Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species

Abstract A Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species is proposed and studied. For non-delay case, such topics as the persistent of the system, the local stability property of the equilibria, the global stability of the positive equilibrium are investigated. For the system with infinite delay, by using the iterative method, a set of sufficient conditions which ensure the global attractivity of the positive equilibrium is obtained. By introducing the density dependent birth rate, the dynamic behaviors of the system becomes complicated. The system maybe collapse in the sense that both the species will be driven to extinction, or the two species could be coexist in a stable state. Numeric simulations are carried out to show the feasibility of the main results.


Introduction
As was pointed out by Berryman [1], the dynamic relationship between predators and their 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. Already, the in uence of the Allee e ect [2][3][4][5][6], the in uence of the mutual interferences [7][8], the in uence of the stage structure [9][10][11][12][13], the stability of the positive equilibrium [12][13][14][15][16][17], the existence and stability of the almost periodic solution [18], the existence of the positive periodic solution [19,20], the persistent of the system [21] have been extensively studied, and many excellent results were obtained.
Allee e ect, which re ects the fact that the population growth rate is reduced at low population size, due to its importance, the ecosystem subject to Allee e ect has recently been extensively studied by many scholars, see [2][3][4][5][6][22][23][24][25][26] and the references cited therein.
Hüseyin Merdan [2] investigated the in uence of the Allee e ect on the Lotka-Volterra type predator-prey system. To do so, the author proposed the following predator-prey with Allee e ect system Hüseyin Merdan showed that if r−aβ > hold, the model (1.1) has three steady-state solutions: A( , ), B ( , ) and C x * , y * ). the rst two are locally unstable, while the third one is locally asymptotically stable. By carrying out a series of numeric simulations, the author found the following two phenomenon. (1) The system subject to an Allee e ect takes a longer time to reach its steady-state solution; (2) The Allee e ect reduces the population densities of both predator and prey at the steady-state.
In [17], Guan, Liu and Xie argued that "It seems interesting to consider the in uence of the Allee e ect on the predator species, since generally speaking, the higher the hierarchy in the food chain, the more likely it is to become extinct" and they proposed the following model with the Allee e ect on the predator species: where r, a are positive constants. They showed that if r > a holds, then system (1.2) admits a unique positive equilibrium, and the Allee e ect has no in uence on the nal density of the species. It bring to our attention that in system (1.1) and (1.2), without consider the in uence of the predator species and the Allee e ect, the prey species satis es the traditional Logistic equation where r is the intrinsic growth rate, which is equal to the birth rate minus death rate. Hence system (1.3) could be revised as where a is the birth rate of the species and d is the death rate of the species. Already, Brauer and Castillo-Chavez [26], Tang and Chen [27] and Berezansky, Braverman, et al. [28] had showed that in some case, the density dependent birth rate of the species is more suitable. If we take the famous Beverton-Holt function [28] as the birth rate, then system (1.4) should be revised to System (1.5) combines with the idea of Merdan [2] and Guan et al. [17], will lead to the following Lotka-Volterra type predator-prey system with Allee e ect on the predator species and density dependent birth rate on the prey species It is well known that in a more realistic model the delay e ect should be an average over past populations. This results in an equation with a distributed delay or an in nite delay [29][30][31][32][33][34][35][36][37][38][39][40][41]. Here, if we incorporate the in nite delay to system (1.6), then we will have the following system We shall consider (1.7) together with the initial conditions where ϕ, ψ ∈ BC + . It is well known that by the fundamental theory of functional di erential equations [37], system (1.7) has a unique solution (x(t), y(t)) satisfying the initial condition (1.9). We easily prove x(t) > , y(t) > in maximal interval of existence of the solution. In this paper, the solution of system (1.7) satisfying the initial conditions (1.9) is said to be positive. We mention here that to this day, though there are many scholars investigated the dynamic behaviors of the ecosystem with Allee e ect [1][2][3][4][5][6][22][23][24][25][26], none of them considered the density dependent birth rate of the species. Also, to the best of the authors knowledge, to this day, still no scholars propose a ecosystem with in nite delay and Allee e ect at the same time. It seems that this is the rst time such kind of model are proposed and studied.
The paper is arranged as follows. In section 2 we investigate the persistent and extinct property of the system, based on this, we are able to investigate the locally stability property of the equilibrium solutions of system (1.6). In section 3, by applying the Dulac criterion, we are able to show that under some assumption, the positive equilibrium is globally asymptotically stable. Section 4 presents some numerical simulations concerning the stability of our model. We end this paper by a brie y discussion.

Persistence and local stability of the equilibria
We need several Lemmas to prove the persistent property of the system. Lemma 2.1 [40] Consider the following equation Assume that a > bd, then the unique positive equilibrium y * of system (2.1) is globally asymptotically stable, where y * = −(eb + dc) + (eb + dc) − ec(db − a) ec .

Lemma 2.2 [22]
Consider the following equation The unique positive equilibrium y * = b is global stability.
Let (x(t), y(t)) be any positive solution of system (1.6). From system (1.6) it follows that Consider the equation It follows from Lemma 2.1 that (2.5) admits a unique globally stable positive equilibrium u * , where By using the di erential inequality theory, any solution of (2.5) satis es Hence, there exists a T > such that For t > T , it follows from the second equation of system (1.6) that Consider the equation It follows from Lemma 2.2 that (2.10) admits a unique globally stable positive equilibrium By di erential inequality theory, any solution of (2.9) satis es Hence, there exists a T > T such that For t > T , it follows from the rst equation of system (1.6) that Now let's consider the equation it follows from Lemma 2.1 that system (2.15) admits a unique positive equilibrium v * , which is globally asymptotically stable. Applying the di erential inequality theory to (2.14) leads to It follows from above inequality that there exists an enough large T > T such that and so, from the second equation of system (1.6), we have Consider the equation It follows from Lemma 2.2 that (2.17) admits a unique globally stable positive equilibrium By using the di erential inequality theory, any solution of (2.16) satis es (2.7), (2.12), (2.15) and (2.19) show that system (1.6) is permanent. This ends the proof of Theorem 2.1.
Remark 2.1. By using the software Maple, for the xed coe cients, one could always compute u * easily, however, condition (2.3) could be replaced by some more restricted but easily veri ed condition, indeed, we could have the following results.

holds, then system (1.6) is permanent.
One interesting problem is to investigate the extinction property of system (1.6), for this, we have the following result.
Proof. From the rst equation of system (1.6) we have Hence For any positive constant ε > enough small, there exists a T > such that Hence, from the second equation of system (1.6), we have Consider the equation It follows from Lemma 2.2 that above equation admits a unique globally stable positive equilibrium u * = ε. By using the di erential inequality theory, we have lim sup t→+∞ y(t) ≤ ε.
Since ε is any small positive constant, setting ε → in above inequality leads to lim t→+∞ y(t) = .
This ends the proof of Theorem 2.2.
Now we are in the position of investigate the stability property of steady-state solutions of the model (1.6).
The steady-state solutions of (1.6) are obtained by solving the equations f (x, y) = and g(x, y) = . The model has three steady-state solutions: A( , ), B(u * , ) and C(x * , y * ).

Theorem 2.3. If a > b d holds, then C(x * , y * ) is non-negative equilibrium and it is locally asymptotically stable. If inequality (2.3) holds, then A( , ) and B(u * , ) is unstable.
Proof. The variation matrix of the continuous-time system (1.6) at an equilibrium solution (x, y) is .
So that both eigenvalues of J(x * , y * ) have negative real parts, and hence this steady-state solution is locally asymptotically stable.
From Theorem 2.1 we know that under the assumption (2.3) holds, system (1.6) is permanent, hence no solution could approach to A( , ) and B(u * , ), which means that A( , ) and B(u * , ) are locally unstable.
This ends the proof of Theorem 2.3.

Global stability
We had showed that the positive equilibrium is locally stable, in this section, we further give su cient conditions to ensure the global stability of the positive equilibrium.
Theorem 3.1. Assume that (2.3) holds, then the unique positive equilibrium is globally asymptotically stable.
From Theorem 2.2 system (1.6) admits an unique local stable positive equilibrium C(x * , y * ). Also, from Theorem 2.3, A( , ) and B(u * , ) is unstable. To ensure C(x * , y * ) is globally asymptotically stable, we consider the Dulac function u (x, y) = x − y − , then

Global attractivity of system (1.7)
As far as system (1.7) is concerned, one of the most important topics is to obtain a set of su cient conditions to ensure the global attractivity of the positive equilibrium, since which means the stale coexistence of the two species. Before we state and prove the main result of this section, we need to introduce two lemmas.
To end the proof of Theorem 4.1, it is enough to show that C(x * , y * ) is globally attractive. It follows from (4.1) that there exists a ε > enough small such that Let (x(t), y(t)) be any positive solution of system (1.7). From system (1.7) it follows that Consider the equation It follows from Lemma 2.1 that (4.3) admits a unique globally stable positive equilibrium u * , where u * is de ned by (2.6). By using the di erential inequality theory, any positive solution of (1.7) satis es Hence, there exists a T > such that For t > T , it follows from the second equation of system (1.7) and (4.7) that Consider the equation It follows from Lemma 2.2 that (4.9) admits a unique globally stable positive equilibrium By di erential inequality theory, any positive solution of (1.7) satis es Hence, there exists a T > T such that For t > T , it follows from the rst equation of system (1.7) and (4.14) that Now let's consider the equation it follows from Lemma 2.1 that system (4.16) admits a unique positive equilibrium v * , which is globally asymptotically stable. Applying the di erential inequality theory to (4.15) It follows from above inequality that there exists an enough large T > T such that for all t ≥ T , the following inequalities hold.
From the second equation of system (1.7), for t ≥ T , we have Consider the equation It follows from above inequality that there exists an enough large T > T such that for all t ≥ T , the following inequalities hold For t > T , it follows from (4.23) and the rst equation of system (1.7) that Consider the equation It follows from Lemma 2.1 that (4.24) admits a unique globally stable positive equilibrium u * m ( ) , from Lemma 4.3, one could see that u * m ( ) < u * . By using the di erential inequality theory, any positive solution of (1.7) satis es lim sup and so, from Lemma 4.1 we have Hence, there exists a T > such that For t > T , it follows from the second equation of system (1.7) and (4.28) that Consider the equation It follows from Lemma 2.2 that (4.30) admits a unique globally stable positive equilibrium M ( ) . By using the di erential inequality theory, any positive solution of (1.7) satis es Hence, there exists a T > T such that For t > T , it follows from the rst equation of system (1.7) and (4.34) that Now let's consider the equation Since it follows from (4.1) that a > b d + aM ( ) .
It follows from above inequality that there exists an enough large T > T such that for all t ≥ T , the following inequalities hold.
From the second equation of system (1.7), we have Consider the equation It follows from Lemma 2.2 that (4.40) admits a unique globally stable positive equilibrium m ( ) . By using the di erential inequality theory, any solution of (4.39) satis es and so, from Lemma 4.1 we have It follows from above inequality that there exists an enough large T > T such that for all t ≥ T , the following inequalities hold.
One could easily see that Repeating the above procedure, we get four sequences Obviously We claim that sequences M (n) i , i = , are non-increasing, and sequences m (n) i , i = , are non-decreasing. To prove this claim, we will carry out by induction. Firstly, from (4.44) we have Let us assume now that our claim is true for n, that is, Then, by Letting n → +∞ in (4.45), we obtain (4.46) shows that (x, y) and (x, y) are solutions of (4.1), which (4.1) has a unique positive solution C(x * , y * ). Hence, we conclude that Thus, the unique interior equilibrium C(x * , y * ) is globally attractive. This completes the proof of Theorem 4.1.

Numeric simulations
Now let's consider the following four examples.
( . ) In this system, corresponding to system (1.6), we take a = c = d = e = a = β = , b = , since a < b d , it follows from Theorem 2.2 that the boundary equilibrium A( , ) is globally asymptotically stable. Figure 1 supports this assertion.
( . ) In this system, corresponding to system (1.6), we take b = c = d = e = β = , a = , a = , by computation, u * = √ − , and so, a b = < + ( √ − ) = d + au * , Hence, the conditions of Theorem 2.1 could not satis ed, however, numeric simulation ( Figure 3) shows that the system also admits a unique positive equilibrium which is globally asymptotically stable.
In this paper, we argued that the nonlinear birth rate of the prey species is more suitable, and take Beverton-Holt function [28] as the birth rate, this leads to system (1.6).
We showed that depending on the range of the birth rate parameter, the system maybe collapse or the two species could be coexist in a stable state. That is, the birth rate plays essential role on the dynamic behaviors of system (1.6).  For the system with in nite delay, by using the iterative method, we could able to show that inequality (2.3) is enough to ensure the globally attractive of the positive equilibrium. We mentioned here that with the nonlinear birth rate, the method used in the paper [34] and [36] could not be applied to our system directly, to overcome this di culty, we developing some new analysis technique.
At the end of the paper, we would like to point out that the results obtained in this paper are the su cient ones, as was shown in Example 4.3, there are still have room to improve our results, we leave this for future study. Also, it seems interesting to investigate the dynamic behaviors of the non-autonomous case of system (1.6), specially focus on the permanence, extinction and almost periodic solution, we also leave this for future investigation.