A Mathematical Model with Pulse Effect for Three Populations of the Giant Panda and Two Kinds of Bamboo

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.


Introduction
The giant panda is a highly specialized Ursid, approximately its 99% of their diet is bamboo [1]. Many of these bamboo species sexually reproduce by synchronous semelparity, that is, the bamboos of a given species within a given region flower at the same time and then die. If the particular bamboo species is one that pandas locally depend upon, there can be a great reduction in local carrying capacity. For example, in the middle of the 1970s and the beginning of 1980s, a large area of Fargesia denudata in Minshan Mountains and Bashania fangiana in Qionglai Mountains bloomed and died, causing the death of at least 138 and 144 giant pandas, respectively [2].
Yuan et al. may be the first person who have developed mathematical models for the relationship between giant pandas and bamboo [3]. After that some mathematical models are presented by some scholars [4,5]. Guo et al. described an improved mathematical model for the relationship between the populations of giant pandas (Ailuropoda melanoleuca) and bamboo by adding a correction term which takes into account the effect of a sudden collapse of bamboo as a food source [5]. Modified by the above, we shall establish an ecological model of the population ecology on the three populations of the giant panda and two kinds of 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 [6,7]. It is known that many biological phenomena involving thresholds, bursting rhythm models in biology, do exhibit impulse effects. The differing varieties of bamboo go through periodic die-offs as part of their renewal cycle. The bamboo, at the end of its life cycle, will bloom and drop its seeds and then dies. Often vast areas of the bamboo forest disappear at the same time. Generally died-back bamboo should take from 10 to 20 years before it can support a panda population again [1,8]. So we can use impulse effect to describe bamboo flowering phenomenon. In this paper, we will consider an impulsive differential system of the population ecology on the three populations of the giant panda and two kinds of bamboo: 2 The Scientific World Journal where 1 ( ) and 2 ( ) are the respective densities of two kinds of bamboo at time and 3 ( ) is the density of the giant panda. 0 ( = 1, 2) denote the birthrate of two kinds of bamboo, respectively. ( = 1, 2) denote the density restriction coefficients of the two kinds of bamboo. 3 ( = 1, 2) are the predation rate of giant panda feeding upon two kinds of bamboo, respectively. ( 3 / 3 ) ( = 1, 2) are the transformation rate of giant panda due to predation on bamboo. Most predator-prey relationships are complicated by the predator's use of multiple prey items or by prey being used by multiple predators. The bamboo-panda relationship does, however, simplify to a binary one such as those modelled by the Lotka-Volterra equations. Although giant pandas do eat other items, their limited remaining habitat has reduced their ability to move on to other species of bamboo which are not flowering [9][10][11].
The organization of the paper is as follows. Section 2 deals with some notation and definitions together with a few auxiliary results related to the comparison theorem, positivity, and boundedness of solutions. Section 3 is devoted to studying the stability of the giant panda-free periodic solutions. In Section 4, we find the conditions which ensure the giant panda to be permanent. The paper ends with discussion on the results obtained in the previous sections.

Preliminaries
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.
Note that under appropriate conditions (such that, is locally Lipschitz continuous with in (( − 1) , ( + − 1) ] × R + ∪ (( + − 1) , ] × R + etc. see Remark 2.3 and Theorem 2.3 of [13] for the details) the Cauchy problem (3) has a unique solution and in that case ( ) becomes the unique solution of (4). We now indicate a result which provides estimation for the solution of a system of differential inequalities. Let PC(R + , R)(PC 1 (R + , R)) denote the class if real piecewise continuous (real piecewise continuously differentiable) functions are defined on R + . Lemma 3 (see [12]). Let the function ∈ 1 (R + , R) satisfy the inequalities: The Scientific World Journal 3 where , ∈ (R + , R) and ≥ 0, ℎ and 0 are constants, and { } ≥0 is a strictly increasing sequence of positive real number. Then, for > 0 Using Lemma 3, it is possible to prove that the solution of the Cauchy problem (1) with strictly positive initial value remains strictly positive.

Stability of the Giant Panda-Free Periodic Solutions
First, we will give the basic properties of the following differential equations considering the absence of the giant panda.

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The Scientific World Journal When the giant panda 3 ( ) is eradicated, it is easy to see that the equations in (1) decouple, and then we consider the properties of the subsystems: Lemma 6 (see [14]). Suppose that ln(1 − ) + 10 > 0, then the system (16) has a periodic solution * 1 ( ) with this notation, and the following properties are satisfied: for all solutions 1 ( ) of (16) starting with strictly positive 10 . Similarly, we have the following Lemma 7.

Permanence
We make mention of the definition of permanence before starting the permanence of system (1).
Therefore, it is necessary only to find an 3 > 0 such that 3 ( ) ≥ 3 for sufficiently large . This can be done in the following two steps.
Next, we consider the following two subsystems: Imitating the proof of Theorem 12, we can obtain the following theorems. Remark 15. In this paper, our purpose is that of considering the survival of the giant panda. From Theorems 13 and 14, we can easily obtain that either condition (65) or condition (66) holds, and the giant panda can be persistent. Clearly, this condition is weaker than condition (46). Moreover, it is also weaker than that of only one kind of food bamboo in the habitat of giant panda.

Conclusion
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 * . When < * , the giant panda-free periodic solution is globally asymptotically stable. That is to say, the giant panda will be extinct if the period of bamboo flowering is smaller than the threshold * , because the bamboo cannot be revived to support giant panda again. Comparing Theorem 12 with Theorems 13 and 14, we know that when there are two kinds of staple bamboo in the giant panda habitat, the conditions which guarantee the giant panda to be permanent are weaker than that of only one kind of staple bamboo in the habitat. Our results will provide a theoretical basis for the rejuvenation update of the bamboo forest after bamboo flowering. Once the bamboo forest flowers, we should timely remove flowering bamboo stains or clamps, and we can promote the flowering bamboo to update and restore as soon as possible by manual intervention approach, such as excavating bamboo stump and rhizome of flowering bamboo, loosing soil and fertilizing Nitrogen fertilizer in the whole forest to promote new whip growth, and sprouting bamboo into bamboo. Our results also provide a theoretical basis for the implementation of artificial bamboo forest. We can select giant panda staple bamboo species according to the flowering cycle to implement artificial bamboo forest plan.