Periodic Solutions Generated by Impulses for State-Dependent Impulsive Differential Equation

It is known that many evolutionary processes are characterized by the fact that at certain moments of time the states change abruptly. Such processes often occur in biology, control theory, optimization theory, physics, and mechanics problems (e.g., [1–6]). It is natural to assume that these perturbations act instantaneously, that is, in the formof impulses. The theory of impulsive differential equations (IDEs) is rather rich, especially for impulse at fixed time. There are many classical methods to study impulsive differential equations. For example, Chen et al. [7] obtained some new results concerning the existence of solutions to an impulsive firstorder, nonlinear ordinary differential equation with periodic boundary conditions via differential inequalities and Schaefer’s fixed-point theorem. Wang et al. [8] got the existence of extreme solutions of a periodic boundary value problem for a second-order functional differential equation by using upper and lower solutions. Based on a nonlinear alternative principle of Leray-Schauder, together with a truncation technique, Chu andNieto [9] studied the impulsive periodic solutions of first-order singular ordinary differential equations. By using a variational method and a variant fountain theorem, Dai and Zhang [10] considered the existence and multiplicity of solutions for a class of nonlinear impulsive problem on the half-line. For more related work, the reader is referred to [11–13] and the references therein. As we know, state-dependent IDEs have become a hot topic in recent years due to their extensive application space, but it is also a difficult research field because of their essential properties: uncertainties for impulsive time and collision times. Very recently, many papers have been devoted to the analysis of IDEs with statedependent impulsive effect. By using differential equation geometry theory and the method of successor functions, the existence and stability of periodic solution for pest management model with state feedback control strategy were discussed in [14, 15] and the homoclinic cycle and homoclinic bifurcationwere analyzed for predator-preymodel with statedependent impulsive harvesting in [16, 17]. On the basis of rotated vector fields theory, Dai et al. [18] discussed the order1 positive periodic solution and homoclinic cycles and homoclinic bifurcations for a general semicontinuous dynamic system. Considering the influence of Allee effect on prey species, the authors in [19, 20] investigated a prey-predator model with Allee effect and state-dependent impulsive harvesting and got the sufficient conditions for the existence of order-1 periodic solution and heteroclinic bifurcation via the geometry theory of semicontinuous dynamic systems. Some other related studies can be seen in [21–23] and the references therein. The aforementioned papers all assumed that the predator just lived on the prey.However, in practice, it is very likely that many enemies have some other food sources. Motivated by this, in this paper, we consider the following state-dependent predator-preymodel inwhich the predator species display the logistic growth in the absence of prey species:


Introduction
It is known that many evolutionary processes are characterized by the fact that at certain moments of time the states change abruptly.Such processes often occur in biology, control theory, optimization theory, physics, and mechanics problems (e.g., [1][2][3][4][5][6]).It is natural to assume that these perturbations act instantaneously, that is, in the form of impulses.
The theory of impulsive differential equations (IDEs) is rather rich, especially for impulse at fixed time.There are many classical methods to study impulsive differential equations.For example, Chen et al. [7] obtained some new results concerning the existence of solutions to an impulsive firstorder, nonlinear ordinary differential equation with periodic boundary conditions via differential inequalities and Schaefer's fixed-point theorem.Wang et al. [8] got the existence of extreme solutions of a periodic boundary value problem for a second-order functional differential equation by using upper and lower solutions.Based on a nonlinear alternative principle of Leray-Schauder, together with a truncation technique, Chu and Nieto [9] studied the impulsive periodic solutions of first-order singular ordinary differential equations.By using a variational method and a variant fountain theorem, Dai and Zhang [10] considered the existence and multiplicity of solutions for a class of nonlinear impulsive problem on the half-line.For more related work, the reader is referred to [11][12][13] and the references therein.As we know, state-dependent IDEs have become a hot topic in recent years due to their extensive application space, but it is also a difficult research field because of their essential properties: uncertainties for impulsive time and collision times.Very recently, many papers have been devoted to the analysis of IDEs with statedependent impulsive effect.By using differential equation geometry theory and the method of successor functions, the existence and stability of periodic solution for pest management model with state feedback control strategy were discussed in [14,15] and the homoclinic cycle and homoclinic bifurcation were analyzed for predator-prey model with statedependent impulsive harvesting in [16,17].On the basis of rotated vector fields theory, Dai et al. [18] discussed the order-1 positive periodic solution and homoclinic cycles and homoclinic bifurcations for a general semicontinuous dynamic system.Considering the influence of Allee effect on prey species, the authors in [19,20] investigated a prey-predator model with Allee effect and state-dependent impulsive harvesting and got the sufficient conditions for the existence of order-1 periodic solution and heteroclinic bifurcation via the geometry theory of semicontinuous dynamic systems.Some other related studies can be seen in [21][22][23] and the references therein.
The aforementioned papers all assumed that the predator just lived on the prey.However, in practice, it is very likely that many enemies have some other food sources.Motivated by this, in this paper, we consider the following state-dependent predator-prey model in which the predator species display the logistic growth in the absence of prey species: where () and () denote population densities of prey and predator at time , respectively.All the parameters are positive constants, in addition, ,  ∈ (0, 1), ℎ 1 < ℎ 2 , and (ℎ 1 , ỹ) is the point of intersection of   = 0 and  = ℎ 1 .This paper is organized as follows.In Section 2, we present some preliminaries.Then in Section 3, we discuss the existence of positive periodic solution of system (1) for different cases.At last, in Section 4, some numerical simulations and conclusions are presented.

Lemma 1. Consider Model
which is globally asymptotically stable.
Throughout this paper, we always assume that the condition  1  22 −  2  12 > 0 holds true.Considering the biological background, we only discuss Model (1) in the region {(, ) :  ≥ 0,  ≥ 0}.Obviously, due to Lakshmikantham et al. [25] and Bainov and Simeonov [26], the global existence and uniqueness of solution for Model (1) are guaranteed by the smoothness properties of right-side functions.
To discuss the dynamics of Model (1), we define three cross sections and two regions: Definition 2 (see [27]).Suppose that the impulse set  and its phase set  are both lines, as shown in Figure 1.Assume that the trajectory starting from  in  firstly intersects  at point  and then jumps to  + in  due to the impulsive effect.Then, one defines  + as the successor point of , and the corresponding successor function of point  is that () =   + −   ; here   and   + are the ordinates of  and  + .

Existence of Positive Periodic Solution for System (1)
Considering the biological meaning, here we always assume that ℎ 1 <  * .Therefore, we have four cases to discuss: 3.1.The Case of (1−)ℎ 2 < ℎ 1 < ℎ 2 <  * .About the existence of positive periodic solution, we have the following graph illustrations.Take a point (ℎ 1 , ỹ + ) on ∑ 1 , where  is small sufficiently.Assuming that the trajectory of Model (1) starting from  firstly intersects ∑ 1 at point  1 and then jumps to  + 1 , obviously,  + 1 is above ; that is to say, On the other hand, assume that the trajectory starting from  + 1 intersects ∑ 1 at  2 and then jumps to thus, Model (1) exists as a positive periodic solution whose initial point is between  and  + 1 ; see Figure 2 Figure 2: The possible trajectories in the case of Figure 3: The possible trajectories in the case of ℎ 1 < (1 − )ℎ 2 < ℎ 2 <  * .

The Case of ℎ
For this case, we have the following graph illustrations.
On the other hand, choosing a point  next to -axis on ∑ 0 , the trajectory starting from  firstly intersects ∑ 2 at point  1 and then jumps to  + 1 on ∑ 0 ; obviously,  + 1 is above ; thus, Therefore, Model (1) exists as a positive periodic solution whose initial point is between  and ; this is shown in Figure 3(a).
On the other hand, there must exist a trajectory starting from  on ∑ 0 that tangents ∑ 1 at point (ℎ 1 , ỹ) and then intersects ∑ 0 , ∑ 2 at points ,  1 , respectively; due to impulsive effect,  1 jumps to Therefore, Model (1) exists as a positive periodic solution whose initial point is between  and ; this can be seen in Figure 3(b).If   + 1 >   , then the trajectory starting from Ω 2 will ultimately stay in Ω 1 .(Here we always assume the impulsive phase set with initial point on ∑ 0 will ultimately exceed point  after one or finite times impulses.In fact, the assumption is reasonable as  should be very small in practical problem.) exists as a positive periodic solution.
Obviously, there must exist a trajectory Γ 1 starting from  1 at Σ 2 that tangents Σ 0 at point  1 and then intersects   = 0, Σ 2 at  1 ,  ≤   , then Model (1) exists as a positive periodic solution whose initial point is between  1 and ; this can be seen in Figure 4(b); >   , then the trajectory starting from Ω 0 = {(, ) : (, ) ∈ Â} will tend to equilibrium and the trajectory starting from Ω 2 \ Ω 0 will ultimately stay in Ω 1 .(In this case, we still assume the impulsive phase set with initial point on ∑ 0 will ultimately exceed point  2 after one or finite impulses.) exists as a positive periodic solution.
Assuming that the trajectory Γ intersects Σ 0 at points  1 ,  2 with   1 >   2 , due to impulsive effect,  jumps to  + .For  + , one has the following four situations to discuss.
(ii) If   + <   2 , then it is easy to get that there exists an order-1 periodic solution; here we omit the details.
(a) If   ≥ ℎ 1 , then Model (1) exists as a positive periodic solution whose initial point is between  1 and  + (see Figure 5(a)).(b) If   < ℎ 1 , then there must exist a trajectory Γ 1 starting from  at Σ 2 that firstly intersects Σ 0 at point  1 and then tangents to Σ 1 at point  1 , and intersects   = 0, Σ 2 at points  1 ,  1 , respectively.Due to impulsive effect,  1 jumps to  + 1 .For  + 1 , we have the following two cases to discuss: 1) exists as a positive periodic solution whose initial point is between  1 and  1 (see Figure 5 , then the trajectory starting from Ω 0 = {(, ) : (, ) ∈ Â 1 } will tend to equilibrium and the trajectory starting from Ω 2 \ Ω 0 will ultimately stay in )) will tend to equilibrium and the trajectory starting from Ω 2 \ Ω 0 will ultimately stay in Ω 1 .

Theorem 8. Assume that
exists as a positive periodic solution.
Remark 9.The positive periodic solutions for Model (1) obtained in Theorem 5∼Theorem 8 are generated by impulses.Here, we say that a solution is generated by impulses if this solution is nontrivial when impulsive effect exists, but it is trivial when there does not exist impulsive effect.For example, when  1  22 −  2  12 > 0, by Lemma 1 we know that Model (2) does not possess any positive periodic solution; then positive periodic solutions of Model (1) under state-dependent impulsive conditions are called positive periodic solution generated by impulses.
Remark 10.As we know, the previous papers concerning state-dependent impulsive effect all assumed that the predator just lived on the prey; here we point out that the predator has some other food resources; this is more practical.On the other hand, the existing state-dependent impulsive differential systems mainly discussed the properties of solutions, including existence, uniqueness, and orbitally asymptotical stability.Here, not aiming at the properties of solutions, we Figure 4: The possible trajectories in the case of ℎ Figure 5: The possible trajectories in the case of ℎ are focused on considering the influence of impulsive effect on the system itself.The theoretical results imply that if impulses do not exist, then the predator and prey species will tend to a point; if impulsive effect occurs, then the predator and prey species will be maintained at a periodic oscillation; that is, both the densities of these two species can change periodically.Therefore, our results demonstrate that impulsive effect takes an important role in ecological system.

Simulations and Conclusions
In this paper, we propose and analyse a state-dependent impulsive predator-prey model in which the predator species display a logistic growth.By using geometrical analysis methods, the existence of positive periodic solutions of Model ( 1) is given.Here we should point out that the positive periodic solutions are generated by impulses.For system (2), which does not exist as impulsive effect, the interior equilibrium is globally asymptotically stable, the phase trajectory and time series chart can be seen in Figures 6 and 7; therefore, system (2) does not exist as positive periodic solution and all the phase trajectories will tend to the interior equilibrium.
When the impulsive effects are operated, system (1) can be gotten, the theoretical results demonstrate that system (1) exists as positive periodic solutions for some cases, and the numerical simulations also illustrate the existence of the periodic solutions; please see Figures

Figure 1 :
Figure 1: Illustration of the successor function.