1)}\) .

Example 3

In (1.2), let \(a=1\), \(a_{1}=2\), \(a_{2}=2\), \(d_{1}=d_{2}=1\), \(d_{3}=d_{4}=1\), \(r=3\), \(r_{1}=1\), \(r_{2}=3/2\) and \(m=2\). System (1.2) with the above coefficients has a predator-extinction equilibrium \(E_{1}(3, 6, 0, 0)\). It is easy to show that \(a_{2}r_{2}-d_{4}(r_{2}+d_{3})=-4\), \(\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}-\frac {1}{m}=1\). By Theorem 3.2, \(E_{1}\) is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 3).
Figure 3
Figure 3

The temporal solution found by numerical integration of system ( 1.2 ) with \(\pmb{\tau=3}\) ; \(\pmb{(\phi_{1}, \phi_{2}, \varphi_{1}, \varphi_{2})=(1, 1, 1, 1)}\) .

4 Discussion

In this paper, we have incorporated stage structure for both the predator and the prey into a predator-prey model with time delay due to the gestation of the predator. By using the iteration technique and comparison arguments, we have established sufficient conditions for the global stability of the coexistence equilibrium and the predator-extinction equilibrium. As a result, we have shown the threshold for the permanence and extinction of the system. By Theorems 3.1 and 3.2, we see that: (i) if (H6) holds, the predator population will go to extinction; (ii) if (H3) and (H5) hold, then both the prey and the predator species of system (1.2) are permanent.

Declarations

Acknowledgements

We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research was funded by the National Natural Science Foundation of China (11101117), the Scientific Research Foundation of Hebei Education Department (No. QN2014040) and the Foundation of Hebei University of Economics and Business (No. 2015KYQ01).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, P.R. China
(2)
Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang, 050003, P.R. China

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© Wang and Feng 2015

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