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A mathematical model for the relationship between the populations of giant pandas and two kinds of bamboo is established. We use the impulsive perturbations to take into account the effect of a sudden collapse of bamboo as a food source. We show that this system is uniformly bounded. Using the Floquet theory and comparison techniques of impulsive equations, we find conditions for the local and global stabilities of the giant panda-free periodic solution. Moreover, we obtain sufficient conditions for the system to be permanent. The results provide a theoretical basis for giant panda habitat protection.

The giant panda is a highly specialized Ursid, approximately its 99% of their diet is bamboo [

Yuan et al. may be the first person who have developed mathematical models for the relationship between giant pandas and bamboo [

Impulsive differential equations, that is, differential equations involving an impulse effect, appear as a natural description of observed evolution phenomena of several real-world problems [

The organization of the paper is as follows. Section

In this section we will introduce some notations and definitions together with a few auxiliary results related to the comparison theorem, which will be useful for establishing our results.

Let

Let

The solution of system (

Suppose

(H):

defined on

Note that under appropriate conditions (such that,

Let the function

where

Using Lemma

The positive octant

Let us consider

All solutions

Let

By Lemma

which yields

First, we will give the basic properties of the following differential equations considering the absence of the giant panda.

When the giant panda

Suppose that

Similarly, we have the following Lemma

Suppose that

It follows from Lemmas

Now, we study the local stability of the giant panda-free periodic solution

Suppose that

The local stability of the periodic solution

where

For the upper triangular matrix, there is no need to calculate the exact forms of

The resetting impulsive condition of system (

All of the eigenvalues of

are

If the reverse of (

If the conditions of Theorem

Choose

Next, we prove that

Similarly, we can get that

Condition (

We make mention of the definition of permanence before starting the permanence of system (

System (

Suppose that

Let

Therefore, it is necessary only to find an

This step will show that

Since

Next, we consider the following two subsystems:

Imitating the proof of Theorem

Subsystem (

Subsystem (

In this paper, our purpose is that of considering the survival of the giant panda. From Theorems

In this paper, we consider an impulsive differential system of the population ecology on the three populations of the giant panda and two kinds of bamboo. The local and global stability of the giant panda-free periodic solution are obtained and we find the threshold value

This work is supported by the National Natural Science Foundation of China (no. 11171284), Basic and Frontier Technology Research Program of Henan Province (nos. 132300410025 and 132300410364), and Key Project for the Education Department of Henan Province (no. 13A110771).