An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator

An impulsive two-prey and one-predator model with square root functional responses, mutual interference, and integrated pest management is constructed. By using techniques of impulsive perturbations, comparison theorem, and Floquet theory, the existence and global asymptotic stability of prey-eradication periodic solution are investigated.We use somemethods and sufficient conditions to prove the permanence of the systemwhich involvemultiple Lyapunov functions and differential comparison theorem. Numerical simulations are given to portray the complex behaviors of this system. Finally, we analyze the biological meanings of these results and give some suggestions for feasible control strategies.


Introduction and Model Formulation
In real world, the study on models of three or more species is very popular, such as food-chain and food webs systems, which have extremely rich dynamics [1,2].For predator-prey model, in portrayal of the relationship between predator and prey, a crucial element is the classic definition of a predator's functional response.In the past few decades, many different functional responses have been extensively investigated [3][4][5][6][7].For example, Liu et al. [5] gave the following Holling type II functional response which describes the relations of one prey and one predator: where  1 () is the density of prey and  2 () is the density of predator at time . is the intrinsic growth rate of prey. represents the rate of intraspecific competition or density dependence. is the death rate of predator. is transformation rate for the predator to prey;  1 () 2 ()/(1 +  1 ()) denotes the Holling type II functional response.However, in the actual ecosystem, if examining more complicated ecological case, some preys show herd behavior.
That is to say, the predator interacts with the prey along the outer corridor of the herd of prey.Hence actual dynamic behaviors of individuals have not been described in detail by the predation term of Holling type II functional response.Ajraldi et al. [8] pointed out that, by using the terms of the square root of the prey population, the response functions of prey that exhibited herd behavior are more properly modeled.In this respect, Braza [9] gave the following predator-prey model: where √ 1 ()/(1 + √ 1 ()) is the square root functional response.Ma et al. [10] also investigated a predator-prey system with square root functional response.Their results showed that square root functional response brought about large influence to the dynamical behaviors.
On the other hand, few researchers consider the mutual interference between predators, but mutual interference between predators always exists in the actual ecosystem.In 1971, Hassell set about studying the capturing behavior between hosts and parasites; he discovered that hosts or parasites had the tendency to depart from each other when they met, which affected the hosts capturing.If the size of parasite became larger and larger, then the mutual interference would be stronger and stronger.Hence he introduced the mutual interference of predator [11].Considering the effect from mutual interference between predators, the dynamic behaviors were more complex.For example, He et al. [12] and Zhang et al. [13] investigated the mutual interference of the predator in detail and obtained much different dynamics with those models without mutual interference.Hence, for predator-prey system, it is necessary to consider the mutual interference of predator.
Based on above discussion, we give the following preypredator system with square root functional response and mutual interference of the predator: where 0 <  ≤ 1 (see [8] for the details).
As is known to all, insects have a profound impact on the survival and development of human beings.Most insects are beneficial to human beings; only a few insects are harmful to human life and agricultural development when they reach a certain amount.Hence it is necessary to kill the harmful pests or control them in a certain quantity.Chemical control and biological control are two most commonly used methods.Chemical control is often applied by spraying pesticides, which are used widely because they can kill pests quickly and reduce economic losses in a short time, but they also produce serious environmental pollution.For less pollution to the environment, by stocking or releasing natural enemies, biological control appears, but the effects are not very great.In order to combine different approaches to control pests at the same time, integrated pest management is given to maximize control efficiency and reduce pollution.During the last two decades, ecological pest control is a complex project [14,15].For predator-prey system, pest control strategy has been an important topic for many researchers [16,17].
The main purpose of this paper is to investigate the dynamical behaviors of an impulsive one-predator twoprey model with mutual interference, square root functional response, and integrated control methods.The model is described by the following differential equations: where  1 (),  2 (), and  3 () are densities of two preys and one predator at time , respectively.  ( = 1, 2) is intrinsic increasing rate;  3 is the death rate of predator. represents the mutual interference of the predator: 0 <  ≤ 1.   ( = 1, 2) is transformation rate for the predator to prey.  ( = 1, 2) is death rate of prey; 0 <   < 1 ( = 1, 2, 3) represents the percent of prey-predator that dies at time  = ( +  − 1), 0 <  < 1.  > 0 is the releasing number of predators at  = .Parameters ,  are competitive effects between two preys, respectively.Parameter  is the moment period of impulsive effect.The integer  ∈ ;  is the set of all nonnegative integers.All parameters are positive constants.We aim to investigate the dynamical behaviors of (4).From the biological point of view, we only consider system (4) in the biological meaningful region: and the initial conditions for system (4) are The structure of this paper is as follows.In Section 2, we give some definitions, notations, and lemmas.In Section 3, by using techniques of impulsive perturbations, Floquet theory, and comparison theorem, we discuss stability, extinction, and permanence of system (4).We give corollaries for single chemical control in Section 4. Then we give some examples and numerical analysis of system (4) in Section 5. Finally, we conclude this paper with a brief discussion in Section 6.

Preliminaries
In this section, some helpful remarks, notations, definitions, and lemmas are introduced which are useful for our main results.
Next, we introduce some fundamental properties about the following subsystem of (4): System ( 10) is a periodically forced linear system easily used to obtain ) is a positive periodic solution of system (10).

Single Chemical Control
If  = 0, system (4) concerns the single chemical control.Then we have the following corollary.
Thirdly, by numerical analysis, we aim to investigate the bifurcation diagrams of impulsive period .Let  1 (0) = 2,  2 (0) = 2, and  3 (0) = 3, and  1 = 0.6,  2 = 0.8,  3 = 0. From these diagrams, we can see that impulsive period  heavily affects the dynamical behaviors.For example, Figure 5 shows the complex dynamic behaviors of  3 .Figure 5 is the magnified parts of Figure 4(c).From Figure 5(a), we can see the following: when impulsive period 107 <  < 117, it is stable; if  = 117, 142.4,148, and 149.1, then bifurcation appears, respectively.That is to say, when  > 117, there is a cascade of period-doubling bifurcations leading to 2period solution (Figure 5 As  increase beyond 149.5, the phenomenon of "crisis" emerges.When  = 151.6,there is a typical chaotic oscillation (Figure 5(c)).When  is near 153.5, we can see in the neighborhood of  = 153.5, after the period-doubling bifurcations, the symmetry-breaking bifurcations appear (Figure 5(d)), which are specially simple bifurcations that come into being multiplicity of steady states [5].It implies that when 107 <  < 117, that is, spraying pesticides and releasing natural enemies are frequent, the solution of this system is stable in this situation; when 117 <  < 149.5, periodic behaviors of prey and the predator will appear; if 149.5 <  < 155, then dynamical properties of this system are complex and the development of this system may be unpredictable.

Conclusion
In this paper, an impulsive two-prey and one-predator system with square root functional responses, mutual interference, and integrated pest management is constructed.Numerical simulations are given to portray the complex behaviors of this system.From Theorem 9, the existence and global asymptotic stability of prey-eradication periodic solution of (4) are obtained.Some methods and sufficient conditions are given to prove the permanence of system (4) in Theorem 10.
From Theorems 9 and 10 and simulations, we know dynamical properties of system (4) are very complex which depend on impulsive period , the releasing amount of predator , the mutual interference of predator , and the parameter   ( = 1, 2, 3) of pests or predator which dies from the chemical control.Figures 1(c) and 3(c) show that different values of mutual interference of predator have different dynamical properties for system (4).Figures 1(a 4 implies that impulsive period  heavily influences the dynamic behavior of system (4).As  changes, periodic behaviors, bifurcations, "crisis" phenomenon, chaotic phenomenon, chaotic oscillation, and symmetry-breaking bifurcations appear, respectively.Hence, we can choose moderate value of  for some different control strategies.
In this paper, by considering mutual interference and square root functional responses, our constructed model is new and complex, which more rationally reflects the real world.Furthermore, we also give the corresponding results on strategies of integrated pest management and classical chemical control.By our obtained results, if combining biological control and chemical control to eradicate preys, we can choose moderate impulsive period  and moderate parameters  and   ( = 1, 2, 3) to effectively eliminate preys and reduce environmental pollution.In particular, we analyze the influence from impulsive period .These theories have some guidance to our real life and the natural balance.

Figure 2 :
Figure 2: Dynamic behavior of the prey-eradication periodic solution with different initial values and the parameters are same as Figure 1.(a) Phase portrait of  1 () and  3 () with different initial values.(b) Phase portrait of  2 () and  3 () with different initial values.

Figure 3 :
Figure 3: (a) is the existence of the positive periodic solution of system (4) with initial values of  1 () = 2,  2 () = 2,  3 () = 3, which implies that system (4) is permanent; (b) is phase portrait of positive periodic solution of system (4); (c) is dynamic behavior of system (4) with the mutual interference of the predator  = 0.66; other parameters are also the same as (a).

Figure 5 :
Figure 5: The magnified parts of Figure 4(c) of dynamic behavior of  3 .(a) and (b) are period solutions.(c) is a typical chaotic oscillation when  = 151.6.(d) is the symmetry-breaking bifurcations when  = 153.5.

Figure 6 :
Figure 6: Dynamical behaviors of prey-eradication periodic solution and permanence of the system with single chemical control: (a) is the prey-eradication periodic solution of the single chemical control; (b) is the permanence of the single chemical control.
), 1(b), and 2 show the existence and attractivity of prey-eradication periodic solution.Figures3(a) and 3(b) show the existence of positive periodic solution.