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Two predator-prey models with nonmonotonic functional response and state-dependent impulsive harvesting are formulated and analyzed. By using the geometry theory of semicontinuous dynamic system, we obtain the existence, uniqueness, and stability of the periodic solution and analyse the dynamic phenomenon of homoclinic bifurcation of the first system by choosing the harvesting rate

Predator-prey interaction is one of the most important relationships in the ecosystem, so it has long been a focus of study in mathematical ecology. The functional response of the predator to the prey describes how the predator density changes with the prey density. The well-known response functions, which have appeared in a lot of literatures, include linear function, Holling type-II function, Holling type-III function, and so on. The above mentioned response functions are all monotonic, and they are really accurate to describe the predator-prey interactions in many cases. But in the recent decades, scholars in different fields found through experiments that nonmonotonic functional response occurs in some predator-prey interactions. For example, if the prey exhibits group defense [

Harvesting strategy of biological resources is also a focus topic in mathematical bioeconomics because it relates to the optimal management of renewable resources. In many literatures, impulsive differential equations are used to model the human action of harvesting. Predator-prey systems with periodic impulsive harvesting have been studied extensively and important results have been achieved [

A lot of research has shown that the population size of the predator is influenced not only by the prey populations but also by the relative rate of prey population growth. If we take the relative growth rate effect into consideration, the above model needs some changes. To this end, we propose the following system:

In this paper, we mainly discuss the dynamics properties of the systems (

In this section, we give some notations and definitions of the geometric theory of semicontinuous dynamical systems which will be useful for the following discussion.

Consider the state-dependent impulsive differential equations

We define the dynamic system consisting of the solution mapping of the system (

For the systems (

For the semicontinuous dynamical system defined by the state-dependent impulsive differential equations (

Let

If there exists a time point

(a) The solution mapping of the system (

For (2) in Definition

If

If there exists a point

If there exists a point

Suppose

Suppose the impulse set

For Definition

For the systems (

Successor function

For the systems (

By Lemma

For the systems (

In this section, we firstly discuss the existence, uniqueness, and stability of the periodic solution of the system (

For the system (

If

If

According to the model (

If

According to Lemma

According to the impulsive conditions of the system (

When

In the following, we prove the uniqueness of the order one periodic solution. Arbitrarily choose two points

Besides, for any point

Existence of order one periodic solution of (

Uniqueness of order one periodic solution of (

Under the conditions of Theorem

According to Theorem

Repeat the above steps, the trajectory from point

Illustration of the orbital asymptotic stability of the order one periodic solution of the system (

Under the conditions of Theorem

For any point

For any point

Analogously, for any point

From the above discussion, we know that the order one periodic solution is the unique periodic solution of the system (

If

when

if

if

For

When

If

When

Illustration of homoclinic bifurcation of system (

By above analysis, we know there exists a

The perturbed system of system (

For any point on the trajectory of system (

In the positive (negative) rotated vector fields of system (

If

From the discussion in Theorem

For the system (

Obviously, point

Similar to the discussion of Theorems

Illustration of homoclinic bifurcation of system (

In this paper, we have proposed two predator-prey models with nonmonotonic functional response and state dependent impulsive harvesting. If we take no account of the predator harvesting, Ruan and Xiao (2001) showed that the system (

The time series and the portrait phase of the system (

The time series and the portrait phase of the system (

Besides, we also prove that the system (

Under the conditions listing in Theorem

The time series and the portrait phase of the system (

Time series and portrait phase of the system (

Time series and portrait phase of the system (

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

This work is supported by the National Natural Science Foundation of China (11171284, 11371306, and 11301453), the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (104200510011), Program for Innovative Research Team (in Science and Technology) in University of Henan Province (2010IRTSTHN006), and Sci-Tech Program Project of Henan Province (122300410034).