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

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.


Introduction
In recent years, many authors [1][2][3][4][5][6][7] have extensively considered different types of predator-prey system.One of the typical systems is the following system: ẋ () =  () −  ()   , ẏ () =  (− +  ()  −1 −  ()) , which was introduced by Hassell in 1975 (see [8] for more details).The character of (1) is that it has the mutual interference constant  (0 <  < 1).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][10][11][12] for more details.
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 manmade 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][17][18][19][20][21][22][23][24][25][26][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;  1 () is the growth rate of the prey in the absence of predator;  2 () is the death rate of predator in the absence of prey;  1 () is the decay rate of the prey in the competition among the preys;  2 () is the decay rate of the predator in the competition among the predators;  1 () is the predation rate of predator, and  2 () is the coefficient of transformation from preys to predators;  1 (  ) and  2 (  ) 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.

Preliminaries
Definition 1.A function () = ((), ()) T ∈ R 2 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 {  } ∩ [0, ], and each discontinuity point is continuous on the left, (ii) () satisfies system (6) in the interval [0, ], Throughout this paper, the following assumptions hold.

Existence of Positive Periodic Solutions
Theorem 3.Besides ( 1 ) and ( 2 ), if there hold the following conditions: then system (6) has at least one positive -periodic solution.
Let () =  () , () =  V() , it follows from (6) that we have It is easy to see that if system (9) has one periodic solution T 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 | ⋅ | 2 denotes the Euclidean norm of R 2 .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 {(     )}  =1 are arbitrary constant vector groups, if ) , where From  is a Fredholm operator with index zero, we get that  has a unique inverse.We define   : Im  → Ker  ∩ Dom  as the generalized inverse to , that is, Then by simply calculating we obtain ) .
(65) Denote  : Dom  × [0, 1] →  as the form where  is a parameter.We will show that when (, V) T ∈ Ω∩ Ker , (, V, ) ̸ = 0 for any  ∈ [0, 1].Assume the conclusion is not true; that is, there is a constant vector (, V) T with || + |V| =  satisfying (, V, ) = 0, that is, By (67) we easily see and we also get Case 2. If V > , there exists a  * ∈ , such that  < V <  * ; from (70), we have Obviously, the open set Ω satisfies all conditions in Lemma 2, and therefore we claim that system (9) has at least one periodic solution on Ω ∩ Dom ; 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).

An Illustrative Example
The following illustrative example demonstrates the effectiveness of our main result.(87) Thus, by Theorem 3, system (86) has at least one positive 2periodic solution.

Conclusion
In this work, we have considered a more general predatorprey 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 (0, 0) is obviously the solution of (6) and that zero solution stands for the extinction of the species.
satisfies | * | + |V * | <  0 and which shows that condition [ 2 ] in Lemma 2 is satisfied.Finally, we prove that condition [ 3 ] in Lemma 2 is satisfied.The isomorphism  of Im  onto Ker  can be defined by