Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 7037245, 15 pages
https://doi.org/10.1155/2017/7037245

Research Article

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

Yantao Luo, Long Zhang, Zhidong Teng, and Tingting Zheng

College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China

Correspondence should be addressed to Long Zhang; longzhang_xj@sohu.com

Received 26 July 2017; Accepted 23 October 2017; Published 6 December 2017

Academic Editor: Guang Zhang

Copyright © 2017 Yantao Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 [2–14]). 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, 17–24]) 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 for . The system evolves from its initial state without being further affected by diffusion until the next pulse appears. , 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 , . 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 but not in . 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 and represents the predator population density in channels between two patches. When 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 , 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 . Meanwhile, the predator species preys on the species in the channels. When , 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 . Obviously, the predation behavior only happens in the time interval ; that is, it is intermittent. Here, represent the intrinsic growth rates of the species in the th patch over the time intervals and , respectively; denote the intrinsic growth rates of the species over the time intervals and , respectively; represent the intercompetition rates of the species in the th patch over the time intervals and , respectively; denote the intercompetition rates of the species over the time intervals and , respectively; represents the survival rates of switching from stage 1 (without dispersal) to stage 2 (dispersal movement); denotes the survival rates of switching from stage 2 to stage 1; is the dispersal rates from the th patch to the th patch during the time interval ; and is the time for the population to disperse from patch to . During the whole process, the predator species never disperses.

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.

2. Preliminaries

Let and denote the set of real numbers and the 2-dimensional Euclidean linear space, respectively, and be a time sequence, satisfying , with as . Here, . Define , is continuous everywhere except at , and exist with . Define ; the norm of is defined by . Let for all and for .

In this paper, we assume that all solutions of system (4) satisfy the following initial conditions:

where . 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 satisfying the initial conditions (5). Obviously, the solution 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 is said to be uniformly almost periodic if, for arbitrary , 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 there exists a real number such that if the points belong to the same interval of continuity of and satisfy the inequality , then (3)For any there exists a relatively dense set such that if , then for all satisfying the condition . The elements of are called -almost periods.

We claim that function is almost periodic, and we denote .

Definition 3 (see [31]). System (4) is said to be permanent if there exist positive constants and , such that for any positive solutions of system (4) with initial value that satisfy condition (5)

Definition 4 (see [24]). Let , then is said to belong to class if

(i) is continuous in for each ,

(ii) is locally Lipschitzian in .

In this paper, there are some notations and assumptions that shall be used:

If , is an almost periodic function, we define

where is a positive constant.

We let be an inverse function of the function .Functions , and are -almost periodic and bounded continuous functions for all and ..There exists a constant , such that, for any ,

There exists a constant , such that, for any ,

There exist constants , such that , where and is the integer part of The set of sequences , is uniformly almost periodic, and , where belongs to the relatively dense set .There exists a constant , such that, for any ,

Now, we give some useful lemmas which will be used in the proofs of the main results.
If there is no predator in system (4), we have the following predator-free system:

where

For system (12), we have the following result.

Lemma 5 (see [26] Theorem ). Suppose that assumptions – hold; then system (12) has a unique globally attractive positive -almost periodic solution .

If there is no prey in system (4), we have the following prey-free system:

where

For system (14), we have the following result.

Lemma 6 (see [26]). Suppose assumptions - and - hold; then system (14) has a unique globally attractive positive -almost periodic solution .

For , we define

where is the right-hand side of system (12). We give the following vector comparison results of the impulsive differential equations.

Lemma 7 (see [1]). Let and . Assume that

where is continuous in and quasi-monotone nondecreasing in , for exists, and is nondecreasing. Let be the maximal solution of the following vector impulsive differential system:

existing on . Then implies that

where is any solution of system (12) existing on . Here, we state that function is quasi-monotone nondecreasing in ; if , and for some , then .

3. Main Results

First, in terms of the ultimate boundedness for system (4), we obtain the following result.

Theorem 8. Suppose that assumptions – hold. Then there is a constant such that

for any positive solutions of system (4).

Proof. Let be any positive solution of system (4) satisfying the initial condition (5). From system (4), we have

for all . Then by the Lemmas 5 and 7, we obtain

where is the solution of system (12) with the initial condition (5). Under assumptions –, from Lemma 5 we obtain , where is the globally asymptotically stable positive -almost periodic solution of system (12). Hence, is bounded on Next, we choose a constant , where Since , there is a pair of and such that

Hence, by (22) we have

Consequently,

Further, we prove that there is a constant such that

From systems (4) and (24) we obtain

for all , where

From the comparison theorem of the impulsive differential equations [16, 28, 29], we have for all , where is the solution of the following auxiliary equation:

with initial value . According to assumptions - and -, we also have

Hence from Lemma 6, system (29) has a unique globally attractive positive -almost periodic solution . For any constant , there is a such that for all . Therefore, we have

for all . Consequently,

Therefore, (26) holds. Choose a constant ; then we can see

this completes the proof of Theorem 8.

Next, on the permanence of system (4), we have the following results.

Theorem 9. Suppose that assumptions - and - hold. Then the predator species of system (4) is permanent.

Proof. Let be any positive solutions of system (4) satisfying the initial condition (5). By system (4), we easily obtain

Using comparison theorem of the impulsive differential equations [16, 28, 29] and Lemma 6, we can easily obtain , for all , where is the solution of system (14) with condition (5). Under the assumptions - and -, by Lemma 6, we can obtain as , where is the globally asymptotically stable positive -almost periodic solution of system (14). Hence, we can easily obtain that, for , there exists a such that

for all . Together with Theorem 8, we have that the predator species of system (4) is permanent. This completes the proof of Theorem 9.

Remark 10. Set

In system (4), based on the assumptions and the actual biological meanings of the parameters and , 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 , which means that the predator species has another food resource. As a result, Theorem 9 implies that if , 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.

Theorem 11. Suppose that assumptions – and the following inequality

for hold. Then the prey species of system (4) is permanent.

Proof. Owing to condition (37), we have that there is a small enough constant , 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 and Lemma 5, for arbitrary constant the following system

has a unique globally attractive positive -almost periodic solution . For above and , there is a , such that, for any and ,

for all , where is the solution of system (42) with the initial condition .
According to the theorem of the continuity of solutions with respect to parameters of the impulsive differential equations [17], there exist and such that

for all . Also owing to the almost periodic property of and , we further have

for all .
Suppose is arbitrary positive solution of system (4); from above inequalities (24), (31), and (35), we have

for all . If there exists a , such that for all , then from system (4), we can obtain

for all . By comparison theorem of impulsive differential equations [16, 28, 29], we also have for all , where is the solution of system (42) satisfying and with the initial value . For (43), let and . From , we have for

for all . Then for all . Thus, by condition (45), we further get

Taking into account system (4) again, from above inequality (49) and system (4), we have

for all . From Lemma 7, we also have for all , where is the solution of system (39) satisfying the initial value . Because of the globally attractivity of positive -almost periodic solution of system (39), we further have that, for previous constant , there exists a such that

for all Hence, we have for all , which leads to a contradiction.
Hence, there exists a constant such that . We now prove that

where

If inequality (52) is not true, then from , there are constants and satisfying , such that There are two cases for .
Case  1. .
Case  2. .
For Case  1, we see that is an impulsive time. Hence, there is an integer such that . We have

which leads to a contradiction.
For Case  2, we have , for all . Assume , then for any we have

where

Integrating (55) from to , for any , we obtain

Assume ; then when , according to above discussion we directly have

Particularly,

Since for all Then we have

for . From comparison theorem of impulsive differential equations [16, 28, 29], we also have for all , where is the solution of system (42) with and satisfies . In (43), choose a and , since , we have for

Hence,

Thus, from (45) we further have

From (63), we have

For arbitrary . We choose an integer such that ; then integrating (64) from to , we have

which leads to a contradiction. According to the discussion above, we finally have

Obviously, is independent of any positive solutions of system (4). This completes the proof of Theorem 11.

Remark 12. Set

Since is the globally asymptotically stable periodic solution of the prey-free subsystem (14) of system (4), we obtain that the integral is the net total growth rate of species within a whole almost period, while is the maximal total survivability of species within the process in which species disperses between patches. Therefore, is the net total growth rate of species during a whole period of transference. Theorem 11 tells us that if , then species in system (4) is permanent.

4. 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.

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

Parameter
Value

Due to system (4) being an almost periodic system, we will show the numerical simulation on the following eight intervals in Table 2.

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

Number
Interval

Interval length
Process

5
1

4
2

6
1

5
2

3
1

7
2

6
1

4
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 – of Lemma 5 hold. Therefore, system (12) has a unique positive globally attractive almost periodic solution . Numerical simulations of these results can be observed in Figures 1(a) and 1(b). From numerical simulations, we obtain the minimum values of , which are (0.6988, 0.8355).

(a)

(b)

Figure 1: (a) The globally stable positive almost periodic solution of the species of system (12). (b) The portrait of the globally stable positive almost periodic solution of the species of system (12). Here, we take the initial values for all .

Then, for system (14), when we also take the above parameters , we have

We can easily demonstrate that the assumptions - and - 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 , then we have

This is contrary to the assumptions - and - of Theorem 9, that is, in the whole period, if 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 and the minimum values of 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 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.

(a)

(b)

Figure 2: (a) The globally stable positive almost periodic solution of the species of system (14), and we take the initial condition for all . (b) Extinction of the species of system (14), and we take the initial values for all .

(a)

(b)

(c)

Figure 3: (a) The globally stable positive almost periodic solution of the species of system (14), and the predator species is permanent just relying on the predation; here, we take the initial condition for all . (b) The time series of almost periodic solution of the species of system (72). (c) The 3-dimension time series of almost periodic solution of the species of system (72).

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 for the purpose of contrast.

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

Parameter
Value

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

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.

(a)

(b)

Figure 4: (a) The time series of almost periodic solution of the species of the predator-prey model with continuous predation. (b) The time series of almost periodic solution of the species of the predator-prey model with intermittent predation. Here, we take the initial condition for all .

Finally, for system (4), to test our main results, we use the above parameters in Table 1. We can simply calculate that the assumptions – in Theorem 11 hold, and for (37), we let 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) and 5(b)).

(a)

(b)

Figure 5: (a) The time series of the globally stable positive almost periodic solution of the species . (b) The portrait of the globally stable positive almost periodic solution of the species , and we take the initial condition for all .

Conflicts of Interest

All the authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11361059, 11702237), the Key Project of Laboratory of Xinjiang Province of China (2016D03022), the NSF of Jiangsu Province of China (BK20130799), and the Doctoral Scientific Research Foundation of Xinjiang University (no. BS160204).

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