The state-dependent impulsive control for a general predator–prey model

In this paper, a general predator–prey model with state-dependent impulse is considered. Based on the geometric analysis and Poincaré map or successor function, we construct three typical types of Bendixson domains to obtain some sufficient conditions for the existence of order-1 periodic solutions. At the same time, the existing domains are discussed with respect to the system parameters. Moreover, the Analogue of Poincaré Criterion is used to obtain the asymptotic stability of the periodic solutions. Finally, to illustrate the results, an example is presented and some numerical simulations are carried out.


Introduction
Impulsive control is widely utilized in biology [6,7,25]. For instance, in the treatment of HIV-1 infection, the combination of vaccination and stimulation delays viral rebound following ART (Antiretroviral therapy) discontinuation [2,11,12]. Compared with the process of disease, the impact of taking drugs, vaccination and stimulation is short enough to be assumed that the therapy leads to an impulsive effect. Usually, from the economy point of view, the corresponding treatment is proposed to receive when the infected system reaches a critical state, on which the amount CD4 + T decreases to 350 or 500 mm −3 . Similarly, in the controlling of insects and other animals, to minimize damage to non-target organisms, and to preserve the environment, the integrated pest management (IPM) was introduced [3,17], which is a strategy that reduce pests to tolerable levels by releasing natural enemies, spraying pesticides, isolating or harvesting the pests when they reach a threshold. Likewise, in fishery industry, for the sake of ecological balance and maximum economic benefit, people usually take a sustainable strategy by harvesting and releasing according to the densities of certain species. For example, sometimes we need suppress the predator-fish by harvesting, and to stimulate the prey-fish by releasing, and other time it's just on the opposite purpose [9]. The discontinuity of human intervention leads to population changing very rapidly, which can be taken into account impulsive differential equations.
In [21], Wang et al. consider an impulsive Kolmogorov predator-prey model with nonselective harvesting given by where x(t) and y(t) denote the densities of the prey and predator species at time t. Suppose one expects to perform an economical harvesting on the prey and predator (e.g. two kinds of food fish). When the density of prey increases to h, one harvests the prey and the predator at a rate p and q respectively. Because the reduction of prey may prevent predators from getting enough growth, and the harvesting on predator may also make it be on the verge of extinction, therefore to maintain the ecological balance among two populations and to obtain a sustainable economic benefit, one can release the predator on a scare of τ . The authors incorporated two discriminants 1 and 2 to discuss the existence, nonexistence and multiplicity as well as the stability of order-1 periodic solutions. Motivated by the work of [20,21], we consider the following state-dependent impulsive predator-prey population ecological model: where x(t), y(t), h and τ are in the same meaning as those in (1); n and m represent the rates at which the prey and the predator are harvested, respectively. Assume that h is positive constant, and n, m ∈ [0, 1),τ ≥ 0, F ∈ C 2 (R 2 , R), G ∈ C(R, R). The initial value conditions satisfy Moreover, the following assumptions hold: There exist two positive numbers w < k such that U(k, 0) = 0 and V(w) = 0, and In this paper, based on the theory of Bendixson domains established in [20,21], some results on the existence and stability of order-1 periodic solutions for system (2) are established. Since the isolines of system (2) are different from that of system (1), the discriminants 1 and 2 are not always valid for system (2). We classify the structure of Bendixson domains into three categories, and combine them with Poincaré maps and successor functions, which help us to obtain the existence and location of impulsive order-1 periodic solution(s). Further, by using the Analogue of Poincaré Criterion, the stability of the order-1 periodic solution for system (2) is considered. In the last, we focus on the numerical simulations for the theory results.

Preliminaries
Let the impulse function be I, then For the sake of simplicity, it is also denoted by I(y) = (1 − m)y + τ . Further, we denote the system without impulse that corresponding to (2) by where Q also represents its coordinate y Q . For convenience, if a point A is above point B, that is y A > y B , then we write A > B. Obviously, The geometric interpretation of Poincaré map P, P N and successor function F is shown in the Figure 1. It is known from the properties of U and V, for system (4), that the existence and uniqueness of the solutions hold true, and the solutions are continuously dependent on and differentiable with respect to the initial value. It is also easy to verify that the solution of (4) with a positive initial value must be positive.  Proof: Obviously, the point (0, 0) is an equilibrium. Since U(k, 0) = 0, the point (k, 0) is a boundary equilibrium of the system (4). In addition, according to assumption (A 1 ), the equation U(x, y) = 0 determine an implicit function, named y = σ (x). From the differential rule for implicit function, we have which means that the function y = σ (x) is decreasing with x, and σ (k) = 0 because of U(k, 0) = 0. Since V(w) = 0 and w < k, so the curve of y = σ (x) must intersect with the line x = w at the point (w, y * ), where y * is the solution of the equation U(w, y) = 0. Hence (w, y * ) is a positive equilibrium of the system (4). The Jacobian matrix along the system (4) is Calculating the eigenvalues of the Jacobian matrix at E 0 = (0, 0), E 1 = (k, 0) and E 2 = (ω, y * ), respectively, it is known that the equilibria E 0 , E 1 are unstable saddle points and E 2 is an asymptotically stable node or focus. Taking a Dulac function D(x, y) = 1 xy (x > 0, y > 0), we have Therefore, the unique positive equilibrium point E 2 is globally asymptotically stable and there is no positive periodic solution in the first quadrant.
When O + (Q) M = , we called the function P,P N , F are well defined. Following the discussion in [21] and [20], we have Further similar analysis to Lemma 2.2 in [21] shows that the vector field of the system (4) is counterclockwise.
Let the isoline y = σ (x) intersects with M and N at points R = (h, y R ) and T = (h, y T ) respectively. Then Because of σ < 0 and h < h, we have y R < y T . The possible location of impulse set, image set, isolines and related characterized points is shown in Figure 2. (ii) the trajectory L 1 intersects with N and M at A andĀ successively; the trajectory L 2 intersects with N, M at B andB successively; (iii) line segments AB andĀB have no tangent points with the trajectory of the system (4) except for the end points.
then D is called a parallel rectangular domain (see Figure 3a).
and L 2 is tangent to N at B, then D is called a sub-parallel rectangular domain (see Figure 3b).
If A > B ⇒ A < B, L 1 intersects with N at A and A successively, and L 2 intersects with N at B and B successively, and L 2 is tangent to M at B , then D is called a semi-circle domain (see Figure 3c).   Figure 3).

Main results
Next, we discuss the existence and location of order-1 periodic solutions of the system (2) under the case h < h < x * = w and h < x * < h, respectively.

h < h < x * = w
In this case, for any Q ∈ N, F(Q) is well defined since E 2 (w, y * ) is globally asymptotically stable. Proof: To specify the structure and location of the order-1 periodic solution and probe the effect of parameter m and τ , we discuss case by case.
The domain formed by trajectory TT and T +T+ , segments TT + andTT + is sub-parallel. It follows from the structure of the trajectories of system (2) and the monotonic increasing of impulse function I that F(T + ) < 0. Thus F(T)F(T + ) < 0. According to Lemma 2.3, there is an order-1 periodic solution in this sub-parallel domain. ( is well defined, the domain composed of trajectories TT and Q τQτ , segments TQ τ andTQ τ is a parallel one. It follows from (7) that F(T)F(Q τ ) < 0, which implies that there is a positive order-1 periodic solution starting from the segment TQ τ . Case 2. 0 < τ m ≤ y T . Take the trajectory as a graph of a function, then So in the parallel domain composed of trajectories Q τQτ , TT and segments Q τ T andQ τT , there is a positive order-1 periodic solution initiating from the segment Q τ T.
From the assumption ( Thus for the successor function F, both F(Q) < 0 and F (Q) < 0 are satisfied for Q < T. Since 0 < m < 1 and the solutions of system (4) are positive with positive initial value. Henceforth, lim ε→0 + F(ε) = 0 holds true because of the continuity of F.

Remark 3.1:
In fact, even since h < x * < h, the inequality F(T) < 0 is always true under the case that F(T) is well defined and τ = 0. In view of the uniqueness of the solution, the trajectory Q + (T) intersects with the isoline on the right side of M, thereforeT < T holds, which means F(T) < 0.
When m = 0, the result is similar to the case τ m > y T .

h < x * < h
In this case, F(T) may be not well defined.
If F(T) is well defined, F(Q) is defined for any Q ∈ N. Similar to the proof of Theorem 3.1, there must exist an order-1 periodic solution for system (2).
If F(T) is not well defined (it is only possible when h < x * < h), we only consider the case that O − (R) ∩ N = and there are at least two intersection points, because the impulse control is hardly valid when there is unique or no intersection [20,21]. Let From Lemma 2.3, we have the following theorems. Proof: Since τ ≥ y R − , we have I(y R ) = (1 − m)y R + τ > τ ≥ y R − . From Lemma 3.1, no matter τ m < y R or τ m > y R , the inequality I(R) < R − can be obtained from τ m < y R − and y R < y T < y R − . The trajectories starting from the points below R − on N are all mapped onto segment R − R − under the effect of impulse, so the trajectory initiating on R − R − will no longer reach M, which are determined by system (4). From Lemma 2.3, there is no positive periodic solution for system (4), so does for system (2). Proof: Equation (7) gives F(Q τ ) > 0. Since I(R) < R − , that is F(R) < 0, and hence F(τ )F(R) < 0 holds. It is known from Lemma 2.3 that there is an order-1 periodic solution initiating from the segment Therefore there is an positive order-1 periodic solution initiating from segment Q τ Q m .     Next, we summarize the results in two tables as follows, which are listed according to whether F(T) is well defined, where the symbol 'par.' and 'semi-tri.' are the abbreviations of 'parallel' and 'semi-trivial'. The check mark ' ' indicates the claim is true(see Table 1 and Table 2).

Example and simulations
Consider the state-dependent impulsive differential system Obviously, corresponding to system (2), we have If k > h and (1 − n)h < μ β , then the assumptions (A 1 ) and (A 2 ) hold. In the following, some simulations are carried out by Matlab software, which are somewhat corresponding to Table 1 and Table 2.

F(T) is well defined
Taking the parameters r = 1, k = 10, β = 0.025, a = 0.188, n = 0.3 and h = 5.5, the computation shows that (x * , y * ) = (7.52, 9. Figure 2 illustrates the existence of the order-1 periodic solutions which are the same as h < x * . Figure 1 and Figure 2 show that there must exist an positive order-1 periodic solution for system (2) under the case F(T) is well defined and τ > 0.

F(T) is not well defined
If F(T) is not well defined, then h < x * < h. Let τ = 30. Then τ m = 60 ≥ y R − , I(R) = y R + = 32.6665 > y R − and P N (R + ) = 32.089 > y R − are satisfied. It can be seen from Figure 6(a) that there is an order-1 periodic solution which locates in a semi-ring domain. Take τ = 7, then τ > y R − and τ m = 14 < y R − hold. As can be seen from Figure 6(b) that there does not exist periodic solution because the trajectories of the system ultimately approach the equilibrium (x * , y * ). While τ = 2, we can verify that 0 < τ < y R − , I(y R ) = y R + = 4.6665 < y R − and F(τ ) = 0.91715 > 0. We can see from Figure 6 6.325 = R − , and the system has a semi-trivial order-1 periodic solution (see Figure 7b). Given a small τ = 0.05, even since the initial value of y is zero, there is still a positive order-1 periodic solution (see Figure 7c).  Figure 7, it can be seen that the existence of periodic solutions depends on the initial values, parameters m, τ and the structure of the system without impulse. Moreover, Figure 7(b) and Figure 7(c) show that a small τ can stimulate a positive order-1 periodic solution.

Conclusion and discussion
In the paper, the existence and existing domain, along with the stability of order-1 periodic solutions are considered. Further, we focus on the effects of parameters τ and m on the existence and stability of periodic solutions, which are performed theoretically and numerically. We construct three kinds of Bendixson domains with different structures, which overcome the difficulty, to some extent, that caused by the discontinuity of impulsive impact.
Our results show that the system (4) without a positive periodic solution may have periodic oscillations under the effects of pulses. In this sense, the pulse action can 'activate' the state of periodic oscillation. When h < x * , F(T) must be well defined, and when h < x * < h, F(T) is not necessarily well defined. If F(T) is well defined, then system (2) has an order-1 periodic solution, and when τ > 0, there must exist a positive order-1 periodic solution. If F(T) is not well defined and the parameters τ and m are taken moderate values, the impulse effect will be invalid ultimately (see Theorem 3.4). If τ and τ m are small enough, the population of the predator species, corresponding to y of the periodic solution, is also small. When τ m is large enough, the periodic solutions locate in the sub-parallel domain (when F(T) is well defined) or in the semi-ring domain (when F(T) is not well defined), corresponding to a larger scale of predator species. When τ = 0, no matter h < h < x * or h < x * < h holds, there exist a semi-trivial periodic solution. However, when h < x * < h, there may still exist other positive order-1 periodic solutions (see Figure 7(a)).
From biological control point of view, when the density of the prey reaches a threshold h, the two species are harvested. When τ = 0, the system has a semi-trivial periodic solution, which means the predator becomes extinct, and the ecosystem is destroyed. Comparing  Figure 7(c), respectively, it can be seen that a small τ can 'activate' an positive order-1 periodic solution, which means the releasing of a small scale of predator can 'stimulate' an dynamic equilibrium of the ecosystem. When |μ 1 | < 1, the dynamic equilibrium is stable, and we can implement the control of harvesting or releasing at every ω time, so as to maintain the ecological balance between the two species and to ensure the sustainable economic benefits.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by The Natural Science Foundation of Hunan Province [2019JJ40240, 2018JJ2319].