Stability property of the prey free equilibrium point

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.


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 in nite 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 maybe 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, where N (t),N (t) and N (t) are the population densities of prey, juvenile predator and adult predator, respectively. r , k, α, β, r , r , m and γ are all positive constants. De ne Then system (1.1) can be rewrite as ( . ) The possible non-negative equilibria of system (1.2) are P ( , , ), P (a, , ) and P (x * , x * , x * ). 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 P , . 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 The aim of this paper is to give su cient condition to ensure the global asymptotically stable of the equilibrium P , 2), more precisely, we have the following result.
is globally asymptotically stable.
We mention here that the method we used here is quite di erent with that of the method used in [11]. Indeed, we only use the di erential inequality theory and the comparison theorem of the di erential 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 brie y discussion.

Proof of Theorem 1.1
Now let's consider the system where α, β, δ , δ and γ are all positive constants, x (t) and x (t) are the densities of the immature and mature species at time t. From Theorem 4.1 in Xiao and Lei [17], we have holds, then the positive equilibrium B(x ** , x ** ) of system (2.1) is globally stable, where Now let's consider the system As a direct corollary of Lemma 2.1, we have Now let's consider the system Noting that condition ( Let (x (t), x (t), x (t)) be any positive solution of system (1.2) with initial condition (x ( ), x ( ), x ( )) = (x , x , x ), and let (u (t), u (t)) be the positive solution of system (2.7) with the initial condition (u ( ), u ( )) = (x , x ), it then follows from the di erential inequality theory that The positivity of the solution of system (1.2), (2.8) and (2.9) lead to Condition (1.3) implies that for enough small positive constant ε > , the following inequality holds.
which is equivalent to For ε > enough small, which satis es (2.12), it then follows from (2.10) that there exists an enough large T > such that Hence, for t > T , from the rst equation of system (1.2) and (2.13), we have Consequently, That is, lim t→+∞ x (t) = .
( . ) For ε > enough small, it follows from (2.15) that there exists a T > T such that For t > T , from the second and third equation of system (1.2) and (2.16), we have Now let's consider the system dv dτ it follows from e > c and Lemma 2.1 that (2.18) admits a unique globally asymptotically stable positive , v (t)) be any positive solution of the system (2.18), one has lim Let (x (t), x (t), x (t)) be any positive solution of system (1.2) with initial condition (x (T ), x (T ), x (T )) = (x , x , x ), and let (v (t), v (t)) be the positive solution of system (2.18) with the initial condition (v (T ), v (T )) = (x , x ), it then follows from the di erential inequality theory that The positivity of the solution of system (1. (2.10) and (2.18) show that Since ε could be any enough small positive constant, now, letting ε → in (2.22) leads to

Numeric simulation
Example 3.1. Consider the following stage structure predator prey system ( . ) Here, corresponding to system (1.2), we take a = b = c = d = f = , e = . Since af + bc = < = be, it follows from Theorem 1.1 that P , , is globally asymptotically stable. Numeric simulation (Fig. 1) also supports this assertion.  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 P . In this paper, by using the di erential inequality theory and the comparison theorem of the di erential equation, we could show that under some suitable condition, the boundary equilibrium P is globally asymptotically stable. Our result has signi cant biological meaning. In system (1.2), without consider the relationship of the predator and prey species, then prey species is governed by the equation

Discusion
which is a Logistic equation, and the positive equilibrium x * = a is globally asymptotically stable. That is, without the in uence of the predator species, the prey species could be exist in long run. Also, without consider the in uence of the prey species, the predator species satis es the system From Lemma 2.2 we know that the system admits a unique positive equilibrium E( e − c f , e − c f ), which is globally asymptotically stable, that is, the predator species has other food resource and it could be survive without the prey species x . Theorem 1.1 shows that although the predator species has other food resources and the prey species x 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 nally lead to the extinction of the prey species.

Declarations
Competing interests