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# Permanence of the periodic predator-prey-mutualist system

*Advances in Difference Equations*
**volume 2015**, Article number: 331 (2015)

## Abstract

In this paper, we study the permanence and the periodic solution of the periodic predator-prey-mutualist system. It is well known that mutualist species can reduce the capture rate of the predator species to the prey species. By further developing the analysis technique of Teng, a set of conditions which ensure the permanence of the system are obtained. In addition, sufficient conditions are derived for the existence of positive periodic solutions to the system. An example together with its numerical simulation shows the feasibility of the main results.

## Introduction

As was pointed out by Berryman [1], the dynamic relationship between predator and prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Already the predator-prey model has been studied by several scholars [2–10]. For example, Das *et al.* [8] investigated a three species ecosystem consisting of a prey, a predator, and a top predator. They derived the criteria for local and global stability of all the eight equilibrium points by using a Routh-Hurwitz and Lyapunov function. Wu and Li [9] studied the permanence and global attractivity of the discrete predator-prey system with Hassell-Varley-Holling type III functional response. Chen and Chen [10] proposed a ratio-dependent predator-prey model incorporating a prey refuge. They studied the global stability, limit cycle, and Hopf bifurcation of the system.

Mutualism is one of the most important relationships in the real world, for instance, ants prevent herbivores from feeding on plants (see [11]) and ants prevent predators from feeding on aphids (see [12, 13]). As was pointed out by Murray [14]: ‘this area has not been as widely studied as the others even though its importance is comparable to that of predator-prey and competition interactions.’ To this end, Rai and Krawcewicz [15] proposed the following three species predator-prey-mutualist system:

where \(x(t)\), \(y(t)\), \(z(t)\) denote the densities of prey, mutualist and predator population at any time *t*, respectively. They applied the equivariant degree method to study the Hopf bifurcation phenomenon of the system.

In this paper, we will study the non-autonomous case of system (1.1), *i.e.*,

where *x* is the density of the prey at time *t*, *y* is the density of the mutualist and *z* is the density of the predator at time *t*, respectively. \(a_{i}(t)\) (\(i=1,2,3\)), \(d_{j}(t)\) (\(j=1,2,3,4\)), and \(c_{k}(t)\) (\(k=1,2\)) are all continuously positive *w*-periodic functions. The assumption of periodicity of the parameters is a way to incorporate the periodicity of the environment (*e.g.* seasonal effects of weather condition, food supplies, temperature, mating habits, harvesting, *etc.*). For this system, due to the lack of density restriction of the predator species, one could not investigate the stability property of the system by constructing the suitable Lyapunov function. Hence, to investigate the persistent property of the system has become one of the most important topics, and we try to push forward this topic.

We arrange the rest of the paper as follows: In Section 2, we introduce one lemma and state the main results of this paper. The results are proved in Section 3. In Section 4, a suitable example together with its numeric simulation is present to show the feasibility of the main results. We end this paper by a briefly conclusion. For more works on the non-autonomous predator-prey system, one could refer to [16–19] and the references cited therein.

## Statement of the main results

Let us first consider the logistic equation,

where \(\alpha(t)\) and \(\beta(t)\) are periodic continuous functions on *R* with common periodic \(w>0\).

### Lemma 2.1

[20]

*If*
\(\beta(t)\geq0\)
*for all*
\(t\in R\)
*and*
\(\int_{0}^{w}\beta(t)\, dt>0\), *then* (2.1) *has a unique nonnegative*
*w*-*periodic solution*
\(u^{*}(t)\)
*which is globally asymptotically stable*, *that is*, \(u(t)-u^{*}(t)\rightarrow0\)
*as*
\(t\rightarrow\infty\)
*for any positive solution*
\(u(t)\)
*of* (2.1). *Moreover*, *if*
\(\int_{0}^{w}\alpha(t)\, dt>0\), *then*
\(u^{*}(t)>0\)
*for all*
\(t\in R\)
*and if*
\(\int_{0}^{w}\alpha(t)\, dt\leq0\)
*then*
\(u^{*}(t)\equiv0\).

### Definition 2.2

System (1.2) is said to be permanent if there exist positive constants \(\eta_{i}\), \(M_{i}\), \(i=1, 2, 3\), such that

for any positive solution \((x(t),y(t),z(t))\) of system (1.2).

We first consider the following system:

from Lemma 2.1, (2.2) has a unique nonnegative *w*-periodic solution \(u_{10}(t)\) which is globally asymptotically stable.

Second, we consider the following system:

where \(M^{*}= \max_{0\leq t\leq w}\{u_{10}(t)+1\}\), from Lemma 2.1, (2.3) also has a unique nonnegative *w*-periodic solution \(u_{20}(t)\) which is globally asymptotically stable.

As concerns the persistent property of the system (1.2), we have the following result.

### Theorem 2.3

*System* (1.2) *is permanent if*

*holds*, *where*
\(u_{10}(t)\)
*and*
\(u_{20}(t)\)
*are the unique positive periodic solution of systems* (2.2) *and* (2.3), *respectively*.

As a direct corollary of Theorem 2 in [21], from Theorem 2.3, we have the following.

### Corollary 2.4

*Under the assumption that* (2.4) *holds*, *system* (1.2) *admits of at least one positive*
*w*-*periodic solution*.

## Proof of the main result

We need the following propositions to prove Theorem 2.3.

### Proposition 3.1

*There exist positive constants*
\(M_{1}\)
*and*
\(M_{2}\), *such that*

*for all solutions of system* (1.2).

### Proof

Obviously, \(R_{+}^{3}=\{(x, y, z)|x\geq0, y\geq0, z\geq0\}\) is a positively invariant set of system (1.2). Given any solution \((x(t), y(t), z(t))\) of system (1.2), we have

By Lemma 2.1, the auxiliary equation

has a unique globally attractive positive *w*-periodic solution \(u_{10}(t)\). Let \(u_{1}(t)\) is the solution of (3.3) with \(u_{1}(0)=x(0)\), by the comparison theorem, we have

Moreover, from the global attractivity of \(u_{10}(t)\), for every given *ε* (\(0<\varepsilon<1\)), there exists a \(T_{1}>0\), such that

Equation (3.4) combined with (3.5) leads to

Let \(M_{1}= \max_{0\leq t\leq w}\{u_{10}(t)+\varepsilon\}\), we have

Since \(M^{*}= \max_{0\leq t\leq w}\{u_{10}(t)+1\}\), there exists a large enough \(T_{2}\geq T_{1}\) such that for all \(t>T_{2}\), one has

From (3.8), we have

By Lemma 2.1, the auxiliary equation

has a unique globally attractive positive *w*-periodic solution \(u_{20}(t)\). Similarly, we find that there is a constant \(T_{3}>T_{2}\) such that

Let \(M_{2}=\max_{0\leq t\leq w}\{u_{20}(t)+\varepsilon\}\), we have

This completes the proof of Proposition 3.1. □

### Proposition 3.2

*There is a universal constant*
\(\alpha>0\)
*such that*

### Proof

If (2.4) holds, we can choose the constant \(\varepsilon_{1}>0\) such that

For any constant \(\alpha>0\), we consider the following equation:

Owing to \(\int_{0}^{w}a_{1}(t)\, dt>0\), \(\int_{0}^{w} (a_{1}(t)-\frac {c_{1}(t)2\alpha}{d_{1}(t)} )\, dt>0\) for small enough \(\alpha>0\). By Lemma 2.1, (3.15) has a unique positive *w*-periodic solution \(v_{1\alpha}^{*}(t)\) which is globally asymptotically stable. Let \(\bar{v}_{1\alpha}(t)\) be the solution of (3.15) with initial condition \(\bar{v}_{1\alpha}(0)=u_{10}(0)\), where \(u_{10}(t)\) is the unique positive periodic solution of (2.2). Hence, for above \(\varepsilon _{1}\), there exists a sufficiently large \(T_{4}>T_{3}\) such that

By the continuity of the solution in the parameter, we have \(\bar {v}_{1\alpha}(t)\rightarrow u_{10}(t)\) uniformly in \([T_{4}, T_{4}+w]\) as \(\alpha\rightarrow0\). Hence, for \(\varepsilon_{1}>0\), there exists a \(\alpha_{0}=\alpha_{0}(\varepsilon_{1})>0\) such that

So, we have

Note that \(v_{1\alpha}^{*}(t)\) and \(u_{10}(t)\) are all *w*-periodic, hence

Choosing a constant \(\alpha_{1}\) (\(0<\alpha_{1}<\alpha_{0}\), \(2\alpha _{1}<\varepsilon_{1}\)), we have

If \(\limsup_{t\to\infty}z(t)\leq\alpha_{1}\), then there exists \(\phi\in R_{+}^{3}\) such that

where \((x(t, \phi), y(t, \phi), z(t, \phi))\) is the solution of system (1.2) with \(\phi(0)=(\phi_{1}(0), \phi_{2}(0), \phi_{3}(0)) >0\). So, there exists \(T_{5}>T_{4}\) such that

We have

Let \(v_{1}(t)\) be the solution of (3.15) with \(\alpha=\alpha_{1}\) and \(v_{1}(T_{5})=x(T_{5})\), then

By the global asymptotic stability of \(v_{1\alpha_{1}}^{*}(t)\), for the given \(\varepsilon=\frac{\varepsilon_{1}}{2}\), there exists \(T_{6}\geq T_{5}\), such that

So,

and hence, by using (3.16), it follows that

From (3.8), there exists \(T_{7}\geq T_{6}\) such that \(y(t)< u_{20}(t)+\varepsilon_{1}\). Therefore, by using (3.17) and (3.19), for \(t\geq T_{7}\) it follows that

Integrating the above inequality from \(T_{7}\) to *t* yields

Thus, from (3.14) it follows that \(z(t)\rightarrow\infty\) as \(t\rightarrow\infty\). This is a contradiction. This completes the proof of Proposition 3.2. □

### Proposition 3.3

*There is a universal constant*
\(\eta_{2}>0\)
*such that*

### Proof

From the second equation of system (1.2) it follows that

and we consider the following equation:

By Lemma 2.1, (3.22) has a unique positive *w*-periodic solution \(u_{30}(t)\). Similar to the analysis of (3.4)-(3.7), for *ε* enough small, without loss of generality, \(\varepsilon< \min_{0\leq t\leq w}{u_{30}(t)}\), we find that there is a constant \(T_{8}>T_{7}\) such that

Letting \(\eta_{2}= \min_{0\leq t\leq w}\{u_{30}(t)-\varepsilon\}\), we have

This completes the proof of Proposition 3.3. □

### Proposition 3.4

*There is a universal constant*
\(\eta_{3}>0\)
*such that*

### Proof

Suppose that (3.25) is not true, then there is a sequence \(\{\phi_{m}\} \in R_{+}^{3}\), such that

On the other hand, by Proposition 3.2, we have

Hence, there are time sequences \(\{s_{q}^{m}\}\) and \(\{t_{q}^{m}\}\) satisfying

and

From the third equation of system (1.2) it follows that

Obviously, by integrating (3.28) from \(s_{q}^{m}\) to \(t_{q}^{m}\),

or

Thus, from the boundedness of \(a_{3}(t)\), we have

By (3.14), there are constants \(P> 0\) and \(\gamma> 0\), such that, for any \(t\geq P\) and \(a\geq0\),

For any *m*, *q*, and \(t\in[s_{q}^{(m)}, t_{q}^{(m)}]\), we have

Let \(v_{1}(t)\) be the solution of (3.15) with the initial condition \(v_{1}(s_{q}^{(m)})=\alpha\), then by (3.31), we have \(x(t, \phi _{m})\geq v_{1}(t)\) for all \(t\in[s_{q}^{(m)}, t_{q}^{(m)}]\). By the periodicity of (3.15), it follows that the periodic solution \(v_{1\alpha}^{*}(t)\) is globally uniformly asymptotically stable. Hence, from (3.16), we find that there is a constant \(T_{0}>P\), and \(T_{0}\) is independent on any *q* and *m*, such that

By (3.29), there is a \(N_{0}>0\) such that \(t_{q}^{(m)}> s_{q}^{(m)}+2T_{0}\) for all \(m\geq N_{0}\). Hence,

Since

for all \(t\in[s_{q}^{(m)}+T_{0}, t_{q}^{(m)}]\) and \(m\geq N_{0}\), by integrating from \(s_{q}^{(m)}+T_{0}\) to \(t_{q}^{(m)}\), we obtain

By (3.30), we have

which is a contradiction. This completes the proof of Proposition 3.4. □

### Proposition 3.5

*There is a universal constant*
\(M_{3}>0\)
*such that*

### Proof

Choose the constant \(M>0\) and small enough *ε* such that

We first prove that

If (3.34) is not true, then there is a \(T_{9}\geq T_{8}\) such that \(z(t)>M\) and \(y(t)\leq M_{2}+\varepsilon\) (*ε* is small enough) for all \(t\geq T_{9}\). Since

By Lemma 2.1 we easily obtain \(x(t)\rightarrow0\) as \(t\rightarrow \infty\). Choose a constant \(0<\varepsilon_{2}<\eta_{2} \) such that

Then there is a \(T_{10}>T_{9}>0\) such that \(x(t)\leq\varepsilon_{2}\) and \(y(t)>\eta_{2}-\varepsilon_{2}\) for all \(t\geq T_{10}\), and so

Integrating the above inequality from \(T_{10}\) to *t* leads to

Hence, we have \(z(t)\rightarrow0\) as \(t\rightarrow\infty\) which is a contradiction. Now, if (3.32) is not true, then there is a sequence \(\{\phi_{m}\} \subset R_{+}^{3}\) such that \(\limsup_{t\to\infty}z(t, \phi_{m})> (2M+1)m \) for all \(m=1,2,\ldots\) . By (3.34), there are time sequences \(\{s_{q}^{m}\}\) and \(\{t_{q}^{m}\}\) satisfying

and

Since there is a \(T_{1}^{(m)}>0\) such that \(x(t, \phi_{m})< M_{1}\) for all \(t\geq T_{1}^{(m)}\), we have

Obviously, there is a \(K_{1}(m)>0\) such that \(s_{q}^{m}>T_{1}^{(m)}\) for all \(q\geq K_{1}(m)\). Hence, we obtain

Consequently,

By (3.35), there is a \(p_{1}>0\) such that for any \(t\geq p_{1}\) and \(a\geq0\),

By (3.33), there is a \(p_{2}>0\) such that for all \(t\geq p_{2}\) and \(b\geq0\),

By (3.37), there is a \(L>0\) such that \(t_{q}^{(m)}-s_{q}^{(m)}>p_{1}+p_{2}\) for all \(m\geq L\), \(q\geq K_{1}(m)\). For any \(m\geq L\), \(q \geq K_{1}(m)\) and \(t\in[s_{q}^{(m)}+p_{1}+p_{2}, t_{q}^{(m)}]\), we have

Hence,

Since

we obtain

which is contradictory with \(z(t_{q}^{(m)}, \phi_{m})=(2M+1)m\). Therefore, there is a constant \(M_{3}>0\) such that (3.32) holds. □

### Proposition 3.6

*There is a universal constant*
\(0<\eta<\eta_{2} \)
*such that*

*and*

*hold*, *where*
\(\eta_{2}\)
*is obtained in Proposition *
3.3.

### Proof

If \(\limsup_{t\to\infty}x(t) \leq \eta\), then by Proposition 3.3, there is a \(T_{11}>T_{10}\) such that \(x(t)<2\eta\) and \(y(t)\geq\eta_{2}-\varepsilon\) (*ε* is small enough) for all \(t\geq T_{11}\). Since

and since

and for a small enough *ε*,

by integrating from \(T_{11}\) to *t*, we can obtain

Hence, inequality (3.38) implies \(z(t)\rightarrow0\) as \(t\rightarrow \infty\), which is a contradiction and Proposition 3.6 is proved. □

### Proposition 3.7

*There is a universal constant*
\(\eta_{1}>0\)
*such that*

### Proof

We first choose the constant \(0<\varepsilon_{3}<\eta\) such that

Here the constants \(\eta_{2} >0\) and \(\eta>0\) are obtained in Propositions 3.3 and 3.6. Suppose that (3.39) is not true. Then there is a sequence \(\{\phi_{m}\}\subset R_{+}^{3}\) such that \(\lim\inf_{t\to\infty}x(t, \phi_{m} )<\frac{\varepsilon_{3}}{m+1}\) for all \(m=1,2,\ldots\) . By Proposition 3.6, there are two time sequences \(\{s_{q}^{(m)}\}\) and \(\{t_{q}^{(m)}\}\) satisfying the following conditions:

and

By Propositions 3.1 and 3.3, there is a \(T^{(m)}>0\) such that \(z(t, \phi_{m})\leq M_{3}\) and \(x(t, \phi_{m})\leq M_{1}\) for all \(t\geq T^{(m)}\), and further there is a \(K^{(m)}>0\) such that \(s_{q}^{m}\geq T^{(m)}\) for all \(q\geq K^{(m)}\). Hence, for any \(t\in[s_{q}^{m}, t_{q}^{m}]\) and \(q\geq K^{(m)}\), we have

By integrating from \(s_{q}^{m}\) to \(t_{q}^{m}\), we obtain

Consequently,

Hence, we obtain

By (3.40), there is a constant \(T_{12}>0\), and \(T_{12}\) is independent on any *m* and *q*, such that

By (3.41), there is a \(M_{0}>0\) such that \(t_{q}^{(m)}>s_{q}^{(m)}+M_{0}\) for all \(m\geq M_{0}\) and \(q\geq K^{(m)}\). Hence, for any \(t\in[s_{q}^{m}, t_{q}^{m}]\), \(m\geq M_{0}\), and \(q\geq K^{(m)}\), we have \(y(t)\geq\eta_{2}-\varepsilon \) and

By integrating from \(s_{q}^{(m)}\) to \(s_{q}^{(m)}+M_{0}\), we obtain

which is a contradiction, and Proposition 3.7 is proved. □

### Proof of Theorem 2.3

The results of Theorem 2.3 now follow from Propositions 3.1-3.7.

## Example

In this section, we shall give an example to illustrate the feasibility of the main results.

### Example

Consider the following predator-prey-mutualist system (see Figure 1):

Corresponding to system (1.2), one has

One easily sees that

has a unique positive periodic solution \(u_{10}(t)\approx10.667\). So \(M^{*}=11.667\),

has a unique positive periodic solution \(u_{20}(t)\approx17.503\).

By simple computation, one has

Condition (2.4) is satisfied. Thus, corresponding to Theorem 2.3 and Corollary 2.4, we know that system (4.1) is permanent and admits at least one positive 2*π*-periodic solution.

## Conclusion

In this paper, we studied a periodic predator-prey-mutualist system. From system (1.2), we see the mutualist species *y* can reduce the capture rate of the predator species *z* to the prey species *x*. By further developing the analysis technique of Teng [20], we obtain a set of conditions which ensure the permanence of system (1.2). Note that \(u_{10}(t)\) and \(u_{20}(t)\) are the globally attractive periodic solution of (2.2) and (2.3), respectively, which, as shown by Lemma 2.1, always exists. Hence, the left side of condition (2.4) implies that if the death rate of the predator species is enough small and the cooperation effect between species *x* and *y* is not very strong, the system is permanent.

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## Acknowledgements

The authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. Also, the research was supported by the Natural Science Foundation of Fujian Province (2015J010121, 2015J01019).

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### Cite this article

Yang, L., Xie, X., Chen, F. *et al.* Permanence of the periodic predator-prey-mutualist system.
*Adv Differ Equ* **2015, **331 (2015). https://doi.org/10.1186/s13662-015-0654-9

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### MSC

- 34C25
- 92D25
- 34D20
- 34D40

### Keywords

- predator-prey-mutualist system
- permanence
- periodic solution