Analysis of an Impulsive One-Predator and Two-Prey System with Stage-Structure and Generalized Functional Response

An impulsive one-predator and two-prey systemwith stage-structure and generalized functional response is proposed and analyzed. By reasonable assumption and theoretical analysis, we obtain conditions for the existence and global attractivity of the predatorextinction periodic solution. Sufficient conditions for the permanence of this system are established via impulsive differential comparison theorem. Furthermore, abundant results of numerical simulations are given by choosing two different and concrete functional responses, which indicate that impulsive effects, stage-structure, and functional responses are vital to the dynamical properties of this system. Finally, the biological meanings of the main results and some control strategies are given.


Introduction and Model Formulation
In real world, the properties of one-predator and one-prey system have been studied widely and many valuable results have been obtained.If examining the cases that there are two preys for a predator, then the above system cannot reflect the real behaviors of individuals accurately, so scholars proposed three-species predator-prey system.The relationship between species in three-species system may take many forms, such as one prey and two predators [1], a food chain [2,3], or two preys and one predator [4,5].On the other hand, for predator-prey model, in description of the relationship between predator and prey, a crucial element is the classic definition of a predator's functional response.Recently, the dynamics of predator-prey systems with different kinds of functional responses have been studied in relevant literature, such as Holling type [6], Crowley-Martin type [7][8][9], Beddington-DeAngelis type [10,11], Watt type [12,13], and Ivlev type [14].For example, Gakkhar and Naji [15] investigated the dynamical behaviors of the following threespecies system with nonlinear functional response: where  1 () and  2 () represent the two preys densities, respectively, and () represents the density of predators depending on the two preys.However, as Pei et al. [16] pointed out that system (1) could not provide an effective approach because there was no impulsive spraying pesticides or harvesting pest at different fixed moment.We know that pests may bring disastrous effects to their existing system when their amount reaches a certain level.For preventing large economic loss, chemical pesticides are often used in the process of pest management.As a matter of fact, the control on pests often makes pests reduce instantaneously in a short time.In the modeling process, these perturbations are often assumed to be in the form of impulses.Based on traditional models, impulsive differential equations are proposed and extensively used in some applied fields, especially in population dynamics; see [17][18][19].The theory of impulsive differential equation is now being recognized richer than the corresponding differential equation without impulses, which plays a key role in the development of biomathematics; see monographs [20,21] and references cited therein.
On the other hand, the stage-structure for predator was also not considered in system (1).In real world, many species go through two or more life stages when they proceed from birth to death.For many animals, their babies are raised by their parents or are dependent on the nutrition from the eggs they stay in.The babies are too weak to produce babies or capture their prey; hence their competition with other individuals of the community can be ignored.Therefore, it is reasonable to introduce stage-structure into competitive or predator-prey models.Many researchers have incorporated it into biological models, where stage-structure is modeled by using a time delay [22][23][24].Authors [5] pointed out that when the system contained time delay, it had more interesting behaviors.Their results showed that time delay could cause a stable equilibrium to become unstable and Hopf bifurcation could occur as the time delay crossed some critical values.These obtained results have shown that stage-structure plays a vital role in predator-prey models and stage-structured systems exhibit complicated properties.Moreover, Xu [25] showed that an important factor in modeling of predator-prey is the choice of functional response.Model with generalized functional response exhibited many universal properties, which could be applied to many fields because of its flexibility.Shao and Li [26] considered a predator-prey system with generalized functional response.Their results indicated that generalized functional response caused dynamical behaviors of the system to be very complex.
Based on these backgrounds, in this paper, developing system (1) with stage-structure, generalized function response, and impulsive spraying pesticides, we will consider the following one-predator and two-prey system: where  1 () and  2 () represent the densities of two different preys, respectively, and we assume that there is no competition between the two preys. 1 () and  2 () denote the densities of immature predator and mature predator, respectively.  is the natural growth rate of   () ( = 1, 2 By use of impulsive differential equation theory and some analysis techniques, we aim to investigate the existence and global attractivity of predator-extinction periodic solution and the permanence of (2).Further, by numerical analysis, we try to find out the effects of impulsive and stage-structure on this system.
Since  1 () does not appear in the first, the second, and the fourth equation of system (2), we can simplify (2) and restrict our attention to the following system: with initial conditions: From biological point of view, without loss of generality, in this paper, we assumed that   () ( = 1,2) is strictly increasing, differential with   (0) = 0, satisfying 0 <   ()/  <   (a constant) for all  > 0. Further, we only consider (3) in the following biological meaning region: The rest of this paper is organized as follows.In Section 2, we give some notations, definitions, and lemmas.By using lemmas and impulsive comparison theorem, we discuss the existence of predator-extinction solution and permanence of system (3) in Sections 3 and 4, respectively.In Section 5, numerical simulations are given to show the complicated dynamical behaviors of (3).Finally, we end this paper by a brief discussion in Section 6.
Proof.Since (H3) holds and  1 (),  2 () are differential for all  > 0, we can choose two positive constants  1 and  2 to be sufficiently small such that From the first equation of system (3), we have Consider the following impulsive comparison system: In view of Lemma 2, we obtain that with which is unique and globally asymptotically stable positive periodic solution of (12).By use of comparison theorem of impulsive differential equation, there exists  1 ∈  such that, for the sufficiently small constant  1 and all  ∈ (, ( + 1)] ( >  1 ), we have Similarly, there exists  2 ∈  such that, for the sufficiently small constant  2 and all  ∈ (, ( + 1)] ( >  2 ), we have Through observation of the third equation of (3), we have Consider the following differential comparison system: According to (11) and Lemma 1, we have lim In view of the positivity of  2 (), we have lim  → ∞  2 () = 0.It implies that for arbitrarily small positive constant  3 and  large enough, we have Further, from the first and the fourth equation of (3), we have Considering the following comparison system of (20), by Lemma 2, we get the positive periodic solution of system (21) as follows: with By comparison theorem, for given constant  1 > 0 and  large enough, we have  * 3 ()− 1 <  1 ().Let  3 → 0, then  * 3 () →  * 1 (), so we have  * 1 () −  3 <  1 ().It follows from ( 15) that  1 () <  * 1 () +  1 for  sufficiently large, which implies that  1 () →  * 1 () as  → ∞.Similarly, we can obtain  2 () →  * 2 () as  → ∞.This is the end of the proof.

Permanence of System (3)
Now we investigate the permanence of system (3).Before stating the theorem, we give the definition of permanence for system (3).
On one hand, from the first and the fourth equation of (3), combining inequality (27), we have Consider the following comparison system: According to Lemma 2 and (H5), by using comparison theorem, there exists an arbitrarily small constant  4 > 0, such that  1 () ≥  * 5 () −  4 for  large enough, where  * 5 () is the unique and globally stable positive periodic solution of (29) with the following form: for  ∈ (, ( + 1)], and By using comparison theorem of impulsive differential equation, we can derive from (30) that for  ∈ (, ( + 1)].Similarly, we have On the other hand, in order to prove the stability of  2 (), we define a Lyapunov function as follows: Calculating the derivative of () along solution  2 () of system (3), we get According to (H4), we can choose a positive constant  5 small enough such that For some constant  * 2 (0 <  * 2 <  3 ), we claim that  2 () <  * 2 cannot be true for all  >  0 .Suppose that the claim is invalid, then there exists a positive constant  0 such that  2 () <  * 2 for all  >  0 .From system (3), we have From the unique solution  * 6 () of the comparison system of (37), we have () ≥  * 6 () −  5 , for  large enough, where is the unique solution of the following system: for  ∈ (, ( + 1)], with holds for  sufficiently large.Similarly we have In view of (35), combining (41) and (42), we get , and   2 ( 1 +  +  2 ) ≤ 0. However, from (43), we have This is a contradiction.Hence, for all  >  1 , we have  2 () ≥   2 > 0.

Numerical Simulation
For the generalized functional response of (3), there are many functional responses that meet the condition, such as Holling type I, Holling type II, Holling type III, Crowley-Martin type, Beddington-DeAngelis type, Watt type, and Ivlev type.In this section, we choose two concrete functional responses to illustrate the rationality of our results and try to find more dynamical behaviors of system (3).We choose such function response as Holling type II and Beddington-DeAngelis type as follows: Firstly, let and  = 1.By calculation, all parameters satisfy conditions of Theorem 3; then we obtain from Theorem 3 that a predatorextinction solution of system (3) exists, which is globally attractive.By numerical analysis with MATLAB, we get the following simulation figures of a predator-extinction solution and its global attractivity.Figure 1 shows the existence of a predator-extinction solution with only one initial value and Figure 2 shows the attractivity of the predator-extinction solution; that is, regardless of different initial values, species  1 ,  2 , and  2 converge to the predator-extinction solution.Secondly, we choose another set of parameters to illustrate the permanence of system (3).Take 1,  = 0.25,  = 1, and  = 1.One can verify that conditions of Theorem 5 are satisfied; then from Theorem 5, system (3) is permanent.By simulation, the results can be indicated clearly by Figure 3. Figure 3(a) shows the permanence of (3) and Figure 3(b) gives a positive periodic solution of this system.
Thirdly, in view of (H4), we know that pest population will die out if  1 and  2 are larger than the corresponding threshold.In order to investigate the influence of  1 ,  2 and time delay , we fix the same parameters in Figure 3 as follows.Consider that  1 = 0.65,  2 = 1,  1 = 0.65,  2 = 1,  1 = 0.5,  2 = 0.2,  1 = 1,  2 = 1,  1 = 1,  2 = 1,  1 = 1,  2 = 0.8,  3 = 1,  4 = 1,  5 = 1,  = 0.25, and  = 1.If  1 = 0.5, by simulation, pest  1 is driven to extinction (see Figure 4(a)), and if  2 = 0.65, then, similarly, pest  2 becomes extinct (see Figure 4(b)).If  1 = 0.5 and  2 = 0.65 at the same time, then not only both pests are going to extinct but also their predator dies out due to lack of food (see Figure 4(c)), which is contrary to the conservation of biological diversity.From biological point of view, we only need to control these two pests at a rational level by adjusting the value of  1 and  2 , respectively.Furthermore, by simulation, if time delay  between immature predator and mature predator goes up to a threshold ( = 4), the predator will die out (see Figure 4(d)), so we claim that the stage-structure also plays an important role in the permanence of system (3).

Discussion
In this paper, considering the complicated effects from the real world, we introduce impulsive spraying pesticides, stagestructure for predator, and generalized functional response into one-predator and two-prey system.Firstly, we investigate the existence and global attractivity of predator-extinction periodic solution under the condition that  − 1  ( 1  1  1 ( 1 )+  2  2  2 ( 2 )) <  2 .Secondly, we obtain the sufficient conditions of the permanence.Finally, by numerical simulation with MATLAB, we further discuss some complicated dynamical behaviors of the system.Our obtained results imply that if  1 or  2 is larger than a threshold (because of lack of food or catching the pest that died from insecticide), the predator will be extinct (see Figure 1), and if pesticides are used too much or harvesting is excessive on two pests, three species will all die out (see Figure 4(c)).In order to keep biological balance or biological diversity, some protective measures can be taken to ensure  2 is less than the threshold (such as disease prevention and releasing immature or mature predator); then the system will be permanent (see Figures 1-3).By comparing Figure 3 with Figures 4(a) and 4(b), if we change parameters  1 and  2 , respectively,  1 and  2 will die out effectively, but the rest of population will still survive, which can be used to provide a reliable control strategy: if impulsive period  is given, we can adjust  1 ,  2 to give a protection for the predator.It will not only reduce the economic loss but also protect environment from damage.Finally, impulsive period  affects the dynamical behaviors of the system heavily, which may  [118.1, 147.4], more than one periodic solution appears.If a moderate pulse is given ( > 147), then the system shows chaotic phenomenon.The bifurcation diagrams include stable solutions, cycles, cascade, and chaos.bring chaotic phenomena, including stable solutions, cycles, cascade, and chaos (see Figure 5).
In a word, our obtained results show that all parameters  1 ,  2 , , and  bring great effects on the properties of system (3), which can be applied to ecological resource management.The complicated dynamical behaviors imply that the influence from parameters  1 ,  2 , , and  is worthy of being studied and we will continue to study the potential dynamical properties in the near future.

Figure 2 :
Figure 2: Dynamical behavior of system (3) with different initial values.These initial values are chosen randomly, and other parameters are the same as those in Figure1.One can find that the solutions are globally attractive.The difference between Figures1 and 2is that more initial values are chosen in Figure2to show that the solutions are globally attractive.

Figure 3 :
Figure3: The permanence of system(3) with initial values of  1 (0) = 0.1,  2 (0) = 0.8,  2 (0) = 0.5, and  2 = 0.2, and other parameters are the same as those in Figure1.Obviously, all these species can coexist and their densities go into a bounded region.(a) Time series of  1 ,  2 , and  2 , which indicate that the solution of (3) goes into a bounded region to be permanent.(b) Phase portrait of system (3), which implies a positive periodic solution.
).   and  are coefficients of internal competition of prey   ()