Complex Dynamical Behaviors in a Predator-Prey System with Generalized Group Defense and Impulsive Control Strategy

A predator-prey systemwith generalized group defense and impulsive control strategy is investigated. By using Floquet theorem and small amplitude perturbation skills, a local asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than some critical value. Otherwise, the system is permanent if the impulsive period is larger than the critical value. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest eradication lost its stability. Numerical examples show that the system considered has more complicated dynamics, including (1) high-order quasiperiodic and periodic oscillation, (2) period-doubling and halving bifurcation, (3) nonunique dynamics (meaning that several attractors coexist), and (4) chaos and attractor crisis. Further, the importance of the impulsive period, the released amount of mature predators and the degree of group defense effect are discussed. Finally, the biological implications of the results and the impulsive control strategy are discussed.


Introduction
In population dynamics, a functional response of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes and it is assumed to be monotonically increasing in most predator-prey systems.For example, Holling type I, II, and III functional response [1]  1 (, ) = ,  2 (, ) =   +  , and the sigmoidal type response function [2]  4 (, ) =  2 ( + ) ( + ) , and Ivlev type response function [3]  5 (, ) =  (1 −  − ) .
The previous functional responses are prey dependent.But, both predator and prey densities have an effect on the response, such as Beddington-DeAngelis functional response [4,5]  6 (, ) =   +  +  (4) and modified Holling type II and type III response functions [6]  7 (, ) =  ( + ) ( + ) ,  8 (, ) =  2 ( +  2 ) ( + ) . ( However, some experimental and observational evidence shown that the functional response is not always monotonically increasing, such as Holling type IV [7]  9 (, ) =   +  +  2 (6) and  10 (, ) =  − [8].Group defense is a term used to describe the phenomenon whereby predation is decreased, or even prevented altogether, due to the increased ability of the prey to better defend or disguise itself when it exists in enough large numbers [9][10][11].The buffalo group defense was modeled using a generalized group defense in [12], 11 (, ) =  1 + ℎ  , (7) where  is a positive integer whose value determines the degree of antipredator behavior and group defense.Recently, it is of great interest to investigate complex dynamics for impulsive perturbations in populations dynamics.In particular, the impulsive prey-predator population models have been investigated by many researchers.The results of studies of the dynamics of a predator-prey model with nonmonotonic functional response, such as Holling type IV functional response with respect to an impulsive control strategy, were presented in [13][14][15][16][17][18][19][20][21][22][23][24].To the best of our knowledge, there are few papers studying the group defense predator-prey with impulsive effect, where the antipredator behavior and group defense effect described by nonmonotonic functional response.Zhang et al. [25] considered a predator-prey system with defensive ability of prey by Holling type IV functional response and impulsive perturbations on the predator:   () =  () ( The conditions for the local asymptotically stable preyeradication periodic solution and permanence of the system are obtained; a series of complex phenomena are displayed by numerical simulation.Furthermore, based on this work, Pei et al. [26] investigated a one-prey multi-predator model with defensive ability of the prey by introducing impulsive biological control strategy: And it shown that the multi-predator impulsive control strategy is more effective than the classical one and makes the dynamical behaviors of the system more complex.Recently, a predator-prey system with impulsive effect and group defense with the nonmonotone function  10 (, ) =  − was studied by Li et al. [27], They proved that there exists a locally stable pest-eradication periodic solution when the impulsive period is less than certain critical values; otherwise, the system is permanent.Some complicated dynamics, such as quasiperiodic oscillation, bifurcation, and attractor crisis, were shown by numerical simulations.
In this paper, we study a predator-prey system with impulsive effect and generalized group defense with the nonmonotone function  11 (, ) = /(1 + ℎ  ): where () and () represent the prey and the predator populations at time , respectively; , , , ℎ, , and  are positive. is the intrinsic rate of increase of the prey and  is the death rate of the predator, / is the carrying capacity of the prey,  > 1 is the degree of anti-predator behavior and group defense, and  (0 <  < 1) is the rate of conversing prey into predator.Δ() = ( + ) − (), Δ() = ( + ) − (),  is the periodic of the impulse for predator in order to eradicate target pests, protect nontarget pest (or harmless insect) from extinction and drive target pest to extinction, or control target pest at acceptably low level to prevent an increasing pest population from causing an economic loss. ∈ N + , N + = {1, 2, . ..},   > 0 ( = 1, 2) is the proportionality constant which represents the rate of mortality due to the applied pesticide; for example, impulsive reduction of the population is possible by harvesting or by poisoning with chemicals used in agriculture. > 0 is the number of predators released each time, for example, by artificial breeding of the species or release of some species.The paper is arranged as follows.In Section 2, some notations and Lemmas are given.In Section 3, using the Floquet theory of impulsive equation and small amplitude perturbation skills, we will prove the local stability of preyeradication periodic solution when the impulsive period is less than some critical value and give the condition of permanence.In Section 4, by using bifurcation theory, the existence and stability of positive periodic solution are studied when  is close to the critical value  0 .In Section 5, the results of numerical examples are shown, and some rich dynamic behaviors are obtained; the effects of the impulsive period, the released amount of mature predators and the coefficient of group defense effect are discussed.Finally, the conclusions are discussed briefly in Section 6.
And we will use the following important comparison theorem on impulsive differential equation [29].

Extinction and Permanence
Firstly, we study the stability of prey-eradication periodic solution.
Theorem 6.Let () = ((), ()) be any solution of system (11); then () = (0,  * ()) is locally asymptotically stable provided that Proof.The local stability of periodic solution () = (0,  * ()) may be determined by considering the behavior of small amplitude perturbations of the solution.Consider There may be written where and Φ(0) = , the identity matrix.The linearization of the third and fourth equations of system (11) becomes Hence, if both eigenvalues of have absolute values less than one, then the periodic solution According to Floquet theory [28] of impulsive differential equation, the prey-eradication solution () = (0,  * ()) is locally stable.This completes the proof.
In the following, we investigate the permanence of system (11).
From Theorems 6 and 8 we know that  max is a threshold.If  <  max , then pest-eradication periodic solution (0,  * ()) is asymptotically stable; if  >  max , then system (11) is permanent.
Remark 10.If  1 =  2 = 0,  = 0; that is, there are without taking any pest-management strategy, large numbers of preys (pest) would coexisting with predators (natural enemy).If  = 0, 0 <  1 ,  2 < 1, that is, there is periodic spraying pesticide (or harvesting) only.Thus, we can easily obtain that   max = ln(1/(1 −  1 ))/ <  max is the threshold.If  1 =  2 = 0,  > 0; that is, there is periodic releasing of predator (natural enemy) only, without periodic spraying pesticide (or harvesting).We can easily get that   max = /() <  max is the threshold.Comparing with the classic methods (such as biological control or chemical control), the integrated pest management (IPM) is a better one, since  max >   max and  max >   max .Some numerical examples will be given in Section 5.

Bifurcation and Existence of Positive Periodic Solution
In this section, we deal with the existence of a nontrivial periodic solution to system (11) near the prey-eradication periodic solution (0,  * ()) via bifurcation.
(a) If  ̸ = 0, then one has a bifurcation.Moreover, one has a bifurcation of a nontrivial periodic solution of (53) if  < 0 and a subcritical case if  > 0.
(b) If  = 0, then one has an undetermined case.
In order to apply Lemma 11, we compute the following: ] . (56) If   0 = 0, this corresponds to  0 satisfying ) .(57) Further, we can get Note that Since it is easy to verify that  > 0 and In order to determine the sign of , let We have Thus, we can conclude that ( 0 ) > 0, since and () is strictly increasing.Therefore, we have  < 0. In view of  0 =  max and according to Lemma 11, we obtain the following result.
Theorem 12. System (11) has a positive periodic solution if  >  0 and  is close to  0 , where  0 satisfies ) , (66) and the nontrivial periodic solution is supercritical case via bifurcation, which means that the positive periodic solution is stable.

Numerical Analysis
In this section, we will study the impulsive effect on system (11) and show that the impulsive perturbations cause complicated dynamical behavior for system (11).The influence of , , and  may be documented by stroboscopically sampling one of the variables over a range of their values.Stroboscopic map is a special case of the Poincaré map for periodically forced system or periodically pulsed system.Fixing points of the stroboscopic map correspond to periodic solutions of system (11) having the same period as the pulsing term; periodic points of period  about stroboscopic map correspond to entrained periodic solutions of system (11) having exactly  times the period of the pulsing; invariant circles correspond to quasi-periodic solutions of system (11); system (11) possibly appear chaotic (or strange) attractors.
According to Theorem 12, if the impulsive periodic  >  max and is close to  max , the prey eradication solution becomes unstable, there is a supercritical bifurcation, then the prey and predator can coexist on a stable positive periodic solution when  = 0.68 >  max ≈ 0.6776 (Figure 2).Therefore, in order to control the pest populations, we would choose an appropriate impulsive periodic . >  max and close to  max would be a better one.
Let  = 0.55 and fix other parameter sets of values; we have displayed bifurcation diagrams for the pest population  and the predator population  for impulsive period  over [1,11] and [6,11].We find that by increasing the impulsive period , system (11) undergoes a process of period-doubling cascade → chaos → crisis and high-order periodic oscillations (Figure 3).When  increases from 6 to 7, there is a cascade of period-doubling bifurcations leading to chaos (Figure 4).When  = 8.62, the chaos suddenly disappears and a -periodic solution appears, then the periodic solution abruptly disappears and the chaos abruptly appears again when  = 9.08, these constituting several types of crises (Figure 5).However, when  = 8.62 and  = 9.08, it appears that attractors are nonunique, coexistence of stranger attractor with -periodic solution (Figure 6).Obviously, which one of the attractors is reached depends on the initial values.
From bifurcation diagrams in Figures 3, 7, and 9, we can easily see that the dynamical behavior of these three cases is very complicated, which includes (1) high-order quasi-periodic and periodic oscillations, (2) period-doubling bifurcation, (3) period-halving bifurcations, (4) nonunique dynamics (meaning that several attractors coexist), and ( 5) cries (the phenomenon of "crisis" in chaotic attractors can suddenly appear or disappear, or change size discontinuously as a parameter smoothly varies).

Conclusion
In this paper, we have investigated a predator-prey system with generalized group defense and concerning impulsive control strategy for pest control in detail.We have shown that there exists an asymptotically stable pest-eradication periodic   solution if the impulsive period is less than the critical value  max .If we choose our impulsive control strategy, in order to drive the pest to extinction, we can determine the impulsive period  according to the effect of the chemical pesticides on the populations and the cost of releasing natural enemies such that  <  max .
But, in a real world, complete eradication of pest populations is generally not possible, nor is it biologically or economically desirable.A good-pest control program should reduce pest population to levels acceptable to the public.When  >  max , the stability of the pest-eradication periodic solution is lost, system (11) is permanent, and there exists a nontrivial periodic solution when  is close to  max .The smaller the period, the fewer the pests.Therefore, we can control the pest population below some economic threshold ( is defined as the pest population level that produces    damage equal to the costs of preventing damage) by choosing appropriate impulsive period  and the number of mature predator released , according to the degree of antipredator behavior and group defense , making an integrated pestmanagement strategy every period .Then, the periodic releasing of natural enemies and spraying pesticides change the properties of the system without impulses and our results suggest an effective approach in the pest control.
Numerical results show that system (11) can take on various kinds of periodic fluctuations and several types of attractor coexistence and is dominated by high-order periodic oscillations, quasi-periodic oscillations, and chaotic oscillations.These results imply that the presence of pulses destroys equilibria, initiates multiple attractors, quasi-periodic oscillations, and chaos, and makes the dynamical behaviors more complex.

Figure 5 :
Figure 5: Crises are shown.There is a crisis that the chaos suddenly disappears when  = 8.61, 8.62, and there is a crisis that the chaos suddenly appears when  = 9.07, 9.08.