Periodic Solutions and Homoclinic Bifurcations of Two Predator-Prey Systems with Nonmonotonic Functional Response and Impulsive Harvesting

Two predator-prey models with nonmonotonic functional response and state-dependent impulsive harvesting are formulated and analyzed. By using the geometry theory of semicontinuous dynamic system, we obtain the existence, uniqueness, and stability of the periodic solution and analyse the dynamic phenomenon of homoclinic bifurcation of the first system by choosing the harvesting rate β as control parameter. Besides, we also study the homoclinic bifurcation of the second system about parameter δ on the basis of the theory of rotated vector field. Finally, numerical simulations are presented to illustrate the results.


Introduction
Predator-prey interaction is one of the most important relationships in the ecosystem, so it has long been a focus of study in mathematical ecology.The functional response of the predator to the prey describes how the predator density changes with the prey density.The well-known response functions, which have appeared in a lot of literatures, include linear function, Holling type-II function, Holling type-III function, and so on.The above mentioned response functions are all monotonic, and they are really accurate to describe the predator-prey interactions in many cases.But in the recent decades, scholars in different fields found through experiments that nonmonotonic functional response occurs in some predatorprey interactions.For example, if the prey exhibits group defense [1,2], the predator growth rate will be inhibited when the prey density reaches a high level.This phenomenon is known to exist widely in nature and considerable work on it has been studied [3][4][5][6][7][8].To study the predator-prey interaction when the prey exhibits group defense, Ruan and Xiao in [4] proposed the following model: where  and  represent the population densities of prey and predator, respectively;  > 0 and  > 0 represent the carrying capacity and the intrinsic birth rate of the prey, respectively, and  > 0 is the death rate of the predator.The function () = /( +  2 ) denotes the predator functional response which can model the phenomenon of group defense because there exists ℎ > 0 such that   () > 0 for 0 ≤  < ℎ and   () < 0 for  > ℎ.
Harvesting strategy of biological resources is also a focus topic in mathematical bioeconomics because it relates to the optimal management of renewable resources.In many literatures, impulsive differential equations are used to model 2 Journal of Applied Mathematics the human action of harvesting.Predator-prey systems with periodic impulsive harvesting have been studied extensively and important results have been achieved [9][10][11][12].In recent years, more and more scholars begin to consider state dependent impulsive harvesting instead of the periodic impulsive form [13][14][15][16][17]. In point of ecology, we would better know some information about the amount of the species when we harvest them, thus we can avoid excessive exploitation and resource exhaustion.To this end, we introduce a reliable real-time monitoring system to estimate the number of the species.Such monitoring systems exist in many cases.According to the feedback information from the monitoring system, we can manage our resources better.For predator-prey system (1), we assume the predator has high commercial value for human beings, and people increase its production mainly through replenishing its prey.In such an ecosystem, we suppose the amount of the prey can be estimated by a monitoring system, and the monitoring data can help us to decide if we harvest the predator or not.To model this harvesting behavior, we can propose the following statedependent impulsive differential equations: where ℎ > 0 is a threshold.When the amount of the prey is large than ℎ, which means the food of the predator is abundant, and the development of the system coincide with our economic interest.When the amount of the prey drops to the threshold ℎ, which means the nutrition of the predator will be deficient, we harvest the predator at rate  ∈ (0, 1) and replenish some prey at the same time.We denote the recruitment of the prey as .
A lot of research has shown that the population size of the predator is influenced not only by the prey populations but also by the relative rate of prey population growth.If we take the relative growth rate effect into consideration, the above model needs some changes.To this end, we propose the following system: where the term [(1 − (/)) − /( +  2 )] 2 represents the effect of the prey relative growth rate on the predator.In this paper, we mainly discuss the dynamics properties of the systems (2) and (3).The paper is organized as follows.In Section 2, some notations and definitions of the geometric theory of semicontinuous dynamical systems are provided.In Section 3, by using the geometry theory of semicontinuous dynamic system, we firstly study the existence, uniqueness, and orbital stability of periodic solutions and analyse the dynamic phenomenon of homoclinic bifurcation of system (2) by choosing the harvesting rate  as control parameter.Then, we study the homoclinic bifurcation of system (3) about parameter  on the basis of the theory of rotated vector field.The paper ends with a brief discussion and some numerical simulations.

Preliminaries
In this section, we give some notations and definitions of the geometric theory of semicontinuous dynamical systems which will be useful for the following discussion.
Definition 1 (see [9]).Consider the state-dependent impulsive differential equations We define the dynamic system consisting of the solution mapping of the system (4) a semicontinuous dynamical system, denoted as (Ω, , , ).We require that the initial point  of the system (4) should not be in the set {, }, that is,  ∈ Ω =  2 + \ {, }, and the function  is a continuous mapping that satisfies () = .Here  is called the impulse mapping, where {, } and {, } represent the straight lines or curves in the plane  2 + , {, } is called the impulse set, and {, } is called the phase set.Definition 3 (see [9]).For the semicontinuous dynamical system defined by the state-dependent impulsive differential equations (4), the solution mapping (, ) : Ω → Ω consists of two parts.
Definition 6 (see [9]).If there exists a point  ∈ {, } and a time point  1 such that (,  1 ) =  ∈ {, }, () =  1 ∈ {, } and  ̸ =  1 , and there also exists a time point  2 such that ( 1 ,  2 ) =  1 ∈ {, } and ( 1 ) =  ∈ {, }, then (, ) is called an order two periodic solution of the system (4) whose period is  1 +  2 (see Figure 1(c)).Analogously, we can define the order  periodic solution of the system (4).Definition 7. Suppose Γ = (, ) is an order one periodic solution of the system (4).If for any  > 0, there must exist  > 0 and  0 ≥ 0, such that for any point  1 ∈ (, ) ∩ {, }, we have (( 1 , ), Γ) <  for  >  0 ; then we call the order one periodic solution Γ orbitally asymptotically stable.Definition 8 (see [9]).Suppose the impulse set  and phase set  of the system (4) are straight lines and a coordinate system can be defined in the phase set .Let point  ∈  and its coordinate is .Assume that the trajectory from the point  intersects the impulse set  at a point   , and, after impulsive effect, the point   is mapped to the point  1 ∈  with the coordinate  1 ; then we call point  1 the order one successor point of point , and the order one successor function of point  is  1 () =  1 − .
Remark 9.For Definition 8, if the trajectory from the point  1 intersects the impulse set  again at a point   1 , and, after impulsive effect, the point   1 is mapped to the point  2 ∈  with the coordinate  2 , then the point  2 is obviously the order one successor point of point  1 ; we also call point  2 the order two successor point of point , and the order two successor function of point  is  2 () =  2 − .If the process can be repeated over and over again, then we can define the order  successor point of point  (which we denote as   and its coordinate is   ) and the order  successor function of  which we denote as   () =   − .
Remark 10.For the systems ( 2) and ( 3), for any point  ∈ , we define the directed distance between point  and the axis as the coordinate of point .In this paper, we denote the coordinate of point  as   .
Lemma 11 (see [9]).Successor function   () is continuous.Lemma 12.For the systems (2) and (3), if there exists two points  ∈ ,  ∈  such that  1 () 1 () < 0, then there must exist a point  ∈  which is between the points  and  such that  1 () = 0; thus, the system must have an order one periodic solution which passes through the point .
Proof.By Lemma 11, we can easily get that there must exist a point  ∈  which is between the points  and  such that  1 () = 0.According to Definition 6, we know Γ = (, ) is an order one periodic solution.That completes the proof.Lemma 13.For the systems (2) and (3), suppose point  ∈  and   () ̸ = 0, then the system does not have an order  periodic solution which passes through the point .

Periodic Solution and Homoclinic Bifurcation
In this section, we firstly discuss the existence, uniqueness, and stability of the periodic solution of the system (2) by using differential equation geometry theory then study the dynamic phenomena of homoclinic bifurcation of both systems ( 2) and ( 3).We now list some results about the system (1) that are given by Ruan and Xiao in [4].
For the system (1), there is always a hyperbolic saddle point at the origin (0, 0) and an equilibrium (, 0) in the axis.When  2 − 4 2 > 0, the system (1) may have one or two positive equilibria.If the positive equilibria exist, we denote them as ( 1 ,  1 ), ( 2 ,  2 ), where Now, we give some results appeared in [4].

Periodic Solution and Homoclinic Bifurcation of System
(2) about Parameter .According to the model (2), the threshold ℎ and the recruitment  of the prey should satisfy the condition 0 < ℎ < ℎ +  <  by ecological significance.For this consideration, we have the following results.
, and  1 < ℎ < ℎ +  < , then there must exist a fixed value  0 ∈ (0, 1) such that for every  ∈ ( 0 , 1), the system (2) has a unique order one periodic solution in region Ω 1 , where region Ω 1 is the region enclosed by the -axis, the impulse set  = ℎ, and the unstable flow of the saddle (, 0).
Proof.According to Lemma 14, the system (1) has three equilibria: two hyperbolic saddles (0, 0) and (, 0) and a globally asymptotically stable equilibrium ( 1 ,  1 ) in the interior of the first quadrant.For convenience, we suppose the -axis intersects the impulse set  = ℎ and the phase set  = ℎ +  at point   and point , respectively.The unstable flow of (, 0) also passes through the impulse set and the phase set, we denote the intersections by point  and point , respectively.Besides, we suppose the vertical isocline / = 0 intersects the impulse set and the phase set at point  and point , respectively.Then the region Ω 1 is the interior of the closed curve Ê0  ∪   ∪    0 , where  0 denotes the equilibrium (, 0) (see Figure 2).According to the impulsive conditions of the system (2), there must exist a fixed value  0 ∈ (0, 1), when  =  0 , point Figure 2: Existence of order one periodic solution of (2).
is mapped to the point  after impulsive effect, that is to say, (1 −  0 )  =   .When  0 <  < 1, after impulsive effect, point  is mapped to a point between points  and .Select a point  0 next to  in the phase set  = ℎ + , the trajectory of the system (2) from point  0 must intersect the impulse set  = ℎ at a point   0 which is next to point , and after impulsive effect, point   0 is mapped to a point  1 which is the order one successor point of  0 ; then, we have That is to say point  1 is between points  and  0 .The trajectory from point  1 must intersect the impulse set  = ℎ again at a point    1 , and after impulsive effect, point    1 is mapped to a point  2 .Since distinct trajectories do not intersect, we can easily have   <    1 <    0 and Obviously, point  2 is the order one successor point of point  1 , and then we have the following results of the order one successor function: By Lemma 12, we know that there must exist a point  in the phase set  = ℎ +  which is between the points  0 and  1 such that  1 () = 0. Then we know the system (2) has an order one periodic solution which passes through the point .
In the following, we prove the uniqueness of the order one periodic solution.Arbitrarily choose two points  1 and  2 in the phase set  = ℎ + , where 0 ≤   1 <   2 ≤   .The trajectories of the system (2) through points  1 and  2 must intersect the impulse set  = ℎ at some points   1 and   2 , respectively, and after impulsive effect, the points   1 and   2 must be mapped to the phase set  = ℎ +  at some points   1 and   2 , respectively (see Figure 3).Because distinct trajectories do not intersect, we can easily have then we get the order one successor functions must satisfy Figure 3: Uniqueness of order one periodic solution of (2).which means the order one successor function  1 is monotone decreasing in the line segment , thus there exists only one point  ∈  such that  1 () = 0.
Besides, for any point  ∈ , it is easy to know that  1 () < 0. So the system (2) has only one order one periodic solution in the region Ω 1 .That completes the proof.

Theorem 17. Under the conditions of Theorem 16, the order one periodic solution of the system (2) is orbitally asymptotically stable.
Proof.According to Theorem 16, the system (2) has a unique order one periodic solution that passes through the point  which is in the phase set  = ℎ + , where   0 <   <   1 .The trajectory of the system (2) through point  1 intersects the impulse set  = ℎ at point   1 , and after impulsive effect, the point    1 is mapped to point  2 , where   <   2 <   .Besides, the trajectory of the system (2) from point  2 must intersect the impulse set  = ℎ again at a point   2 , and after impulsive effect, the point   2 is mapped to a point  3 in the phase set  = ℎ + , where   1 >   3 >   (see Figure 4).
Proof.For any point  ∈  1 ,   1 <   <   , the trajectory from point  will undergo impulsive effect infinite times.Denote the impulsive point and phase point corresponding to the th ( = 1, 2, . ..) impulsive effect by    and   , respectively.Similar to the discussions in Theorem 17, we can easily know   <     <    0 ,  = 1, 2, . .., then we have Obviously, point   is the order  successor point of point , then the order  successor function   () =    −  ̸ = 0.By Lemma 13, there does not exist an order  ( = 1, 2, . ..) periodic solution through point .
From the above discussion, we know that the order one periodic solution is the unique periodic solution of the system (2) and there is no order  ( ≥ 2) periodic solution in region Ω 1 .That completes the proof.
According to Lemma 15, the system (1) has four equilibria: two hyperbolic saddles (0, 0) and ( 2 ,  2 ), a hyperbolic stable node (, 0), and a node or focus ( 1 ,  1 ).Obviously, the axis, the vertical isocline / = 0, and the unstable and stable flow of ( 2 ,  2 ) are all intersected with the impulse set  = ℎ and phase set  = ℎ + .For convenience, we denote the intersections by points   and   ,  and ,  and , and  1 and , respectively (see Figure 5).Then the region Ω 2 is the interior of the closed curve Ê0  2  ∪   ∪      0 , where  0 and  2 denote the equilibria (, 0) and ( 2 ,  2 ), respectively.According to the impulsive conditions of the system (2), there must exist a fixed value  0 ∈ (0, 1), when  =  0 , point  is mapped to the point  after impulsive effect.Also, there must exist a fixed value  * ∈ ( 0 , 1), when  =  * , point  is mapped to the point  after impulsive effect.Thus we have (1 −  0 )  =   and (1 −  * )  =   .When  =  * , we have (1 − )  =   ; we know the curve B 2  ∪  is an order one circle which has the saddle  2 ( 2 ,  2 ) in it; that is to say, the system (2) has an order one homoclinic cycle.
If  ∈ ( 0 ,  * ), after impulsive effect, point  is mapped to a point between points  and .Similar to the discussion of Theorems 16 and 17, we can prove that the system (2) has an order one periodic solution which passes through a point  ∈  and is orbitally asymptotically stable.Besides, for any point  ∈   , the trajectory of the system (2) from point  must ultimately tend to the equilibrium (, 0) and does not undergo any impulsive effect, that is to say the system (2) has no periodic solution passing through the point ; so we can know the system (2) has a unique order one periodic solution in the region Ω 2 which is orbitally asymptotically stable.
When  ∈ ( * , 1), we have 0 < (1 − )  <   .Then the trajectory of the system (2) through point  ∈  must tend to the equilibrium (, 0) after coming across once impulsive effect and the trajectory passing through point  ∈    must tend to the equilibrium (, 0) without undergoing any impulsive effect.So we get that the system (2) has no periodic solution in region Ω 2 .That completes the proof.

Homoclinic Bifurcation of System
(3) about Parameter .By above analysis, we know there exists a  * ∈ (0, 1) such that system (2) has an order one homoclinic cycle and when the parameter  is appropriately changed, the homoclinic cycle disappears and bifurcates a unique stable order one periodic solution.In the following, we will choose  as a control parameter and study homoclinic bifurcation of system (3) by using the theory of rotated vector fields.For the sake of convenience, we give some properties about rotated vector fields.
The perturbed system of system (4) with parameter  is as follows: then we have the following definitions.
Proof.From the discussion in Theorem 19, we know that when  =  * and  = 0, the system (3) has an order one homoclinic circle B 2  ∪ .
Divided by the vertical isocline [(1 − (/)) − /( +  2 )] = 0, the rotated direction of vector fields below the vertical isocline is counterclockwise, but above the vertical isocline, the rotated direction of vector fields of system (3) is clockwise.When the parameter  changes from 0 (the value in which heteroclinic cycle exists) to 0 <  1 ≪ 1, both the unstable and stable flow of ( 2 ,  2 ) move away from their original Journal of Applied Mathematics positions.We denote the new intersections of the unstable and stable flow of ( 2 ,  2 ) and the impulse set  = ℎ and phase set  = ℎ +  are  and , and  1 and , respectively (see Figure 6).According to Theorem 19, we have (1− * )  =   ; then there must exist a point  which is in the impulse set  = ℎ and below the point  such that (1 −  * )  =   .Obviously, system (3) must have a trajectory passing through the point , and we suppose it intersects with the phase set  = ℎ +  at point .point  is the order one successor point of point , then the order one successor function  1 () =   −   < 0. Besides, select a point  0 next to  in the phase set  = ℎ + ; the trajectory of the system (3) from point  0 must intersect the impulse set  = ℎ at a point   0 which is next to point  and above point , and after impulsive effect, point   0 is mapped to a point  1 which is the order one successor point of  0 , then we have is to say point  1 is above point .Then we have the successor function  1 ( 0 ) =   1 −   0 > 0. By Lemma 12, we know that there must exist a point  in the phase set  = ℎ +  which is between the points  and  such that  1 () = 0. Then we know the system (3) has an order one periodic solution which passes through the point .
Similar to the discussion of Theorems 16 and 17, we can prove the uniqueness and stability of the order one periodic solution.That completes the proof.

Numerical Simulations and Discussions
In this paper, we have proposed two predator-prey models with nonmonotonic functional response and state dependent impulsive harvesting.If we take no account of the predator harvesting, Ruan and Xiao (2001) showed that the system (1) has no periodic solution under the conditions listing in the Theorem 16.Our numerical simulations also confirm such conclusions (see Figure 7).However, under the same conditions, if we further consider an impulsive predator harvesting, we get that the system (2) has a periodic solution in some cases.We also showed that the existence is mainly dependent on the harvesting rate .We found that there exists a threshold  0 ∈ (0, 1), and the system must have a unique orbitally asymptotically stable order one periodic solution when the harvesting rate  is higher than  0 .When the periodic solution exists, our numerical simulations show that, with a bigger harvesting rate, the solution has a longer period and a larger amplitude (see Figure 8).Besides, we also prove that the system (2) does not have order , ( = 2, 3, . ..) periodic solutions in Theorem 18; then the order one periodic solution is the unique periodic solution.These results illustrate that if we can satisfy the conditions listing in Theorem 16, we can not only form a sustainable ecological system but also obtain large amounts of valuable biological resources.
Under the conditions listing in Theorem 19, the system (1) also does not have any periodic solution (see Figure 9) and we also give the conditions under which the system (2) has an order one periodic solution.Besides, according to the conclusions of Theorem 19, we can choose the parameter  as a bifurcation parameter such that the impulsive differential equation system (2) exhibits the phenomenon of homoclinic bifurcation.Similar to many ordinary differential equation systems, we can see that there exists a bifurcation point  =  * for the system (2).When  =  * , the system (2) has an order one homoclinic cycle which is the unique order one circle in region Ω 2 .When  gradually changes from  =  * to  0 <  <  * , the order one homoclinic cycle is broken and a new order one periodic solution which is orbitally asymptotically stable is generated (see Figure 10).When  gradually changes from  =  * to  >  * , the order one homoclinic cycle is also broken but no new order one periodic solution is generated at the same time; that is to say, the system (2) will have no periodic solution (see Figure 11).These results illustrate that it demands reasonable control of the harvest yield to form a good ecological environment.

Figure 1 :
Figure 1: (a) The solution mapping of the system (4).(b) The order one periodic solution of the system (4).(c) The order two periodic solution of the system (4).

Figure 4 :
Figure 4: Illustration of the orbital asymptotic stability of the order one periodic solution of the system (2).

Figure 5 :
Figure 5: Illustration of homoclinic bifurcation of system (2) by choosing the harvesting rate  as control parameter.