Stability Analysis of a Ratio-Dependent Predator-Prey Model

Abstract

In this study, a ratio-dependent predator-prey model is investigated. The local stability and global stability of the nonnegative boundary equilibrium and positive equilibrium of the model are discussed, respectively. Sufficient condition is obtained for the existence of Hopf bifurcation at the positive equilibrium.

1. Introduction

Recently, the predator-prey models have been studied by many authors [1–8]. In general, a predator-prey model has the following forms:where and are the densities of the prey and predator population at time t, respectively. The function represents the growth of the prey population rate, represents the growth rate of predator population, and represents the functional response function of predator population to prey population. In [1], Xu et al. used the function as the functional response function of predator population to prey population. The time delay due to the gestation of the predator is discussed in [1].

It is noted that in model (1), each individual’s prey admits the same risk to be attacked by predators and each individual predator admits the same ability to feed on prey. This assumption seems not to be realistic for many animals. In natural world, there are many species whose individuals pass through an immature stage. Stage structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. In the last two decades, stage-structured models have received great attention [3–7, 9].

Based on above discussion, we study the following predator-prey model:where and are the densities of the immature and mature prey at time and and are the densities of the immature and mature predators at time . In model (2), all parameters are positive constants. is the time delay due to the gestation of the predator. is the ratio-dependent functional response.

Model (2) is of the following initial conditions:

The organization of this study is as follows. In Section 2, we discuss the local stability of the nonnegative boundary equilibrium and the positive equilibrium of models (2) and (3). The existence of a Hopf bifurcation for models (2) and (3) at the positive equilibrium is also established. Sufficient conditions are derived for the global stability of the nonnegative boundary equilibrium and positive equilibrium of models (2) and (3) in Section 3, respectively.

2. Local Stability and Hopf Bifurcation

In this section, by analyzing the corresponding characteristic equations, we study the local stability of each of nonnegative equilibria and the existence of a Hopf bifurcation at the positive equilibrium of models (2) and (3).

If , model (2) has a nonnegative boundary equilibrium , where

If , model (2) has a positive equilibrium , where

The characteristic equation of model (2) at takes the following form:where . When , all roots of equation,are negative. Now, we consider the roots of the following equation. . By calculating, we obtain

When , we get . Therefore, is locally stable for all . When , we get . Thus, is unstable.

The characteristic equation of model (2) at is of the formwhere

Let ; then, (9) has the following form:

Note that . Whenthen positive equilibrium is locally asymptotically stable.

Let and hold. If is a solution of (9), by calculation, we can obtainwherewhen , is locally asymptotically stable for all . When , is the positive root of (12); in this case, (9) has a pair of roots . By (12), we obtain

Therefore, remains stable for .

Differentiating (9) with respect to , we obtain that

Hence, we get

Therefore, as , there is Hopf bifurcation.

From above discussion, we have the following results.

Theorem 1. For model (2) with (3), we have the following:(i)Let ; if , then is locally asymptotically stable; if , then is unstable.(ii)Assume and hold; if , then is locally asymptotically stable for all ; if , then there exists a , s.t., is locally asymptotically stable if and unstable if . When , models (2) and (3) undergo Hopf bifurcation at .

3. Global Stability

In this section, by using an iteration technique, we discuss the global stability of the nonnegative equilibria and of models (2) and (3), respectively.

Theorem 2. Lethold; then, the nonnegative boundary equilibrium of model (2) is globally stable.

Proof. It follows from the positive solution of model (2), and we can obtainBy Lemma 2.2 of [5] and comparison, we haveTherefore, there is a positive number , for sufficiently small positive number , such that as , . Hence, for , we derive thatBy Lemma 2.2 of [5] and comparison, we can obtainTherefore, there is a positive number , such that if , .
For , we derive from model (2) thatBy Lemma 2.2 of [5] and comparison, we haveBy model (2), it follows thatBy Lemma 2.4 of [3] and comparison, we obtain thatwhich together with (19) and (21) yieldsHence, the equilibrium of model (2) is globally stable.

Theorem 3. Assume , , and hold; ifthen the positive equilibrium of model (2) is global stability.

Proof. LetBy the first two equations of model (2), we can obtain thatBy Lemma 2.2 of [5] and comparison, we haveSo, for sufficiently small positive number , there exists a positive number , such that if , then .
For , by the last two equations of model (2), we getBy Lemma 2.2 of [5] and comparison, we obtainHence, , in whichTherefore, for sufficiently small positive number , there is , such that if , .
For , by the first two equations of model (2), we haveBy Lemma 2.4 of [3] and comparison, we derive thatHence, for sufficiently small positive number , there is , such that if , .
For , it follows from the last two equations of model (2) thatBy Lemma 2.4 of [3] and comparison, we can obtainTherefore, for sufficiently small positive number , there is a positive number , such that if , . In this case, by the first two equations of model (2), we haveFor sufficiently small positive number , if holds, by Lemma 2.2 of [5] and a comparison argument, we can obtainTherefore, for sufficiently small positive number , there is , such that if , .
From the last two equations of model (2), we obtain that for ,By Lemma 2.2 of [5] and comparison, if holds, we haveHence, for sufficiently small, there is a , such that if , .
Again, for sufficiently small positive number and , by the first two equations of model (2), we haveBy Lemma 2.4 of [3] and comparison, if holds, we can obtainSo, there is a positive number , for , .
For sufficiently small positive number and , from the last two equations of model (2), we can deriveBy Lemma 2.4 of [3] and comparison, if , we haveRepeat the above process; for , we can obtain eight sequences:in whichIt is noted thatDirect calculation, we have and as nonincreasing, and and as nondecreasing. Therefore, the limits of sequences in , , , and exist. LetWe haveNow, we prove that . By (50), we can obtainFrom above two equations, we haveIf , then we obtainSince , therefore, . This is a contradiction. So, . By (50), we have and . Therefore, the positive equilibrium is globally stable.

4. Discussion

In this study, we have studied a ratio-dependent predator-prey model with stage structure for the prey and predator. A time delay due to the gestation of the predator is considered. By using the eigenvalue theory, we have obtained the sufficient conditions for the local stability of the nonnegative equilibria of model (2). The existence of Hopf bifurcation is given. By the iteration technique and comparison arguments, sufficient conditions have been established for the global stability of the nonnegative equilibria. From Theorem 2, we know that if holds, the predator population will go to extinction. By Theorem 3, we learn that if and hold, then both the predator and prey species of model (2) are permanent [10, 11].

Data Availability

The [DATA TYPE] data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (12105073), the Natural Science Foundation of Hebei Province (A2019207070), and the Scientific Research Foundation of Hebei University of Economics and Business(2021ZD07).