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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 17, Issue 1

# Stability property of the prey free equilibrium point

Qin Yue
• Corresponding author
• College of Finance and Mathematics, West Anhui University, Liuan 237000, Anhui, People’s Republic of China
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Published Online: 2019-07-09 | DOI: https://doi.org/10.1515/math-2019-0051

## Abstract

We revisit a prey-predator model with stage structure for predator, which was proposed by Tapan Kumar Kar. By using the differential inequality theory and the comparison theorem of the differential equation, we show that the prey free equilibrium is globally asymptotically stable under some suitable assumption. Our study shows that although the predator species has other food resource, if the amount of the predator species is too large, it could also do irreversible harm to the prey species, and this could finally lead to the extinction of the prey species. Our result supplement and complement some known results.

MSC 2010: 34D23; 92B05; 34D40

## 1 Introduction

During the last decades, many scholars investigated the dynamic behaviors of the stage structured ecosystem, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and the references cited therein. In their series papers, Chen et al[1, 2, 3] studied the stability, persistence and extinction property of a stage-structured predator-prey system, and they found that despite the extinction of the prey species, the predator species could still be permanent, they argued that the reason maybe relies on the predator species has other food resources. In [4], Chen et al showed that stage structure plays important roles on the persistent and extinction property of the cooperation system. The results of [3] is generalized by Pu et al[10] to the infinite delay case. Also, many scholars[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] investigated the extinction property of the ecosystem. For example, Zhao et al[27] proposed a cooperative system with strong and weak partner, and they showed that the strong species may be driven to extinction under some suitable assumption. Chen et al[29] studied the extinction property of a two species nonlinear competition system. Yang et al[28] showed that single feedback control variable could lead to the extinction of the species.

Kar[11] proposed the following prey-predator model with stage structure for predator,

$dN1dt=r1N1(1−N1k)−αN1N3,dN2dt=βN3−r2N2,dN3dt=−r3N3+mαN1N3+γN2−δN32,$(1.1)

where N1(t),N2(t) and N3(t) are the population densities of prey, juvenile predator and adult predator, respectively. r1, k, α, β, r2, r3, m and γ are all positive constants. Define

$N1=Kr2r1x1,N2=βx2mα,N3=r2x3mα,t=τr2.$

Then system (1.1) can be rewrite as

$dx1dτ=ax1−x12−bx1x3,dx2dτ=x3−x2,dx3dτ=−cx3+dx1x3+ex2−fx32.$(1.2)

The possible non-negative equilibria of system (1.2) are P0(0, 0, 0), P1(a, 0, 0) and $\begin{array}{}{P}_{2}\left({x}_{1}^{\ast },{x}_{2}^{\ast },{x}_{3}^{\ast }\right).\end{array}$ Under the assumption ce, the author investigated the stability property of the above three equilibria. The author also pointed out “ We remark that if e > c, then there exists another equilibrium in the absence of prey. But it is not feasible since prey is the only source of food for the predator.” Indeed, this equilibrium could be expressed as $\begin{array}{}{P}_{3}\left(0,\frac{e-c}{f},\frac{e-c}{f}\right)\end{array}$. We mention here that generally speaking, predator may have many resources as its food, and seldom did predator species take only one kind of prey species as its food resource. For example, the Chinese Alligator can be regarded as a stage-structured predator species since the mature is more than ten years old, and the Chinese Alligator almost eat all acquatic animals. Certainly, if one kind of prey species is scare, it will take other prey species as its food resource. Hence, we argue that it is necessary to reconsider the declaration of the T. K. Kar, and we should investigate the stability property of the equilibrium $\begin{array}{}{P}_{3}\left(0,\frac{e-c}{f},\frac{e-c}{f}\right)\end{array}$.

The aim of this paper is to give sufficient condition to ensure the global asymptotically stable of the equilibrium $\begin{array}{}{P}_{3}\left(0,\frac{e-c}{f},\frac{e-c}{f}\right)\end{array}$ of system (1.2), more precisely, we have the following result.

#### Theorem 1.1

Assume that

$af+bc(1.3)

holds, then $\begin{array}{}{P}_{3}\left(0,\frac{e-c}{f},\frac{e-c}{f}\right)\end{array}$ is globally asymptotically stable.

We mention here that the method we used here is quite different with that of the method used in [11]. Indeed, we only use the differential inequality theory and the comparison theorem of the differential equation. We will prove Theorem 1.1 in the next section, and a numeric example is presented in Section 3 to show the feasibility of the main results. We end this paper by a briefly discussion.

## 2 Proof of Theorem 1.1

Now let’s consider the system

$dx1dt=αx2−βx1−δ1x1,dx2dt=βx1−δ2x2−γx22,$(2.1)

where α, β, δ1, δ2 and γ are all positive constants, x1(t) and x2(t) are the densities of the immature and mature species at time t. From Theorem 4.1 in Xiao and Lei[17], we have

#### Lemma 2.1

Assume that

$α>δ2(1+δ1β)$(2.2)

holds, then the positive equilibrium $\begin{array}{}B\left({x}_{1}^{\ast \ast },{x}_{2}^{\ast \ast }\right)\end{array}$ of system (2.1) is globally stable, where

$x1∗∗=αx2∗∗β+δ1,x2∗∗=αβ−δ2(β+δ1)γ(β+δ1).$(2.3)

Now let’s consider the system

$dx2dτ=x3−x2,dx3dτ=−cx3+ex2−fx32,$(2.4)

As a direct corollary of Lemma 2.1, we have

#### Lemma 2.2

Assume that

$e>c$(2.5)

holds, then the positive equilibrium $\begin{array}{}E\left(\frac{e-c}{f},\frac{e-c}{f}\right)\end{array}$ of system (2.4) is globally stable.

#### Proof of Theorem 1.1

Let (x1(t), x2(t), x3(t)) be any positive solution of the system (1.2). From the second and the third equation of system (1.1), we have

$dx2dτ=x3−x2,dx3dτ≥−cx3+ex2−fx32,$(2.6)

Now let’s consider the system

$du2dτ=u3−u2,du3dτ=−cu3+eu2−fu32,$(2.7)

Noting that condition (1.3) implies that e > c, and so, from Lemma 2.2, (2.7) admits a unique globally asymptotically stable positive equilibrium $\begin{array}{}E\left(\frac{e-c}{f},\frac{e-c}{f}\right)\end{array}$. That is, let (u1(t), u2(t)) be any positive solution of the system (2.7), one has

$limt→+∞u2(t)=limt→+∞u3(t)=e−cf.$(2.8)

Let (x1(t), x2(t), x3(t)) be any positive solution of system (1.2) with initial condition (x1(0), x2(0), x3(0)) = (x10, x20, x30), and let (u1(t), u2(t)) be the positive solution of system (2.7) with the initial condition (u2(0), u3(0)) = (x20,x30), it then follows from the differential inequality theory that

$xi(t)≥ui(t)forallt≥0,i=1,2.$(2.9)

The positivity of the solution of system (1.2), (2.8) and (2.9) lead to

$lim inft→+∞xi(t)≥limt→+∞ui(t)=e−cf,i=1,2.$(2.10)

Condition (1.3) implies that for enough small positive constant ε > 0, the following inequality holds.

$af+bc−be+bfε<0,$(2.11)

which is equivalent to

$a−b(e−cf−ε)<0.$(2.12)

For ε > 0 enough small, which satisfies (2.12), it then follows from (2.10) that there exists an enough large T1 > 0 such that

$xi(t)>e−cf−ε,i=1,2,forallt>T1.$(2.13)

Hence, for t > T1, from the first equation of system (1.2) and (2.13), we have

$dx1dτ=ax1−x12−bx1x3,≤ax1−x12−bx1(e−cf−ε)≤(a−b(e−cf−ε))x1.$

Consequently,

$x1(t)≤x1(T1)exp⁡{(a−b(e−cf−ε))(t−T1)}→0ast→∞.$(2.14)

That is,

$limt→+∞x1(t)=0.$(2.15)

For ε > 0 enough small, it follows from (2.15) that there exists a T2 > T1 such that

$x1(t)<εforallt≥T2.$(2.16)

For t > T2, from the second and third equation of system (1.2) and (2.16), we have

$dx2dτ=x3−x2,dx3dτ≤−cx3+dεx3+ex2−fx32,$(2.17)

Now let’s consider the system

$dv2dτ=v3−v2,dv3dτ=−cv3+dεv3+ev2−fv32,$(2.18)

it follows from e > c and Lemma 2.1 that (2.18) admits a unique globally asymptotically stable positive equilibrium $\begin{array}{}{E}_{1}\left(\frac{e-c+d\epsilon }{f},\frac{e-c+d\epsilon }{f}\right),\end{array}$ That is, let (v1(t), v2(t)) be any positive solution of the system (2.18), one has

$limt→+∞v1(t)=e−c+dεf,limt→+∞v2(t)=e−c+dεf.$(2.19)

Let (x1(t), x2(t), x3(t)) be any positive solution of system (1.2) with initial condition (x1(T2), x2(T2), x3(T2)) = (x10, x20, x30), and let (v2(t), v3(t)) be the positive solution of system (2.18) with the initial condition (v2(T2), v3(T2)) = (x20,x30), it then follows from the differential inequality theory that

$xi(t)≤vi(t)forallt≥T2.$(2.20)

The positivity of the solution of system (1.2), (2.19) and (2.20) lead to

$lim supt→+∞xi(t)≤limt→+∞vi(t)=e−c+dεfi=1,2.$(2.21)

(2.10) and (2.18) show that

$e−cf≤lim inft→+∞xi(t)≤lim supt→+∞xi(t)≤e−c+dεf,i=1,2.$(2.22)

Since ε could be any enough small positive constant, now, letting ε → 0 in (2.22) leads to

$limt→+∞xi(t)=e−cf,i=1,2.$(2.23)

(2.23) together with (2.15) shows that $\begin{array}{}{P}_{3}\left(0,\frac{e-c}{f},\frac{e-c}{f}\right)\end{array}$ is globally asymptotically stable. This completes the proof of Theorem 2.1.

## 3 Numeric simulation

#### Example 3.1

Consider the following stage structure predator prey system

$dx1dτ=x1−x12−x1x3,dx2dτ=x3−x2,dx3dτ=−x3+x1x3+3x2−x32.$(3.1)

Here, corresponding to system (1.2), we take a = b = c = d = f = 1, e = 3. Since af + bc = 2 < 3 = be, it follows from Theorem 1.1 that P3(0, 2, 2) is globally asymptotically stable. Numeric simulation (Fig. 1) also supports this assertion.

Figure 1

Dynamics behaviors of of system (3.1), the initial conditions (x1(0), x2(0), x3(0)) = (3, 0.5, 2), (3, 1, 2), (1, 0.5, 2) and (3, 2, 2), respectively.

## 4 Discusion

Kar[11] proposed a stage structured predator prey system (i.e., system (1.1)), he investigated the global stability property of the equilibria, however, he did not investigated the stability property of the prey free equilibrium P3. In this paper, by using the differential inequality theory and the comparison theorem of the differential equation, we could show that under some suitable condition, the boundary equilibrium P3 is globally asymptotically stable. Our result has significant biological meaning. In system (1.2), without consider the relationship of the predator and prey species, then prey species is governed by the equation

$dx1dτ=ax1−x12,$(4.1)

which is a Logistic equation, and the positive equilibrium $\begin{array}{}{x}_{1}^{\ast }=a\end{array}$ is globally asymptotically stable. That is, without the influence of the predator species, the prey species could be exist in long run.

Also, without consider the influence of the prey species, the predator species satisfies the system

$dx2dτ=x3−x2,dx3dτ=−cx3+ex2−fx32,$(4.2)

From Lemma 2.2 we know that the system admits a unique positive equilibrium $\begin{array}{}E\left(\frac{e-c}{f},\frac{e-c}{f}\right)\end{array}$, which is globally asymptotically stable, that is, the predator species has other food resource and it could be survive without the prey species x1.

Theorem 1.1 shows that although the predator species has other food resources and the prey species x1 is only one of the food resources of the predator species, however, if the amount of the predator species is too large, it could also do irreversible harm to the prey species, and this could finally lead to the extinction of the prey species.

## Acknowledgment

The author would like to thank Dr.Rongyu Han for bringing our attention to the paper of Kant and Kumar.

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Accepted: 2019-04-18

Published Online: 2019-07-09

Competing interests: The authors declare that there is no conflict of interests.

Funding: The research was supported by the Key projects for supporting outstanding young talents in Universities in Anhui under Grant(gxyqZD2016240) and the Natural Science Foundation of Anhui Province(1808085MG224).

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

Citation Information: Open Mathematics, Volume 17, Issue 1, Pages 646–652, ISSN (Online) 2391-5455,

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