Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 138984, 12 pages
http://dx.doi.org/10.1155/2015/138984

Research Article

Existence of Positive Periodic Solutions for a Predator-Prey System of Holling Type IV Function Response with Mutual Interference and Impulsive Effects

Haidong Liu and Fanwei Meng

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received 19 August 2014; Revised 17 January 2015; Accepted 18 January 2015

Academic Editor: Xue Z. He

Copyright © 2015 Haidong Liu and Fanwei Meng. 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.

We investigate the existence of periodic solutions for a predator-prey system with Holling function response and mutual interference. Our model is more general than others since it has both Holling type IV function and impulsive effects. With some new analytical tricks and the continuation theorem in coincidence degree theory proposed by Gaines and Mawhin, we obtain a set of sufficient conditions on the existence of positive periodic solutions for such a system. In addition, in the remark, we point out some minor errors which appeared in the proof of theorems in some published papers with relevant predator-prey models. An example is given to illustrate our results.

1. Introduction

In recent years, many authors [1–7] have extensively considered different types of predator-prey system. One of the typical systems is the following system:

which was introduced by Hassell in 1975 (see [8] for more details). The character of (1) is that it has the mutual interference constant . When Hassell studied the capturing behavior between the hosts (some bees) and parasite (a kind of butterfly), he noted that the hosts had the tendency to leave each other when they met, which interfered the hosts capturing effects. He also found that the mutual interference would be stronger while the populations of the parasite became larger and therefore he introduced the concept of mutual interference constant . From then on, many authors began to study some kinds of predator-prey systems with mutual interference; see [9–12] for more details. Recently, Wang and Zhu [13] investigated a Volterra model with mutual interference and a Holling II type functional response

And Wang et al. [14] discussed a Volterra model with mutual interference and a Holling III type functional response:

where

is the Holling III type predation function. We can easily get for which shows that the predation rate increases with the increasing prey population density. But some experiments and observations indicate that the nonmonotonic response occurs at a level: when the nutrient concentration reaches a high level, an inhibitory effect on the specific growth rate may occur. That means that the predation function may not always increase. To describe such inhibitory effect, Andrews in 1968 (see [15] for more details) suggested another type of Holling function called Holling type IV function

On the other hand, because of many natural and man-made factors, such as fire, drought, flooding, hunting, and harvesting, the intrinsic discipline of biological species usually undergoes some discrete changes of relatively short duration at some fixed times. More appropriate mathematical models for those situations are probably systems with impulsive effects. In recent years, many researchers have investigated several kinds of impulsive differential equations (see [16–27] and the references therein).

In this paper, we consider the following predator-prey system of Holling type IV function response with mutual interference and impulsive effects:

where denotes the density of the prey population and denotes the density of the predator population; is the growth rate of the prey in the absence of predator; is the death rate of predator in the absence of prey; is the decay rate of the prey in the competition among the preys; is the decay rate of the predator in the competition among the predators; is the predation rate of predator, and is the coefficient of transformation from preys to predators; and represent the populations and at regular harvest pulse.

By use of the continuation theorem in coincidence degree theory and some new analytical tricks, we have derived sufficient conditions for the existence of positive periodic solutions of the general system (6). In proving the theorem, we have avoided the errors that exist in the existing articles. We also provide an example to illustrate our theorem.

2. Preliminaries

Definition 1. A function is said to be a -periodic solution of system (6), if it satisfies the following conditions:

(i) is a piecewise continuous map with first-class discontinuity points in , and each discontinuity point is continuous on the left,(ii) satisfies system (6) in the interval ,(iii) satisfies , .

Throughout this paper, the following assumptions hold.

, , and are all continuous positive periodic functions with a common period . and , , are positive constants, , and there exists a positive integer , such that , , for .

Let and be two Banach spaces, is a linear map, and is a continuous map. If dim codim and are closed, then we call operator a Fredholm operator with index zero [28]. If is a Fredholm operator with index zero and there exist continuous projects and such that = , = = , then has an inverse function, which we set as . Assume is any open bounded set, if is bounded and is relatively compact, then we say is -compact on . Since is isomorphic to , there exists isomorphism . Now we come to the continuation theorem [28, page 40].

Lemma 2 (see [28], Continuation Theorem). Let and be both Banach spaces, let be a Fredholm operator with index zero, let be an open bounded set, and let be -compact on . If all the following conditions hold:

for each , ,for each , ,,

then the equation has at least one solution on .

For the convenience, we denote

3. Existence of Positive Periodic Solutions

Theorem 3. Besides () and (), if there hold the following conditions:

then system (6) has at least one positive -periodic solution.

Proof. Suppose is an arbitrary positive solution of system (6).
Let , , it follows from (6) that we have

It is easy to see that if system (9) has one -periodic solution , then is a positive -periodic solution of (6). Therefore, we need only to prove that (9) has one -periodic solution.
To apply Lemma 2, we take

with the norm

and let

with the norm

be equipped with the norm

where denotes the Euclidean norm of . Then and are both Banach spaces.
Let

and define operators and as follows, respectively:

then

is closed in , and

It follows that is a Fredholm mapping of index zero, and it is easy to know that and are both continuous projectors such that

where and are defined by

where   are arbitrary constant vector groups, if , and satisfy

From is a Fredholm operator with index zero, we get that has a unique inverse. We define as the generalized inverse to , that is,

Then by simply calculating we obtain

By the Lebesqgue convergence theorem, and are both continuous. From the Arzela-Ascoli Theorem, we can get that is relatively compact and is bounded for any open set . So is -compact on for any open bounded set .
Now we consider the operator equation , , that is,

Integrating (24) over the interval leads to

From the first equation of (25), we have

So we get

Multiplying the first equation of (24) by and integrating over , we obtain

From (28) and the integral mean value theorem, there exists a such that

Similarly, multiplying the second equation of (24) by and integrating over , we obtain

from (30) and the integral mean value theorem, there exists a such that

From (29) and (31), we have

From (32), we get

which yields

From (27) and (34), we obtain

Set

and then, from (27), (35), and (36), we get

Since , there exist such that

From (37) and (38), we see that

On the other hand, it follows from (24) that

Thus, from (39)–(41), we have

Meanwhile, the first equation of (25) implies

In view of and (34), we have from (44) that

If , then it follows from (45) that

so we have

From (38) and (47), we have

This, together with (40), leads to

Let ; then from (42) and (49), we have

Let

Case 1. If , for any , then it follows from the second equation of (25) that

and from (38) we get

that is,

Similar to (49), from (41) and (54) we obtain

Case 2. There exists such that , that is,

From (41) and (56) we obtain

So we have

Let , then from (43) and (58), we get

Clearly, and are independent of . Denote , where is taken sufficiently large such that each solution (if the system has at least one solution) of the following system of algebraic equations:

satisfies and

where

Let ; then satisfies condition in Lemma 2. If , then is a constant vector in with . So

which shows that condition in Lemma 2 is satisfied. Finally, we prove that condition in Lemma 2 is satisfied. The isomorphism of onto can be defined by

For , we have

Denote as the form

where is a parameter. We will show that when , for any . Assume the conclusion is not true; that is, there is a constant vector with satisfying , that is,

By (67) we easily see

and we also get

Case 1. If , from (70) and , we have .
Case 2. If , there exists a , such that ; from (70), we have

Case 2.1. If , from (71) and , we obtain .
Case 2.2. If , we have .
So we have

Then from (69) and (72), we get .
From (68), we have

and from (69) and (72), we have

Case 1. If , then we get the lower bounds of .
Case 2. If , then we see , which together with (74) yields

which implies

So we get

From (68) and (69), we obtain

and then we have

Then from (77) and (79), we get .
Therefore,

which leads to a contradiction. Using the property of topological degree, we have

By and , we see that the following system of algebraic equation

has a unique solution in . Thus, a standard and direct calculation shows that

Obviously, the open set satisfies all conditions in Lemma 2, and therefore we claim that system (9) has at least one -periodic solution on ; that is, system (6) has at least one positive periodic solution. Thus we complete the proof.

Remark 4. Our model (6) is more general than those in [24, 25] since there are different types of Holling functions. The results in [24, 25] do not give the decision on existence of positive solution to (6).

Remark 5. In our proof, by new tricks, we avoid the errors that existed in [23–25]. In the proof of Theorem 2.1 [23, page 230], the authors stated that: “let , ,

where

be continuous at , , exist and , .” From this, we can derive that ; that is, and ,  . From the definition of , we obtain that exists, and then we can deduce that and are continuous at . So we have , for any . So we get , for any , which contradicts the conclusion of Theorem 2.1 [23]. Errors similarly reappear in proving Theorem 2.4 [24, page 3393] and Theorem 2.1 [25, page 1047].

4. An Illustrative Example

The following illustrative example demonstrates the effectiveness of our main result.

Example 1. Consider the following predator-prey system of Holling type IV function response with mutual interference and impulsive effects

We fix the parameters , , , . By an easy calculation, we obtain , .
Therefore we have

Thus, by Theorem 3, system (86) has at least one positive -periodic solution.

5. Conclusion

In this work, we have considered a more general predator-prey model with Holling type IV function response and the impulsive effect. By use of the continuation theorem in coincidence degree theory and new anaytical tricks, we have provided the sufficient conditions to ensure the existence of the positive solution to this model. We also point out minor errors in some papers on relevant models. In the future, investigation on the convergence of the positive solutions will be probably very interesting and significant since is obviously the solution of (6) and that zero solution stands for the extinction of the species.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for his/her careful reading and kind suggestions. This research was supported by the National Natural Science Foundations of China (nos. 11171178 and 61104136) and the Science and Technology Project of High Schools of Shandong Province (no. J14LI09).

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