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Global attractivity of Leslie–Gower predator-prey model incorporating prey cannibalism

Abstract

Leslie–Gower predator prey model with Holling II type cannibalism term is proposed and studied in this paper. Local and global stability properties of the system are investigated. Our study indicates that cannibalism has no influence on the local stability property of the equilibrium, which is quite different to the known results. Numeric simulations are carried out to show the feasibility of the main results.

Introduction

The aim of this paper is to investigate the dynamic behaviors of the following Leslie–Gower predator prey model with prey cannibalism:

$$ \begin{aligned} &\frac{dH}{dt}=(r_{1}+c_{1}-a_{1}P-b_{1}H)H- \frac{f H^{2}}{d+H}, \\ &\frac{dP}{dt}= \biggl(r_{2}-a_{2}\frac{P}{H} \biggr)P, \end{aligned} $$
(1.1)

where H and P are the density of prey species and the predator species at time t, respectively. \(r_{i}, i=1, 2\), are the intrinsic growth rate of the prey and predator species, respectively. \(r_{1}/b_{1}\) is the environment carrying capacity of the prey species, f is the cannibalism rate. \(C(H)=f\times H\times \frac{H}{H+d}\) is the generic cannibalism term. \(c_{1}H\) is the new offsprings due to the cannibalism. Obviously, \(c_{1}< f\), since it takes depredation of a number of prey by the cannibal to produce one new offspring.

During the last decades, mathematics biology has become one of the important research areas [140]. Specially, many scholars investigated the dynamic behaviors of the ecosystem with cannibalism (see [2834] and the references cited therein). Cannibalism often occurs in plankton, fishes, spiders [28], and social insect populations [29]. It is a behavior that consumes the same species and helps to provide food sources.

Leslie [35] introduced the following predator-prey model where the “carrying capacity” of the predator’s environment is proportional to the number of prey:

$$ \begin{aligned} &\frac{dH}{dt}=(r_{1}-a_{1}P-b_{1}H)H, \\ &\frac{dP}{dt}= \biggl(r_{2}-a_{2}\frac{P}{H} \biggr)P, \end{aligned} $$
(1.2)

where H and P are the density of prey species and the predator species at time t, respectively. Obviously, the above system admits a unique coexisting fixed point

$$ H^{*}=\frac{r_{1}a_{2}}{a_{1}r_{2}+a_{2}b_{1}}, \qquad P^{*}=\frac{r_{1}r_{2}}{a_{1}r_{2}+a_{2}b_{1}}. $$
(1.3)

By applying linear analysis, one could easily show that this fixed point is stable. By constructing some suitable Lyapunov function, Korobeinikov [36] showed that the fixed point is globally stable.

Chen [37] extended model (1.2) by incorporating a refuge protecting mH of the prey, where \(m\in [0,1)\) is constant. This leaves \((1-m)H\) of the prey available to the predator, and modifying system (1.2) to the following system:

$$ \begin{aligned}& \frac{dH}{dt}=(r_{1}-b_{1}H)H-a_{1}(1-m)HP, \\ &\frac{dP}{dt}= \biggl(r_{2}-a_{2}\frac{P}{(1-m)H} \biggr)P, \end{aligned} $$
(1.4)

where \(m\in [0,1)\) and \(r_{i}, a_{i}, i=1,2, b_{1}\) are all positive constants. He showed that prey refuge has no influence on the persistence property of the system.

Liang and Pan [38] proposed the following ratio-dependent Holling–Tanner model:

$$ \begin{aligned} &\frac{dx}{dt}=rx \biggl(1-\frac{x}{k} \biggr)-\frac{mx}{Ay+x}y, \\ &\frac{dy}{dt}=y \biggl[s\biggl(1-h\frac{y}{x}\biggr) \biggr], \end{aligned} $$
(1.5)

where \(r, k, m, A, s, h\) are all positive constants. Sufficient conditions which ensure the global stability of the positive equilibrium and the existence of a unique limit cycle have been obtained, respectively.

In 2016, Basheer et al. [31] proposed the prey-predator model with prey cannibalism as follows:

$$ \begin{aligned}& \frac{du}{dt}=u(1+c_{1}-u)-\frac{uv}{u+\alpha v}-c\frac{u^{2}}{u+d}, \\ &\frac{dv}{dt}=\delta v \biggl(\beta -\frac{v}{ u } \biggr), \end{aligned} $$
(1.6)

where \(c_{1}< c\), 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 all nonnegative constants. Here the generic cannibalism term \(C(u)\) is added in the prey equation and is given by

$$ C(u)=c\times u\times \frac{u}{u+d}, $$

where c is the cannibalism rate. This term has a clear gain of energy to the cannibalistic prey, and this leads to the increase in reproduction in the prey, modeled via adding a \(c_{1}u\) term to the prey equation. The authors showed that prey cannibalism cannot stabilize the unstable interior equilibrium, in certain parameter regime, but can destabilize the stable interior equilibrium, leading to a stable limit cycle. It brings to our attention that the authors only investigated the local stability property of the equilibrium and did not investigate the global stability property of the positive equilibrium, nor did they investigate the uniqueness of the limit cycle.

Recently, stimulated by the works of Basheer et al. [31, 32], based on the traditional Lotka–Volterra type predator prey system, Deng et al. [33] investigated the dynamic behaviors of the following predator-prey model with cannibalism for predator:

$$ \begin{aligned} &\frac{dx}{dt} = x(b-\alpha x-my), \\ &\frac{dy}{dt} = y(-\beta +c_{1}+nx)-\frac{cy^{2}}{y+d}, \end{aligned} $$
(1.7)

where \(c_{1}< c\), x and y are the density of the prey and predator at time t, respectively. The authors showed that cannibalism has both positive and negative effects 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 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. Also, unlike Basheer et al. [31, 32], by constructing some suitable Lyapunov function, Deng et al. [33] showed that if system (1.7) has a positive equilibrium, it then is globally asymptotically stable.

Stimulated by the works of Basheer et al. [31, 32] and Deng et al. [33], based on model (1.2), we propose the Leslie–Gower predator prey model with prey cannibalism, i.e., system (1.1). As far as system (1.1) is concerned, two interesting issues are proposed: Can we obtain sufficient conditions to ensure the existence of a unique globally stable positive equilibrium? Can we give some positive answer on the influence of the cannibalism on the dynamic behaviors of the system?

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

The existence and local stability of the equilibria of system (1.1)

Concerned with the existence of the equilibria of system (1.1), we have the following result.

Theorem 2.1

System (1.1) admits the boundary equilibrium\(A(H_{0},0)\)and the unique positive equilibrium\(B(H^{*},P^{*})\), where

$$\begin{aligned} &H_{0}=\frac{-A_{2}+\sqrt{A_{2}^{2}-4A_{1}A_{3}}}{2A_{1}}, \\ &H^{*}=\frac{-B_{2}+\sqrt{B_{2}^{2}-4B_{1}B_{3}}}{2B_{1}}, P^{*}=\frac{r_{2}H^{*}}{a_{2}}, \\ &A_{1}=b_{1}>0, \\ \begin{aligned} &A_{2}=b_{1}d-c_{1}+f-r_{1}, \\ &A_{3}=-c_{1}d-dr_{1}< 0, \end{aligned} \\ &B_{1}=a_{1}r_{2}+a_{2}b_{1}>0, \\ &B_{2}=a_{1}dr_{2}+a_{2}b_{1}d-a_{2}c_{1}+a_{2}f-a_{2}r_{1}, \\ &B_{3}=-a_{2}c_{1}d-a_{2}dr_{1}< 0. \end{aligned}$$
(2.1)

Proof

The equilibria of system (1.1) satisfy the equation

$$ \begin{aligned}& (r_{1}+c_{1}-a_{1}P-b_{1}H)H- \frac{f H^{2}}{d+H}=0, \\ &\biggl(r_{2}-a_{2}\frac{P}{H} \biggr)P=0. \end{aligned} $$
(2.2)

From the second equation of (2.2), one has \(P=0\) or \(P=\frac{r_{2}H}{a_{2}}\). Substituting \(P=0\) to the first equation of (2.2) leads to

$$ (r_{1}+c_{1}-b_{1}H)-\frac{f H}{d+H}=0. $$
(2.3)

Equation (2.3) is equivalent to

$$ A_{1}H^{2}+A_{2}H+A_{3}=0, $$
(2.4)

where \(A_{1}, A_{2}, A_{3}\) are defined by (2.1). (2.4) has a unique positive solution \(H_{0}\), hence, system (1.1) has the boundary equilibrium \(A(H_{0},0)\).

Substituting \(P=\frac{r_{2}H}{a_{2}}\) to the first equation of (2.2) leads to

$$ H \biggl( -H{ b_{1}}-{\frac{H{ a_{1}} { r_{2}}}{{ a_{2}}}}+{c_{1}}+{ r_{1}} \biggr) -{\frac{f{H}^{2}}{d+H}}=0. $$
(2.5)

Equation (2.5) is equivalent to

$$ B_{1}H^{2}+B_{2}H+B_{3}=0, $$
(2.6)

where \(B_{1}, B_{2}, B_{3}\) are defined by (2.1). (2.6) has a unique positive solution \(H^{*}\), hence, system (1.1) has the boundary equilibrium \(B(H^{*},P^{*})\).

This ends the proof of Theorem 2.1. □

Theorem 2.2

\(A(H_{0},0)\)is unstable equilibrium, and\(B(H^{*},P^{*})\)is locally asymptotically stable.

Proof

The Jacobian matrix of system (1.1) is calculated as follows:

$$ J(H,P)= \begin{pmatrix} A_{11}& -a_{1}H \\ \frac{P^{2}a_{2}}{H^{2}}& r_{2}-2\frac{a_{2}P}{H}\end{pmatrix}, $$
(2.7)

where

$$ A_{11}=-2 H{ b_{1}}-P{ a_{1}}+{ c_{1}}+{ r_{1}}-2 {\frac{Hf}{d+H}}+{ \frac{f{H}^{2}}{ ( d+H ) ^{2}}}. $$

Then the Jacobian matrix of system (1.1) about the equilibrium \(A(H_{0},0)\) is

$$\begin{aligned} &J \bigl(A(H_{0},0) \bigr) \\ &\quad = \begin{pmatrix} -2 H_{0}{ b_{1}} +{ c_{1}}+{ r_{1}}-2 {\frac{H_{0}f}{d+H_{0}}}+{\frac{f{H_{0}}^{2}}{ ( d+H_{0} ) ^{2}}} &-a_{1}H_{0} \\ 0 & r_{2}\end{pmatrix} \\ &\quad = \begin{pmatrix} - H_{0}{ b_{1}} -{\frac{df{H_{0}}}{ ( d+H_{0} ) ^{2}}} &-a_{1}H_{0} \\ 0 & r_{2}\end{pmatrix}. \end{aligned}$$
(2.8)

The eigenvalues of \(J(A)\) are \(\lambda _{1}=r_{2}>0\), \(\lambda _{2}= - H_{0}{ b_{1}} -{ \frac{df{H_{0}}}{ ( d+H_{0} ) ^{2}}} <0\). Thus, \(A(H_{0},0)\) is a saddle.

The Jacobian matrix of system (1.1) about the equilibrium \(B(H^{*},P^{*})\) is

$$ \begin{aligned} J \bigl(B\bigl(H^{*},P^{*}\bigr) \bigr) = \begin{pmatrix} - H^{*}{ b_{1}}-{\frac{df{H^{*}} }{ ( d+H^{*} ) ^{2}}} &-a_{1}H^{*} \\ r_{2}\frac{P^{*}}{H^{*}}& -r_{2}\end{pmatrix}.\end{aligned} $$
(2.9)

Then we have

$$ \operatorname{Det}J (B\bigl(H^{*},P^{*}\bigr)=r_{2} \biggl( H^{*}{ b_{1}}+{\frac{df{H^{*}} }{ ( d+H^{*} ) ^{2}}} \biggr)+ a_{1}H^{*} r_{2} \frac{P^{*}}{H^{*}}>0 $$

and

$$ \operatorname{Tr}J (B\bigl(H^{*},P^{*}\bigr)=- H^{*}{ b_{1}}-{\frac{df{H^{*}} }{ ( d+H^{*} ) ^{2}}}-r_{2} < 0. $$

So that both eigenvalues of \(J (B(H^{*},P^{*}))\) have negative real parts [38], and \(B(H^{*},P^{*})\) is locally asymptotically stable.

This ends the proof of Theorem 2.2. □

Global attractivity

The aim of this section is to investigate the global attractivity of the positive equilibrium of system (1.1). To do so, we need the following lemma.

Lemma 3.1

If\(a>0, b>0\)and\(\dot{x}\geq x(b-ax)\), when\(t\geq {0}\)and\(x(0)>0\), we have

$$ \liminf_{t\rightarrow +\infty } x(t)\geq \frac{b}{a}. $$

If\(a>0, b>0\)and\(\dot{x}\leq x(b-ax)\), when\(t\geq {0}\)and\(x(0)>0\), we have

$$ \limsup_{t\rightarrow +\infty } x(t)\leq \frac{b}{a}. $$

The above lemma is a direct corollary of Lemma 2.2 of Chen [40], so we omit the detailed proof here.

Concerned with the global attractivity of the positive equilibrium, we have the following result.

Theorem 3.1

The positive equilibrium\(B(H^{*},P^{*})\)is globally attractive provided that

$$ a_{1}r_{2}< a_{2}b_{1} $$
(3.1)

holds.

Proof

From (3.1), we could choose \(\varepsilon >0\) small enough such that

$$ r_{1}+c_{1}-a_{1}\frac{r_{2}}{a_{2}} \biggl( \frac{r_{1}+c_{1}}{b_{1}}+\varepsilon \biggr)-\biggl(2+\frac{a_{2}}{r_{2}}\biggr) \varepsilon >0. $$
(3.2)

Indeed, inequality (3.1) is equivalent to

$$ r_{1}+c_{1}>a_{1}\frac{r_{2}}{a_{2}}\frac{(r_{1}+c_{1})}{b_{1}}, $$

hence, for \(\varepsilon >0\), which satisfies

$$ \varepsilon < \frac{r_{1}+c_{1}-a_{1}\frac{r_{2}}{a_{2}}\frac{(r_{1}+c_{1})}{b_{1}}}{ a_{1}\frac{r_{2}}{a_{2}}+2+\frac{a_{2}}{r_{2}} }, $$

inequality (3.2) holds.

Let \((H(t),P(t))\) be any positive solution of system (1.1). From the first equation of (1.1), we have

$$\begin{aligned} \frac{dH}{dt}&=(r_{1}+c_{1}-a_{1}P-b_{1}H)H- \frac{f H^{2}}{d+H} \\ &\leq (r_{1}+c_{1} -b_{1}H)H. \end{aligned}$$
(3.3)

Applying Lemma 3.1 to (3.3) leads to

$$ \limsup_{t\rightarrow +\infty }H(t)\leq \frac{r_{1}+c_{1}}{b_{1}}. $$
(3.4)

For \(\varepsilon >0\) small enough, which satisfies (3.2), it follows from (3.4) that there exists \(T_{11}>0\) such that

$$ H(t)< \frac{r_{1}+c_{1}}{b_{1}}+\varepsilon \stackrel{\mathrm{def}}{=}M_{1}^{(1)}. $$
(3.5)

From the second equation of (1.1), for \(t>T_{11}\), we have

$$\begin{aligned} \frac{dP}{dt}&= \biggl(r_{2}-a_{2} \frac{P}{H} \biggr)P \\ &\leq \biggl(r_{2}-a_{2}\frac{P}{M_{1}^{(1)}} \biggr)P. \end{aligned}$$
(3.6)

Applying Lemma 3.1 to (3.6) leads to

$$ \limsup_{t\rightarrow +\infty }P(t)\leq \frac{r_{2}M_{1}^{(1)}}{a_{2}}. $$
(3.7)

For \(\varepsilon >0\) small enough, which satisfies (3.2), it follows from (3.7) that there exists \(T_{12}>0\) such that

$$ P(t)< \frac{r_{2}M_{1}^{(1)}}{a_{2}}+\varepsilon \stackrel{\mathrm{def}}{=}M_{2}^{(1)}. $$
(3.8)

From the first equation of (1.1) and (3.8), for \(t\geq T_{12}\), we also have

$$\begin{aligned} \frac{dH}{dt}&=(r_{1}+c_{1}-a_{1}P-b_{1}H)H- \frac{f H^{2}}{d+H} \\ &\geq \biggl(r_{1}+c_{1} -a_{1}M_{2}^{(1)}- \biggl(b_{1}+\frac{f}{d}\biggr)H \biggr)H. \end{aligned}$$
(3.9)

From (3.2) we could see that inequality (3.1) implies that

$$ r_{1}+c_{1} -a_{1}M_{2}^{(1)}>0. $$
(3.10)

Hence, applying Lemma 3.1 to (3.9) leads to

$$ \liminf_{t\rightarrow +\infty }H(t)\geq \frac{r_{1}+c_{1} -a_{1}M_{2}^{(1)}}{b_{1}+\frac{f}{d}}. $$
(3.11)

For \(\varepsilon >0\), which satisfies (3.2), it follows from (3.11) that there exists \(T_{13}>T_{12}\) such that

$$ H(t)>\frac{r_{1}+c_{1} -a_{1}M_{2}^{(1)}}{b_{1}+\frac{f}{d}}- \varepsilon \stackrel{\mathrm{def}}{=}m_{1}^{(1)}. $$
(3.12)

From (3.12) and the second equation of (1.1), for \(t\geq T_{13}\), we have

$$\begin{aligned} \frac{dP}{dt}&= \biggl(r_{2}-a_{2} \frac{P}{H} \biggr)P \\ &\geq \biggl(r_{2}-a_{2}\frac{P}{m_{1}^{(1)}} \biggr)P. \end{aligned}$$
(3.13)

Applying Lemma 3.1 to (3.13) leads to

$$ \liminf_{t\rightarrow +\infty }P(t)\geq \frac{r_{2}m_{1}^{(1)}}{a_{2}}. $$
(3.14)

For \(\varepsilon >0\), which satisfies (3.2), it follows from (3.14) that there exists \(T_{14}>T_{13}\) such that

$$ P(t)>\frac{r_{2}m_{1}^{(1)}}{a_{2}}-\varepsilon \stackrel{\mathrm{def}}{=}m_{2}^{(1)}. $$
(3.15)

From the first equation of (1.1), for \(t>T_{14}\), we have

$$\begin{aligned} \frac{dH}{dt}&=(r_{1}+c_{1}-a_{1}P-b_{1}H)H- \frac{f H^{2}}{d+H} \\ &\leq \biggl(r_{1}+c_{1}-a_{1}m_{2}^{(1)} - \biggl(b_{1}+\frac{f}{d+M_{1}^{(1)}} \biggr)H \biggr)H. \end{aligned}$$
(3.16)

Applying Lemma 3.1 to (3.16) leads to

$$ \limsup_{t\rightarrow +\infty }H(t)\leq \frac{r_{1}+c_{1}-a_{1}m_{2}^{(1)}}{b_{1}+\frac{f}{d+M_{1}^{(1)}}}. $$
(3.17)

For \(\varepsilon >0\) small enough, which satisfies (3.2), it follows from (3.17) that there exists \(T_{21}>0\) such that

$$ H(t)< \frac{r_{1}+c_{1}-a_{1}m_{2}^{(1)}}{b_{1}+\frac{f}{d+M_{1}^{(1)}}}+ \frac{\varepsilon }{2}\stackrel{\mathrm{def}}{=}M_{1}^{(2)}. $$
(3.18)

Obviously,

$$ M_{1}^{(2)}=\frac{r_{1}+c_{1}-a_{1}m_{2}^{(1)}}{b_{1}+\frac{f}{d+M_{1}^{(1)}}}+ \frac{\varepsilon }{2}< \frac{r_{1}+c_{1}}{b_{1}}+\varepsilon =M_{1}^{(1)}. $$
(3.19)

From the second equation of (1.1), for \(t>T_{21}\), we have

$$\begin{aligned} \frac{dP}{dt}&= \biggl(r_{2}-a_{2} \frac{P}{H} \biggr)P \\ &\leq \biggl(r_{2}-a_{2}\frac{P}{M_{1}^{(2)}} \biggr)P. \end{aligned}$$
(3.20)

Applying Lemma 3.1 to (3.20) leads to

$$ \limsup_{t\rightarrow +\infty }P(t)\leq \frac{r_{2}M_{1}^{(2)}}{a_{2}}. $$
(3.21)

For \(\varepsilon >0\) small enough, which satisfies (3.2), it follows from (3.21) that there exists \(T_{22}>0\) such that

$$ P(t)< \frac{r_{2}M_{1}^{(2)}}{a_{2}}+\varepsilon \stackrel{\mathrm{def}}{=}M_{2}^{(2)}. $$
(3.22)

From (3.8), (3.19), and (3.22), one has

$$ M_{2}^{(2)}< M_{2}^{(1)}. $$
(3.23)

From the first equation of (1.1) and (3.8), for \(t\geq T_{12}\), we also have

$$\begin{aligned} \frac{dH}{dt}&=(r_{1}+c_{1}-a_{1}P-b_{1}H)H- \frac{f H^{2}}{d+H} \\ &\geq \biggl(r_{1}+c_{1}-a_{1}M_{2}^{(2)} - \biggl(b_{1}+\frac{f}{d+m_{1}^{(1)}} \biggr)H \biggr)H. \end{aligned}$$
(3.24)

From (3.2) and (3.23) we could see that

$$ r_{1}+c_{1} -a_{1}M_{2}^{(2)}>0. $$
(3.25)

Hence, applying Lemma 3.1 to (3.24) leads to

$$ \liminf_{t\rightarrow +\infty }H(t)\geq \frac{r_{1}+c_{1} -a_{1}M_{2}^{(2)}}{b_{1}+\frac{f}{d+m_{1}^{(1)}}}. $$
(3.26)

For \(\varepsilon >0\), which satisfies (3.2), it follows from (3.26) that there exists \(T_{23}>T_{22}\) such that

$$ H(t)>\frac{r_{1}+c_{1} -a_{1}M_{2}^{(2)}}{b_{1}+\frac{f}{d+m_{1}^{(1)}}}- \frac{\varepsilon }{2}\stackrel{\mathrm{def}}{=}m_{1}^{(2)}. $$
(3.27)

From (3.12), (3.23), and (3.27) one has

$$ m_{1}^{(2)}>m_{1}^{(1)}. $$
(3.28)

From (3.28) and the second equation of (1.1), for \(t\geq T_{23}\), we have

$$\begin{aligned} \frac{dP}{dt}&= \biggl(r_{2}-a_{2} \frac{P}{H} \biggr)P \\ &\geq \biggl(r_{2}-a_{2}\frac{P}{m_{1}^{(2)}} \biggr)P. \end{aligned}$$
(3.29)

Applying Lemma 3.1 to (3.29) leads to

$$ \liminf_{t\rightarrow +\infty }P(t)\geq \frac{r_{2}m_{1}^{(2)}}{a_{2}}. $$
(3.30)

For \(\varepsilon >0\), which satisfies (3.2), it follows from (3.30) that there exists \(T_{24}>T_{23}\) such that

$$ P(t)>\frac{r_{2}m_{1}^{(2)}}{a_{2}}-\varepsilon \stackrel{\mathrm{def}}{=}m_{2}^{(2)}. $$
(3.31)

From (3.15), (3.28), and (3.31) one has

$$ m_{2}^{(2)}>m_{2}^{(1)}. $$
(3.32)

It follows from (3.18), (3.22), (3.27), and (3.31) that, for all \(t\geq T_{24}\),

$$ \begin{aligned} &0< m_{1}^{(1)}< m_{1}^{(2)}< x(t)< M_{1}^{(2)}< M_{1}^{(1)}, \\ &0< m_{2}^{(1)}< m_{2}^{(2)} < y(t)< M_{2}^{(2)}< M_{2}^{(1)}. \end{aligned} $$
(3.33)

Repeating the above procedure, we get four sequences \(M_{i}^{(n)}, m_{i}^{(n)}, i=1, 2, n=1, 2,\ldots\) , such that

$$\begin{aligned} &M_{1}^{(n)}=\frac{r_{1}+c_{1}-a_{1}m_{2}^{(n-1)}}{b_{1}+\frac{f}{d+M_{1}^{(n-1)}}}+ \frac{\varepsilon }{n}, \end{aligned}$$
(3.34)
$$\begin{aligned} &m_{1}^{(n)}=\frac{r_{1}+c_{1} -a_{1}M_{2}^{(n)}}{b_{1}+\frac{f}{d+m_{1}^{(n-1)}}} -\frac{\varepsilon }{n}, \end{aligned}$$
(3.35)
$$\begin{aligned} &M_{2}^{(n)}=\frac{r_{2}M_{1}^{(n)}}{a_{2}}+\frac{\varepsilon }{n}, \end{aligned}$$
(3.36)
$$\begin{aligned} &m_{2}^{(n)}=\frac{r_{2}m_{1}^{(n)}}{a_{2}}-\frac{\varepsilon }{n}. \end{aligned}$$
(3.37)

Now, we will show that the sequences \(M_{i}^{(n)}\) are strictly decreasing, and the sequences \(m_{i}^{(n)}\) are strictly increasing for \(i=1, 2\) by induction. Firstly, from (3.33), we have

$$ m_{i}^{(1)}< m_{i}^{(2)},\qquad M_{i}^{(2)}< M_{i}^{(1)},\quad i=1,2. $$
(3.38)

Let us suppose that

$$ m_{i}^{(n-1)}< m_{i}^{(n)},\qquad M_{i}^{(n)}< M_{i}^{(n-1)},\quad i=1,2. $$
(3.39)

It then follows from (3.34) and (3.39) that

$$ M_{1}^{(n)}>M_{1}^{(n+1)}. $$
(3.40)

By using (3.40), it follows from (3.36) that

$$ M_{2}^{(n)}>M_{2}^{(n+1)}. $$
(3.41)

It then follows from (3.35), (3.39), and (3.41) that

$$ m_{1}^{(n+1)}>m_{1}^{(n)}. $$
(3.42)

From (3.37) and (3.42), we have

$$ m_{2}^{(n+1)}>m_{2}^{(n)}. $$
(3.43)

Therefore, we have

$$ \begin{aligned} &0< m_{1}^{(1)}< m_{1}^{(2)}< \cdots < m_{1}^{(n)}< x(t)< M_{1}^{(n)}< \cdots < M_{1}^{(2)}< M_{1}^{(1)}, \\ &0< m_{2}^{(1)}< m_{2}^{(2)}< \cdots < m_{2}^{(n)}< y(t)< M_{2}^{(n)}< \cdots < M_{2}^{(2)}< M_{2}^{(1)}. \end{aligned} $$
(3.44)

Hence, the limits of \(M_{i}^{(n)}\) and \(m_{i}^{(n)}\), \(i=1, 2, n=1, 2,\ldots \) , exist. Denote that

$$ \lim_{n\rightarrow +\infty }M_{1}^{(n)}=\overline{H},\qquad \lim _{n\rightarrow +\infty }m_{1}^{(n)}=\underline{H},\qquad \lim _{n\rightarrow +\infty }M_{2}^{(n)}=\overline{P},\qquad \lim _{n\rightarrow +\infty }m_{2}^{(n)}=\underline{P}. $$
(3.45)

Then \(\overline{H}\geq \underline{H}, \overline{P}\geq \underline{P}\). Letting \(n\rightarrow +\infty \) in (3.34)–(3.37), we obtain

$$\begin{aligned} &\overline{H}=\frac{r_{1}+c_{1}-a_{1}\underline{P}}{b_{1}+\frac{f}{d+\overline{H}}}, \end{aligned}$$
(3.46)
$$\begin{aligned} &\underline{H}=\frac{r_{1}+c_{1} -a_{1}\overline{P}}{b_{1}+\frac{f}{d+ \underline{H}}}, \end{aligned}$$
(3.47)
$$\begin{aligned} &\overline{P}=\frac{r_{2}\overline{H}}{a_{2}}, \end{aligned}$$
(3.48)
$$\begin{aligned} &\underline{P}=\frac{r_{2}\underline{H}}{a_{2}}. \end{aligned}$$
(3.49)

Substituting (3.48) and (3.49) to (3.46) and (3.47), it then follows

$$\begin{aligned} &b_{1}\overline{H}+\frac{f\overline{H}}{d+\overline{H}}=r_{1}+c_{1}-a_{1} \frac{r_{2}\underline{H}}{a_{2}}, \end{aligned}$$
(3.50)
$$\begin{aligned} &b_{1}\underline{H}+\frac{f\underline{H}}{d+ \underline{H}}=r_{1}+c_{1} -a_{1}\frac{r_{2}\overline{H}}{a_{2}}. \end{aligned}$$
(3.51)

Subtracting the above two equalities leads to

$$ b_{1}(\overline{H}-\underline{H})+\frac{f\overline{H}}{d+\overline{H}} -\frac{f\underline{H}}{d+ \underline{H}} =a_{1}\frac{r_{2} }{a_{2}}(\overline{H}- \underline{H}), $$
(3.52)

which is equivalent to

$$ \biggl(b_{1}-a_{1}\frac{r_{2} }{a_{2}}\biggr) ( \overline{H}-\underline{H})+ \frac{df(\overline{H}-\underline{H})}{(d+ \overline{H})(d+ \underline{H})}=0. $$
(3.53)

Since condition (3.1) implies that

$$ b_{1}-a_{1}\frac{r_{2} }{a_{2}}>0. $$
(3.54)

Also

$$ df>0. $$
(3.55)

Hence, it follows from (3.53) that

$$ \overline{H}=\underline{H}. $$
(3.56)

From (3.48), (3.49), and (3.56), we have

$$ \overline{P}=\underline{P}. $$

Under the assumption of Theorem 3.1, system (1.1) admits a unique positive solution \((H^{*}, P^{*})\), hence \(\overline{H}=\underline{H}=H^{*}, \overline{P}=\underline{P}=P^{*}\). That is to say,

$$ \lim_{t\rightarrow +\infty }H(t)=H^{*}, \qquad\lim_{t \rightarrow +\infty }P(t)=P^{*}. $$
(3.57)

This ends the proof of Theorem 3.1. □

Numeric simulations

Example 4.1

Now let us consider the following model:

$$ \begin{aligned} &\frac{dH}{dt}=(r_{1}+c_{1}-a_{1}P-b_{1}H)H- \frac{f H^{2}}{d+H}, \\ &\frac{dP}{dt}= \biggl(r_{2}-a_{2}\frac{P}{H} \biggr)P. \end{aligned} $$
(4.1)

Here, corresponding to system (1.1), we take \(r_{1}=b_{1}=f=d=r_{2}=a_{2}=1\), \(c_{1}=0.5, a_{1}=0.2\), then one could see that

$$ a_{1}r_{2}=0.2< a_{2}b_{1}=1. $$

Hence, it follows from Theorem 3.1 that system (4.1) admits a unique positive equilibrium, which is globally attractive. Numeric simulations (Figs. 1 and 2) also support this assertion.

Figure 1
figure 1

Dynamic behaviors of the first species in system (4.1), the initial conditions \((H(0), P(0))=(1.5,1.5)\), \((1.5, 0.3)\), \((0.2, 0.1)\), and \((0.4, 1.5)\), respectively

Figure 2
figure 2

Dynamic behaviors of the second species in system (4.1), the initial conditions \((H(0), \text{P(0))=(1.5,1.5)}\), \((1.5, 0.3)\), \((0.2, 0.1)\), and \((0.4, 1.5)\), respectively

Example 4.2

Now let us consider the following model:

$$ \begin{aligned} &\frac{dH}{dt}=(r_{1}+c_{1}-a_{1}P-b_{1}H)H- \frac{f H^{2}}{d+H}, \\ &\frac{dP}{dt}= \biggl(r_{2}-a_{2}\frac{P}{H} \biggr)P. \end{aligned} $$
(4.2)

Here, corresponding to system (1.1), we take \(r_{1}=b_{1}=f=d=a_{2}=1\), \(c_{1}=0.5, a_{1}=2, r_{2}=2\). Then one could see that

$$ a_{1}r_{2}=4>a_{2}b_{1}=1. $$

Hence, inequality (3.1) does not hold, one could make a conclusion from Theorem 2.1 that system (4.2) admits a unique positive equilibrium, which is locally asymptotically stable. However, one could not draw any conclusion about the global asymptotic stability of the positive equilibrium. Numeric simulation (Fig. 3) shows that the positive equilibrium is globally asymptotically stable.

Figure 3
figure 3

Dynamic behaviors of the second species in system (4.2), the initial conditions \((H(0), P(0))=(1,0.1)\), \((1, 0.)\), \((1, 0.4)\), and \((1, 0.5)\), respectively

Conclusion

Based on the traditional Leslie–Gower predator prey model and the works of Basheer et al. [31, 32] and Deng et al. [33], we proposed a Leslie–Gower predator prey model incorporating the nonlinear cannibalism. Already Basheer et al. [31] incorporated the cannibalism to the Holling–Tanner model with ratio-dependent functional response (i.e., system (1.2)). They showed that cannibalism in the prey cannot stabilize the unstable interior equilibrium in the ODE case, but can destabilize the stable interior equilibrium, leading to a stable limit cycle. In this paper, we focus our attention on the most simple Leslie–Gower predator prey model. Our study shows that the system with cannibalism always admits a positive equilibrium and a predator free equilibrium, the predator free equilibrium is unstable, while the positive equilibrium is locally asymptotically stable. Compared with the Leslie–Gower predator prey system without cannibalism (i.e., system (1.2)), our result shows that cannibalism has no influence on the local stability property of the positive equilibrium, this is quite different to the results of Basheer et al. [31, 32]. Our results are also different to those of Deng et al. [33]: for the system they considered, cannibalism may have both positive or negative effects on the stability of the system. Also, under some very simple conditions, we could also prove that the positive equilibrium is globally attractive. It brings to our attention that inequality (3.1) is independent of the coefficients of the cannibalism term. However, for the system without cannibalism (i.e., system (1.2)), the positive equilibrium is globally attractive without any restriction on the coefficients, and numeric simulation (Fig. 3) also shows that (3.1) could be dropped out. Hence we have a conjecture:

Conjecture

Condition (3.1) is not needed to ensure the positive equilibrium of system (1.1) to be globally attractive.

However, at present, we have difficulty to prove this conjecture, we leave this for future investigation.

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Acknowledgements

The authors would like to thank Dr. Hang Deng for bringing our attention to the paper of A. Basheer.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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The research was supported by the Natural Science Foundation of Fujian Province (2019J01841).

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Lin, Q., Liu, C., Xie, X. et al. Global attractivity of Leslie–Gower predator-prey model incorporating prey cannibalism. Adv Differ Equ 2020, 153 (2020). https://doi.org/10.1186/s13662-020-02609-w

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Keywords

  • Leslie–Gower predator prey model
  • Cannibalism
  • Stability