Stability property of the prey free equilibrium point

Qin Yue

doi:10.1515/math-2019-0051

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Stability property of the prey free equilibrium point

Qin Yue

Veröffentlicht/Copyright: 9. Juli 2019

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Aus der Zeitschrift Open Mathematics Band 17 Heft 1

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.

Keywords: Stage structure; Predator-prey; Global attractivity

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 P2(x1∗,x2∗,x3∗). Under the assumption c ≥ e, 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 P3(0,e−cf,e−cf) . 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 P3(0,e−cf,e−cf) .

The aim of this paper is to give sufficient condition to ensure the global asymptotically stable of the equilibrium P3(0,e−cf,e−cf) of system (1.2), more precisely, we have the following result.

Theorem 1.1

Assume that

af+bc<be (1.3)

holds, then P3(0,e−cf,e−cf) 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 B(x1∗∗,x2∗∗) 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 E(e−cf,e−cf) 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 E(e−cf,e−cf) . 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 E1(e−c+dεf,e−c+dεf), 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 P3(0,e−cf,e−cf) 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.

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

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 x1∗=a 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 E(e−cf,e−cf) , 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.

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.

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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Received: 2018-02-07

Accepted: 2019-04-18

Published Online: 2019-07-09

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Stage structure; Predator-prey; Global attractivity

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