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Stability and Hopf bifurcation for a ratiodependent predatorprey system with stage structure and time delay
Advances in Difference Equations volume 2015, Article number: 255 (2015)
Abstract
A ratiodependent predatorprey 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 predatorextinction 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 predatorextinction 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.
Introduction
Since the pioneering works of Arditi et al. [1, 2], ratiodependent predatorprey models have received much attention from scientists (see, for example, [3–7]). In [4], Kuang and Beretta studied the following ratiodependent predatorprey model with the MichaelisMenten type functional response
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 stagestructured system. Stagestructured models have received great attention in recent years (for example, [6–10]). 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:
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 intraspecific 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
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 predatorextinction 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 predatorextinction 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.
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 predatorextinction equilibrium \(E_{1}(x_{1}^{+}, x_{2}^{+}, 0, 0)\), where
Further, if the following condition holds

(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
The characteristic equation of (1.2) at the equilibrium \(E_{1}(x_{1}^{+}, x_{2}^{+}, 0, 0)\) is of the form
Note that \(rr_{1}>d_{2}(r_{1}+d_{1})\), it is easy to show that the equation
always has two negative real roots. All other roots are given by the roots of equation
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
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
where
When \(\tau=0\), equation (2.2) becomes
Clearly \(p_{3}>0\). If the following condition holds

(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
Hence, the coexistence equilibrium \(E^{*}\) of system (1.2) is locally asymptotically stable.
If iω (\(\omega>0\)) is a solution of (2.2), separating real and imaginary parts, we have the following:
Squaring and adding the two equations of (2.4), it follows that
where
Let (H1) hold, it is easy to show that
If the following condition holds

(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

(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
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
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
From this equation, we can obtain
Hence, we derive that
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
Accordingly, if (H1) holds, then we have that
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:

(i)
Let \(rr_{1}>d_{2}(r_{1}+d_{1})\), if \(a_{2}r_{2}< d_{4}(r_{2}+d_{3})\), then the predatorextinction 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.

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

(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 \(\tau=\tau_{0}\).
Global stability
In this section, we are concerned with the global stability of the coexistence equilibrium \(E^{*}\) and the predatorextinction 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

(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
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
Consider the following auxiliary equations:
Since \(rr_{1}>d_{2}(r_{1}+d_{1})\) holds, by Lemma 2.2 of [10], it follows from (3.1) that
A comparison argument shows that
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 \(t>T_{1}+\tau\),
Consider the following auxiliary equations:
Since (H3) holds, by Lemma 2.4 of [6], it follows from (3.3) that
By comparison, we derive that
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
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 \(t>T_{2}\),
Since (H5) holds, by Lemma 2.2 of [10] and a comparison argument, it follows from (3.5) that
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 \(t>T_{3}+\tau\),
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
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
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 \(t>T_{4}\),
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
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 \(t>T_{5}+\tau\),
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
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 \(t>T_{6}\),
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
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 \(t>T_{7}+\tau\),
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
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 \(k\geq2\),
It is readily seen that
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
We therefore obtain from (3.11) that
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
Assume that \(\bar{x}_{1}\neq\underline{x}_{1}\), then we derive from (3.16) that
Substituting (3.17) into (3.18), it follows that
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 predatorextinction equilibrium \(E_{1}(x_{1}^{+}, x_{2}^{+}, 0, 0)\) of system (1.2) is globally stable provided that

(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
Consider the following auxiliary equations:
Since \(rr_{1}>d_{2}(r_{1}+d_{1})\), by Lemma 2.2 of [10], it follows from (3.20) that
By comparison, we obtain that
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 \(t>T_{1}+\tau\),
Consider the following auxiliary equations:
Since (H6) holds, by Lemma 2.4 of [6], it follows from (3.22) that
A comparison argument shows that
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 \(t>T_{2}\),
Consider the following auxiliary equations:
Since (H6) holds, by Lemma 2.2 of [10], we can obtain
For \(\varepsilon>0\) is arbitrary small, by comparison, we derive that
which, together with (3.21), yields
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).
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, \(\frac{rr_{1}d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}\frac {a_{2}r_{2}d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m}(1+\frac{d_{4}(r_{2}+d_{3})}{a_{2}r_{2}})\approx 0.0988\), \(\frac{rr_{1}d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}\frac {a_{2}r_{2}d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m}(1+2\frac {d_{4}(r_{2}+d_{3})}{a_{2}r_{2}+d_{4}(r_{2}+d_{3})})\approx0.1212\). 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).
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 predatorextinction 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).
Discussion
In this paper, we have incorporated stage structure for both the predator and the prey into a predatorprey 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 predatorextinction 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.
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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).
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MSC
 34K18
 34K20
 34K60
 92D25
Keywords
 predatorprey system
 stage structure
 time delay
 stability
 Hopf bifurcation