# Dynamic behaviors of Lotka–Volterra predator–prey model incorporating predator cannibalism

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## Abstract

A Lotka–Volterra predator–prey model incorporating predator cannibalism is proposed and studied in this paper. The existence and stability of all possible equilibria of the system are investigated. Our study shows that cannibalism has both positive and negative effect on the stability of the system, it depends on the dynamic behaviors of the original system. If the predator species in the system without cannibalism is extinct, then suitable cannibalism may lead to the coexistence of both species, in this case, cannibalism stabilizes the system. If the cannibalism rate is large enough, the prey species maybe driven to extinction, while the predator species are permanent. If the two species coexist in the stable state in the original system, then predator cannibalism may lead to the extinction of the prey species. In this case, cannibalism has an unstable effect. Numeric simulations support our findings.

## Keywords

Predator–prey Stability Predator cannibalism## 1 Introduction

*x*and

*y*are the density of the prey and predator at time

*t*, respectively.

*b*and

*α*denote the intrinsic growth rate and intraspecific competition of the prey, respectively;

*β*is the death rate of the predator;

*m*denotes the strength of intraspecific interaction between prey and predator;

*n*is the conversion efficiency of ingested prey into new predators; \(cy^{2}/(y+d)\) denotes the cannibalism of the predator; \(c_{1}\) is the birth rate from the predator cannibalism. All the coefficients are nonnegative constants.

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. During the last decade, many scholars investigated the dynamic behaviors of the predator–prey species, see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] and the references therein.

### Theorem A

*In system* (1.2), *there are two boundary equilibria*\(O(0,0)\), \(E^{1}( \frac{b}{\alpha }, 0) \). \(O(0,0)\)*is a saddle and*\(E^{1}( \frac{b}{\alpha }, 0)\)*is globally asymptotically stable if*\(\beta > \frac{bn}{\alpha }\). *Assume that*\(\beta < \frac{bn}{\alpha }\), *the positive equilibrium*\(E^{2} ( \frac{\beta }{n}, \frac{bn-\alpha \beta }{mn} )\)*exists*, *which is globally asymptotically stable*.

*t*, respectively. Yu [18] provided two sets of sufficient conditions on the global asymptotic stability of a positive equilibrium. After that, Yue [19] considered the dynamics of a modified Leslie–Gower predator–prey model with Holling-type II schemes and a prey refuge:

*mx*is part of the refuge protecting of the prey, here \(m\in [0, 1)\). Yue [19] found that increasing the amount of refuge can ensure the coexistence and attractivity of the two species more easily.

In recent years, cannibalism as a special phenomenon in nature which often occurs in plankton [22], fishes [23], spiders [24], and social insect populations [26] attracted the attention of many scholars. It is a behavior that consumes the same species and helps to provide food sources. Obviously, cannibalism has a very important effect on the dynamic behaviors of the populations (see [22, 23, 24, 25, 26, 27, 28, 29, 30, 31]).

*t*, respectively, \(z(t)\) is the density of the prey at time

*t*. The term \(\beta xy\) reflecting the intraspecific interaction denotes the cannibalization rate of adult predators to juvenile ones, the term \(\varepsilon xy \) is the rate of the adult predators increase due to being better fed through eating juveniles. Zhang et al. [29] obtained that large cannibalization rate can make the positive equilibrium globally stable although its stability would change with the increase of the cannibalism rate.

Generally speaking, scholars [22, 23, 24, 25, 26, 27, 28, 29] used the bilinear function \(\beta x y\) to describe the cannibalism phenomenon. Only recently did scholars [30, 31] adopted the idea of the functional response of predator–prey model and proposed the nonlinear cannibalism model.

*u*and

*v*represent the densities of prey and predator at time

*t*, respectively. The parameters \(c_{1}\),

*α*,

*c*,

*d*,

*δ*, and

*β*are nonnegative constants. Different from the previous works [24, 25, 26, 27, 28, 29], Basheer et al. [30] used the Holling II type functional response to describe cannibalism. Here the generic cannibalism term \(C(u)\) is added in the prey equation and is given by

*c*is the cannibalism rate. This term is obviously more appropriate with the reality of ecology and has a clear gain of energy to the cannibalistic prey. This gain results in an increase in reproduction in the prey, modeled via adding a \(c_{1}u\) term to the prey equation. Obviously, \(c_{1}< c\), as it takes depredation of a number of prey by the cannibal to produce one new offspring. They obtained that prey cannibalism alters the dynamics of the predator–prey model. System (1.6) is stable with no cannibalism, while it is unstable with prey cannibalism under the same conditions. After that, Basheer et al. [31] studied the predator–prey model with cannibalism in both predator and prey population and obtained more detailed results.

As far as system (1.2) is concerned, if the boundary equilibrium point \(E^{1}\) of system (1.2) is globally asymptotically stable, which means that the predator will eventually become extinct and the prey will survive, then how does cannibalism affect the dynamic behaviors of the system? If the positive equilibrium point \(E^{2}\) of system (1.2) is globally asymptotically stable, then how does cannibalism affect the dynamic behaviors of the system? This motivated us to propose and study system (1.1).

The paper is arranged as follows. In the next section, we investigate the existence and local stability of the equilibria of system (1.1). In Sect. 3, we discuss the global stability of the equilibria. Numeric simulations are presented in Sect. 4 to show the feasibility of the main results. We end this paper with a brief discussion.

## 2 Existence and local stability of equilibria

### 2.1 The existence of equilibria

*y*into the second equation of (2.3), we can get the equation as follows:

- (a)
If \(C\leq 0\), Eq. (2.4) has the unique positive root \(x_{1}= \frac{B+\sqrt{B^{2}-4AC}}{2A}\geq \frac{B}{A}> \frac{b}{\alpha }\). Obviously, \(x_{1}\) does not satisfy the condition of (2.6).

- (b)
If \(C>0\), we have \(\beta > c_{1}\) or \(\beta \leq c_{1}< \beta +\frac{bc}{b+dm}\). Then Eq. (2.4) has two positive roots \(x_{2,3}= \frac{B\pm \sqrt{B^{2}-4AC}}{2A}\).

(1) If \(\beta \leq c_{1}<\beta + \frac{bc}{b+dm}\), we have \(f( \frac{b}{\alpha })<0\), then system (1.1) has a positive equilibrium \(E_{3}(x_{2}^{*}, y_{2}^{*})\), where \(x_{2}^{*}= \frac{B-\sqrt{B^{2}-4AC}}{2A}\), \(y_{2}^{*}= \frac{b-\alpha x_{2}^{*}}{m}\).

(2) If \(\beta > c_{1}\), we cannot determine the size of \(f( \frac{b}{\alpha })\). So we will discuss the following:

Summarizing the above discussion, we obtain the following theorem.

### Theorem 2.1

*For all positive parameters*,

*there are two boundary equilibria*\(E_{0}(0, 0)\), \(E_{1}( \frac{b}{\alpha }, 0)\).

*The boundary equilibrium*\(E_{2} (0, \frac{d(c_{1}-\beta )}{\beta +c-c_{1}} )\)

*exists if*\(c_{1}>\beta \).

*In system*(1.1),

*for the positive equilibrium*,

*we have*:

- (i)
*If*\(0<\beta -c_{1}< \frac{bn}{\alpha }\),*then system*(1.1)*has the unique positive equilibrium*\(E^{*}(x_{2}^{*}, y_{2}^{*})\),*where*\(x_{2}^{*}= \frac{B-\sqrt{B^{2}-4AC}}{2A}\), \(y_{2}^{*}= \frac{b-\alpha x_{2}^{*}}{m}\). - (ii)
*If*\(\beta \leq c_{1}<\beta + \frac{bc}{b+dm}\),*then system*(1.1)*has the unique positive equilibrium*\(E^{*}(x_{2}^{*}, y_{2}^{*})\),*where*\(x_{2}^{*}= \frac{B-\sqrt{B^{2}-4AC}}{2A}\), \(y_{2}^{*}= \frac{b-\alpha x_{2}^{*}}{m}\).

### 2.2 The local stability of equilibria

### Theorem 2.2

*In system*(1.1),

*for the boundary equilibrium*\(E_{0}(0,0)\),

*we have*

- (1)
*If*\(c_{1}<\beta \),*then*\(E_{0}(0, 0)\)*is a saddle*; - (2)
*If*\(c_{1}=\beta \),*then*\(E_{0}(0, 0)\)*is a saddle node*; - (3)
*If*\(c_{1}>\beta \),*then*\(E_{0}(0, 0)\)*is an unstable node*.

### Proof

*τ*is a new time variable, which makes the system into the following form:

The proof of Theorem 2.2 is finished. □

### Theorem 2.3

*In system*(1.1),

*for the boundary equilibrium*\(E_{1}( \frac{b}{\alpha }, 0)\),

*we have*:

- (1)
*If*\(c_{1}\geq \beta \),*then*\(E_{1}( \frac{b}{\alpha }, 0)\)*is a saddle*; - (2)
*If*\(c_{1}<\beta \),*then*:- (i)
*If*\(\beta -c_{1}< \frac{bn}{\alpha }\),*then*\(E_{1}( \frac{b}{\alpha }, 0)\)*is a saddle*; - (ii)
*If*\(\beta -c_{1}> \frac{bn}{\alpha }\),*then*\(E_{1}( \frac{b}{\alpha }, 0)\)*is a stable node*; - (iii)
*If*\(\beta -c_{1}= \frac{bn}{\alpha }\),*then*\(E_{1}( \frac{b}{\alpha }, 0)\)*is a saddle node*.

- (i)

### Proof

If \(\beta -c_{1}\leq 0\), i.e., \(c_{1}\geq \beta \), then \(\lambda _{2}= \frac{bn}{\alpha }-(\beta -c_{1})>0\), so \(E_{1}( \frac{b}{\alpha }, 0)\) is a saddle.

If \(c_{1}<\beta \), we have \(\lambda _{2}>0\), if \(\frac{bn}{\alpha }>\beta -c_{1}\), then \(E_{1}( \frac{b}{\alpha }, 0)\) is a saddle.

*τ*is a new time variable, then we have

According to Theorem 7.1 in Chap. 2 in [32], we have \(m=2\), \(a_{m}= \frac{mn}{b\alpha }+ \frac{c}{bd}>0\), so \(E_{1}( \frac{b}{\alpha }, 0)\) is a saddle node.

The proof of Theorem 2.3 is finished. □

### Theorem 2.4

*In system*(1.1),

*when the boundary equilibrium*\(E_{2} (0, \frac{d(c_{1}-\beta )}{\beta +c-c_{1}} )\)

*exists*,

*we have*

- (1)
*If*\(\beta < c_{1}<\beta + \frac{b(\beta +c)+\beta dm}{b+dm}\),*then*\(E_{2} (0, \frac{d(c_{1}-\beta )}{\beta +c-c_{1}} )\)*is a saddle*; - (2)
*If*\(c_{1}>\beta + \frac{b(\beta +c)+\beta dm}{b+dm}\),*then*\(E_{2} (0, \frac{d(c_{1}-\beta )}{\beta +c-c_{1}} )\)*is a stable node*; - (3)
*If*\(c_{1}=\beta + \frac{b(\beta +c)+\beta dm}{b+dm}\),*then*\(E_{2} (0, \frac{d(c_{1}-\beta )}{\beta +c-c_{1}} )\)*is a saddle node*.

### Proof

If \(C>0\), i.e., \(\beta < c_{1}<\beta + \frac{b(\beta +c)+\beta dm}{b+dm}\), we have \(\lambda _{1}= \frac{C}{c+\beta -c_{1}}>0\), so \(E_{2} (0, \frac{d(c_{1}-\beta )}{\beta +c-c_{1}} )\) is a saddle.

If \(c_{1}=\beta + \frac{b(\beta +c)+\beta dm}{b+dm}\), then \(E_{2} (0, \frac{d(c_{1}-\beta )}{\beta +c-c_{1}} )\) is a saddle node. The proof is similar to Theorem 2.3, we omitted it.

The proof of Theorem 2.4 is finished. □

### Theorem 2.5

*In system* (1.1), *when the equilibrium*\(E^{*}(x^{*}, y^{*})\)*exists*, *it is locally asymptotically stable*.

### Proof

The proof of Theorem 2.5 is finished. □

## 3 Global stability of equilibria

In this section we consider the global asymptotic stability of the equilibria.

### Theorem 3.1

*Assume that*

*holds*,

*then*\(E_{1}(\frac{b}{\alpha }, 0)\)

*is globally asymptotically stable*.

### Proof

We will prove Theorem 3.1 by constructing some suitable Lyapunov function.

The proof of Theorem 3.1 is finished. □

### Theorem 3.2

*Assume that*

*holds*,

*then*\(E_{2} (0, \frac{d(c_{1}-\beta )}{\beta +c-c_{1}} )\)

*is globally asymptotically stable*.

### Proof

We will prove Theorem 3.2 by constructing some suitable Lyapunov function.

The proof of Theorem 3.2 is finished. □

### Theorem 3.3

*When the equilibrium*\(E_{3}(x^{*}, y^{*})\)*exists*, *it is globally asymptotically stable*.

### Proof

We will prove Theorem 3.3 by constructing some suitable Lyapunov functions.

The proof of Theorem 3.3 is finished. □

## 4 Numerical simulations

In this section we consider the dynamics of systems (1.1) and (1.2) under different parameters.

## 5 Conclusion

Based on the traditional Lotka–Volterra predator–prey model, we propose and study a predator–prey model with predator cannibalism in this paper. We have investigated the local and global stability of the possible equilibria of the model. Meanwhile, we can find some interesting phenomenon about the dynamic behaviors of system (1.1). If system (1.2) (no cannibalism, i.e., \(c=0\) and \(c_{1}=0\)) has a boundary equilibrium \(E^{1}( \frac{b}{\alpha }, 0)\), which is globally asymptotically stable (see Fig. 1), a suitable cannibalism rate (\((\beta < c_{1}<\beta + \frac{b(\beta +c)+\beta dm}{b+dm} )\)) leads to system (1.1) admitting a unique positive equilibrium, and it is globally asymptotically stable (see Fig. 3, Fig. 4, and Fig. 5). That is to say, cannibalism within a certain range can make the two species persistent. So in this case, cannibalism in a certain range has a positive effect for the coexistence of the prey and the predator. With the increase of \(c_{1}\), the positive equilibrium will disappear and the boundary equilibrium \(E_{2} (0, \frac{d(c_{1}-\beta )}{c+\beta -c_{1}} )\) will appear (see Fig. 6). That is to say, without other sources of food, predator populations can still survive on cannibalism. For example, salamanders only depend on cannibalism to survive in summer.

If system (1.2) has a positive equilibrium \(E^{2}( \frac{\beta }{n}, \frac{bm-\alpha \beta }{mn})\), which is globally asymptotically stable (see Fig. 7), with the increase of \(c_{1}\), the population density of prey decreases while that of predator increases (see Fig. 8 and Fig. 9). When \(c_{1}\) is large enough, prey populations will be driven to extinction. That is to say, predator cannibalism will make prey extinct (see Fig. 10). Predator cannibalism also changes the type of the equilibria (see Fig. 1, Fig. 5, and Fig. 6; Fig. 7, Fig. 9, and Fig. 10).

That is, by introducing the predator cannibalism, the dynamic behaviors of the system become complicated.

## Notes

### Acknowledgements

The author would like to thank Dr. Liqiong Pu for bringing our attention to the paper of Jiming Zhang.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### Funding

The research was supported by the National Natural Science Foundation of China under Grant (11601085).

### Competing interests

The authors declare that there is no conflict of interests.

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