Coexistence for an Almost Periodic Predator-Prey Model with Intermittent Predation Driven by Discontinuous Prey Dispersal

An almost periodic predator-prey model with intermittent predation and prey discontinuous dispersal is studied in this paper, which differs from the classical continuous and impulsive dispersal predator-prey models. The intermittent predation behavior of the predator species only happens in the channels between two patches where the discontinuous migration movement of the prey species occurs. Using analytic approaches and comparison theorems of the impulsive differential equations, sufficient criteria on the boundedness, permanence, and coexistence for this system are established. Finally, numerical simulations demonstrate that, for an intermittent predator-prey model, both the intermittent predation and intrinsic growth rates of the prey and predator species can greatly impact the permanence, extinction, and coexistence of the population.


Introduction
In real ecosystems, since the spatial distribution and dynamics of a population are greatly affected by their spatial heterogeneity and population mobility, dispersal becomes one of the dominant themes in mathematical biology.In fact, animal dispersal movements between patches are extremely prevalent in ecological environments; for example, many types of birds and mammals will migrate from cold regions to warm regions in search of a better habitat or a breeding site [1].Therefore, to take spatial heterogeneity into account, realistic population models should contain the dispersal process.During the past couple of decades, predator-prey models with diffusion in a patchy environment have attracted significant attention from ecologists, biologists, and biomathematicians.Many important works and monographs about the properties of population dynamics in a spatial idiosyncratic environment, for example, permanence, extinction, and global asymptotic stability of positive periodic solutions, have been written (see [2][3][4][5][6][7][8][9][10][11][12][13][14]).Teng and Chen [6] considered a nonautonomous predator-prey Lotka-Volterra type dispersal system with periodic coefficients and distributed delays: where  is the population density of the predator species confined to the 1st patch.Criteria for the permanence, extinction, and existence of positive periodic solutions for system (1) were established.In this model, the prey dispersal behavior occurs at every point in time and simultaneously between any patches; that is, it is a continuous bidirectional dispersal.
However in practice, it is often the case that diffusion occurs in the form of regular pulses.For example, when winter comes, birds will migrate between patches to seek a better habitat, whereas they do not diffuse in other seasons, and the dispersion of foliage seeds occurs at a fixed period of time every year.For another example, in the Pacific Northwest, Larimichthys polyactis cross over deep water during the winter and migrate to the coast during the spring; then, 3-6 months after spawning, they scatter offshore and return to the depths of the sea during late autumn [15].All these types of migratory behaviors are appropriately assumed to be in the form of pulses in the modeling process.Thus, impulsive diffusion provides a more natural description.Currently, theories of impulsive differential equations [16] have been introduced into population dynamics.A large number of models have been described by impulsive diffusion (see [14,[17][18][19][20][21][22][23][24]) during the past couple of decades.
Shao [23] considered the following predator-prey models with impulsive prey diffusion between two patches: where the pulse diffusion of the species  occurs in every period  (a positive constant) and   is the dispersal rate in the th patch satisfying 0 <   < 1 for  = 1, 2. The system evolves from its initial state without being further affected by diffusion until the next pulse appears.Δ  () =   ( + )−  (), where   ( + ) represents the density of the population of the prey species  in the th patch immediately after the th diffusion pulse at time  = ;   () represents the density of the population of the prey species in the th patch before the th diffusion pulse at time  = ,  = 0, 1, . ... Criteria for the global attractivity and permanence of system (2) were obtained.Furthermore, migration movements of the population will be influenced by many uncertain factors (such as the landscape and weather).Therefore, the dispersal movement of migratory species will have to be suspended when the environment becomes unavailable.In other words, patches would permit normal movement patterns between patches to occur only during certain time intervals instead of all the time.For instance, in the Canary Islands of Spain, Anas platyrhynchos undergo a spring migration from early March to the end of March and a fall migration from late September to the end of October, departing as late as early November, during which they are extensively killed by humans, and other carnivorous animals can prolong the journey [25].As another example, the wild goose will fly to the south when winter comes; in this process, they will stop to rest in some places and at certain time periods.In other words, their diffusion behavior is neither continuous at all times nor impulsive at a fixed time, but it is intermittent within some time intervals.Therefore, it is more reasonable to model this kind of population dynamics with intermittent dispersals.Zhang et al. [26] considered the following nonautonomous almost periodic single species model with intermittent dispersals and dispersal delays between two patches: where all the parameters are almost periodic and the dispersal movement happens only in the time interval  ∈ [ 2+1 ,  2+2 ) but not in  ∈ [ 2 ,  2+1 ).Here,  ∈ .Criteria for the existence, uniqueness, and global attractivity of positive almost periodic solution for system (3) were established.Moreover, in a real ecological system, there always exist natural enemies during the migratory process between patches.For example, annually, at the end of July, with the arrival of the dry season, millions of wildebeests, zebras, and other herbivorous wildlife form a migratory army, migrating from the Serengeti National Park, Tanzania, Africa, to Kenya's Masai Mara National Nature Reserve to find enough water and food.Along the way, they will be preyed upon by lions, leopards, and so on.Additionally, crocodiles and hippopotamus will wait and ambush the migration species in the Mara River.As the seasons alternate, i.e., when the rainy season comes, the migration movement starts again, and these species will return to the Serengeti National Park, and vice versa [27].Obviously, the predation behavior for the above situation is intermittent and only happens in the channels where the dispersals of migratory species occur.Motivated by the above consideration, in this paper, we introduce an almost periodic predator-prey model with intermittent predation and discontinuous prey dispersal between two patches: where   () denotes the prey population density in the th patch ( = 1, 2) and () represents the predator population density in channels between two patches.When  ∈ [ 2 ,  2+1 ) with  ∈ , the prey species   inhabits the th patch and does not disperse.At the same time, the predator species  inhabits the channels between two patches with other food sources.When  →  2+1 , the intrinsic discipline of the species  in each patch changes.The channels between the two patches will open, and the species  disperses bidirectionally from one patch to another; this dispersal movement will continue for the time interval  ∈ [ 2+1 ,  2+2 ).Meanwhile, the predator species  preys on the species  in the channels.When  →  2+2 , the gate of the channels will close, the species   will stop dispersing and inhabit patch .At the same time, the predator species  will also stop preying on species .In allusion to system (4) above, our main purpose in this paper is to establish a series of criteria on the ultimate boundedness, permanence, and coexistence of the two populations for system (4).The methods used in this paper are motivated by the works on the permanence and extinction for periodic predator-prey systems in patchy environments given by Teng and Chen in [6] and the works on the survival analysis for a periodic predator-prey model given by Zhang et al. in [22].
This paper is organized as follows.In Section 2, some definitions, assumptions, and useful lemmas are introduced.In Section 3, we state and prove the main results.Finally, special examples and numerical simulations are illustrated to demonstrate our theoretical results in Section 4.

Preliminaries
Let  and  2 denote the set of real numbers and the 2dimensional Euclidean linear space, respectively, and   be a time sequence, satisfying   <  +1 , with   → ∞ as  → ∞.
In this paper, we assume that all solutions of system (4) satisfy the following initial conditions: where  = ( 1 ,  2 ) ∈  + .It is not hard to prove that the functional of right of system ( 4) is continuous and satisfies the local Lipschitz condition with respect to  in the space × .Therefore, by the fundamental theory of the impulsive functional differential equations with finite delays [16,28,29], system (4) has a unique solution ( 1 (, ),  2 (, ), ()) satisfying the initial conditions (5).Obviously, the solution ( 1 (, ),  2 (, ), ()) is positive in its maximal interval of the existence.
Before going into details, we first draw some very useful definitions and lemmas.
Definition 1 (see [30]).The set of sequences {   |    =  +1 −   , ,  ∈ ,   ∈ } is said to be uniformly almost periodic if, for arbitrary  > 0, there exists a relatively dense set in  of -almost periodic common for all of the sequences; here Definition 2 (see [30]).Assume that the following conditions hold: (1) The set of sequences {   } is almost periodic, ,  ∈ .
(2) For any  > 0 there exists a real number  > 0 such that if the points   ,   belong to the same interval of continuity of () and satisfy the inequality ( We claim that function  ∈ ([ − τ, ],  2 ) is almost periodic, and we denote  ∈ .
In this paper, there are some notations and assumptions that shall be used: where  is a positive constant.
If there is no predator  in system (4), we have the following predator-free system: where For system (12), we have the following result.
If there is no prey  in system (4), we have the following prey-free system: where For system (14), we have the following result.
Next, on the permanence of system (4), we have the following results.
Remark 10.Set In system (4), based on the assumptions and the actual biological meanings of the parameters  3 (),  3 () and ã3 (), b3 (), we can see that the conditions in Theorem 9 are easily satisfied.Additionally, the constant  represents the minimal total growth rate of the predator species .If  > 0, which means that the predator species  has another food resource.As a result, Theorem 9 implies that if  > 0, then the predator species  will be permanent regardless of whether the prey species  exists or not.
Next, on the permanence of the prey species  of system (4), we have the following result.
Proof.Owing to condition (37), we have that there is a small enough constant  1 > 0, such that Consider the following auxiliary system: where By (38), we have Then, from (41) and Lemma 5, we know system (39) has a unique globally attractive positive -almost periodic solution And then, based on assumption  8 and Lemma 5, for arbitrary constant  > 0 the following system has a unique globally attractive positive -almost periodic solution  *  ().For above  1 > 0 and M = max{ 2 ,  −1 2 } > 0, there is a  3 = ( 1 , M) > 0, such that, for any  0 ≥ 0 and for all  ≥  0 +  3 , where   (,  0 ,  0 ) is the solution of system (42) with the initial condition   ( 0 ) =  0 .

Numerical Simulation and Discussion
In this paper, an almost periodic predator-prey model with intermittent prey dispersals and intermittent predation has been investigated.By using the methods of analysis and comparison theorems of the impulsive differential equations, we obtain sufficient conditions of the boundedness and permanence for this system.
To illustrate our results for system (4), we give some numerical simulations by using the following values of parameters in Table 1.
Due to system (4) being an almost periodic system, we will show the numerical simulation on the following eight intervals in Table 2.
Therefore, we can consider almost periodic system as periodic approximately and the − period is 10.First, for system (12), when we take each parameter as in Table 1, we have  We can see that all the assumptions ( 1 )-( 7 ) of Lemma 5 hold.Therefore, system (12) has a unique positive globally attractive almost periodic solution  * () = ( * 1 (),  * 2 ()).Numerical simulations of these results can be observed in Figures 1(a) and 1(b).From numerical simulations, we obtain the minimum values of ( 1 (),  2 ()), which are (0.6988, 0.8355).Then, for system (14), when we also take the above parameters  3 (),  3 (), we have (69) We can easily demonstrate that the assumptions ( 1 )-( 2 ) and ( 7 )-( 8 ) of Lemma 6 and Theorem 9 hold.Obviously, system (14) has a unique almost periodic solution  * (), and the predator species  of system (4) always has the minimum value of 0.3528 and the maximum value of 1.9017 when taking different initial values; that is, it is permanent.All of these results can be observed in Figure 2(a).However, for system (4), when we keep the other parameters unchanged and only just adjust the value of ã3 () = −0.65 + 0.12 sin(1.2)+ 0.12 sin(1.5),then we have This is contrary to the assumptions ( 1 )-( 2 ) and ( 7 )-( 8 ) of Theorem 9, that is, in the whole period, if  < 0 without considering the prey species , then the predator species  of system (4) will be extinct (see Figure 2(b)).
Nevertheless, if we take the prey species  into account, using the above parameters c1 (), c2 () and the minimum values of ( 1 (),  2 ()) with 0.6988, 0.8355 to make then we find an interesting phenomenon where the predator species  of system (4) goes from extinct to permanent (see Figure 3(a)).It turns out that if the predator species  in the time intervals [ 2+1 ,  2+2 ) has no other food resource but can only rely on the prey species , then the more the predation behavior happens, the more likely the predator species will be permanent.Moreover, to illustrate the difference between continuous and intermittent predation of the predator species , we consider the following system [25]: We keep the other parameters unchanged including the intrinsic growth rates of the species , the interspecies competition rates of all species, the survival rates of the species , and the dispersal rates of the species  in system (72).We take another set of parameters in Table 3, and we let   = 0,   =   = 1 ( = 1, 2) for the purpose of contrast.
By simple calculation, we have     Then, we find that continuous predation makes the predator permanent and the prey extinct (see Figures 3(b) and 3(c)).However, if we keep all others parameters unchanged and just change system (72) to system (4), that is, changing the model from continuous predation to intermittent predation, we have (74) Then, there are some interesting phenomena that the predator goes from permanent to extinct, while the opposite occurs for the prey.All of these results can be observed (Figures 4(a) and 4(b)) by numerical simulations.By the above analysis, we can conclude that if the predator continues preying on prey and the predation intensity is higher than the intrinsic growth of the prey, then the prey will be extinct, while the predator will be permanent.We also obtain that if the predation behavior is intermittent and the predation intensity is small, then the predator will be extinct, while the prey will be permanent.So, for an intermittent predator-prey model, both the intermittent predation and the intrinsic growth rates of the prey and predator species can greatly impact the permanence or extinction of the system.Finally, for system (4), to test our main results, we use the above parameters in Table 1.We can simply calculate that the assumptions ( 1 )-( 7 ) in Theorem 11 hold, and for (37), we let  * () = 1.9017 then From numerical simulations, we obtain that the minimum value of  of system (3) is always 0.8085 when using different initial values; that is, the prey  is permanent (see Figures 5(a

Table 1 :
Parameter values used in the simulations of system (4).

Table 2 :
Parameter values used in the simulations of system (4).

Table 3 :
Parameter values used in the simulations of system (4).