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.
The giant panda is a highly specialized Ursid, approximately its 99% of their diet is bamboo . 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 .
Yuan et al. may be the first person who have developed mathematical models for the relationship between giant pandas and bamboo . 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 . 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: where and are the respective densities of two kinds of bamboo at time and is the density of the giant panda. denote the birthrate of two kinds of bamboo, respectively. denote the density restriction coefficients of the two kinds of bamboo. are the predation rate of giant panda feeding upon two kinds of bamboo, respectively. 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 [9–11].
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.
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 , , and . Denote as the set of all of nonnegative integers and as the right-hand sides of the first three equations (1). Let , and then is said to belong to class if(1)is continuous on and exists, where and .(2) is locally Lipschitzian in .
Definition 1. Let , for , and the upper right derivative of with respect to the impulsive differential system (1) is defined as
The solution of system (1) is piecewise continuous function , is continuous on and exists, where and . The smoothness properties of guarantee the global existence and uniqueness of solution of system (1) for details, see . Given a solution of (1), defined on with , we say that a solution of (1) is a proper continuation to the right of if is defined on for some and for . The interval is called the maximal interval of existence of solution (saturated solution) of (1) if is well defined on and it doesnot have any proper continuation to the right. For other results on impulsive differential equations, see [12, 13].
Lemma 2 (see ). Suppose and
(H): is continuous on and the limit exists, where and , and is finite for and , and are nondecreasing for all . Let be the maximal solution for the impulsive Cauchy problem:
defined on . Then implies that , where is any solution of (3).
Note that under appropriate conditions (such that, is locally Lipschitz continuous with in etc. see Remark 2.3 and Theorem 2.3 of  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 denote the class if real piecewise continuous (real piecewise continuously differentiable) functions are defined on .
Lemma 3 (see ). Let the function satisfy the inequalities:
where and , and are constants, and is a strictly increasing sequence of positive real number. Then, for
Lemma 4. The positive octant is an invariant region for system (1).
Proof. Let us consider a saturated solution of system (1) with a strictly positive initial value . By Lemma 2, we can obtain that, for , where represents the largest integer not exceeding . That is, remains strictly positive on .
Lemma 5. All solutions of (1) with initial value are bounded.
Proof. Let be a solution of (1) with a positive initial value and let
Then, if , and , we obtain that
As the right-hand side of (10) is bounded from above denoted by , it follows that
By Lemma 3, it follows that
which yields and since the limit of the right-hand side of (14) for is it easily follows that is bounded in its domain. Consequently, are bounded by a constant “” for sufficiently lager .
3. 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.
When the giant panda 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 ). Suppose that , then the system (16) has a periodic solution with this notation, and the following properties are satisfied:
for all solutions of (16) starting with strictly positive .
Similarly, we have the following Lemma 7.
Now, we study the local stability of the giant panda-free periodic solution by means of the Floquent theory. (We can see the details from Page 26 to 35 of .)
Theorem 8. Suppose that and and hold, and then the giant panda-free periodic solution is locally stable.
Proof. The local stability of the periodic solution may be determined by considering the behavior of small-amplitude perturbations of the solution. Define
Substituting (21) into system (1), it is possible to obtain a linearization of the system as follows:
which can be written as
, and is the identity matrix, so the fundamental solution matrix is
For the upper triangular matrix, there is no need to calculate the exact forms of and as it is not required in the analysis that follows. And .
The resetting impulsive condition of system (1) becomes
All of the eigenvalues of
are Since , , and condition (20) hold, it is obvious that , , and , which implies that is stable.
If the reverse of (20) is satisfied, then and is unstable.
Theorem 9. If the conditions of Theorem 8 are satisfied, the giant panda-free periodic solution is globally asymptotically stable.
Proof. Choose , small enough that if condition (20) holds,
It is seen that
and so by Lemmas 2 and 3,
where is the periodic solution of the system
where is the periodic solution of the system
Integrating (37) over yields
as . This implies that as .
Next, we prove that and as if . For there exists a such that , . Without loss of generality, we may assume that for all . Then we have By Lemmas 2 and 3, we have where is the periodic solution of the system as , and Therefore, for , we have for large enough. Let , and we get for large , which implies as .
Similarly, we can get that as . This completes the proof.
Remark 10. Condition (20) can be rewritten as follows: Denote , and we find that when , the giant panda-free periodic solution is globally asymptotically stable. That is to say, in this case, the giant panda will be extinct. In biology, when the period of bamboo flowing is smaller than the threshold , the bamboo cannot be revived to support giant panda again, so giant panda will die by starvation.
We make mention of the definition of permanence before starting the permanence of system (1).
Proof. Let be a solution of (1). From Lemma 3, there exists a constant such that for each solution of (1) for all sufficiently large . The first equation of (1) implies
By Lemmas 2 and 3, we have
where is the periodic solution of system:
Similarly, if holds, we can get , where is the periodic solution of system:
Therefore, it is necessary only to find an such that for sufficiently large . This can be done in the following two steps.
Step 1. Choose , , small enough that if condition (46) holds,
This step will show that for some . Assuming the contrary, for all , from system (1), we have Consider the corresponding impulsive compare system: By Lemma 2, , and . By Lemma 3, Also, Thus Integrating (58) over yields Therefore
Since , as . This implies that as , which contradicts the boundedness of .
Step 2. If for all , then the proof is complete. If not, let , and then for and . By Step 1, there exists a such that . Set , and then for and . This process can be continued by repeating Step 1. If this process steps after a finite number of repetitions, the proof is complete. If not, there exists an interval sequence , such that . Let . If , there must exist a subsequence such that as . From Step 1, this can lead to a contradiction with the boundedness of ; therefore, . Then Let , and then This completes the proof.
Next, we consider the following two subsystems:
Imitating the proof of Theorem 12, we can obtain the following theorems.
Theorem 13. Subsystem (63) is permanent if holds.
Theorem 14. Subsystem (64) is permanent if holds.
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.
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.
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).
G. B. Schaller, J. Hu, W. Pan, and J. Zhu, The Giant Pandas of Wolong, University Chicago Press, Chicago, III, USA, 1985.
J. C. Hu, “Review on the classification and population ecology of the giant panda,” Zoological Research, vol. 21, no. 1, pp. 28–34, 2000.View at: Google Scholar
C. G. Yuan, J. C. Hu, H. D. Zhang, and S. Dong, The Mathematical Model of the Population Between Giant Pandas and Bamboos and Its Application in Shed 5.1, Research and Progress in Biology of the Giant Panda, Science and Technology press of Sichuan Province, Chengdu, China, 1989.
G. Yang, H. D. Zhang, and J. C. Hu, “A study of the mathematical model of the dynamic relations between giant pandas and two kinds of food bamboo in Liang Shan mountains,” Journal of Biomathematics, vol. 8, no. 4, pp. 177–185, 1993.View at: Google Scholar
J. Guo, Y. Chen, H. Zhang, G. Chen, J. Hu, and Y. I. Wu, “A mathematical model for the population of giant pandas and bamboo in Yele Nature Reserve of Xiangling mountains,” Journal for Nature Conservation, vol. 10, no. 2, pp. 69–74, 2002.View at: Google Scholar
B. Liu, W. Liu, and Z. Teng, “Analysis of a predator-prey model concerning impulsive perturbations,” Dynamics of Continuous, Discrete and Impulsive Systems B, vol. 14, no. 1, pp. 135–153, 2007.View at: Google Scholar
A. H. Taylor and Q. Zisheng, “Bamboo regeneration after flowering in the Wolong Giant Panda Reserve, China,” Biological Conservation, vol. 63, no. 3, pp. 231–234, 1993.View at: Google Scholar
G. B. Schaller, “Secrets of the Wild Panda,” National Geographic, vol. 169, no. 3, pp. 284–309, 1986.View at: Google Scholar
G. B. Schaller, Q. Teng, K. Johnson, X. Wang, H. Shen, and J. Hu, “The feeding ecology of giant pandas in the Tangjiahe Reserve,” in Research and Progress in Biology of the Giant Panda, J. Hu, F. Wei, C. Yuan, and Y. Wu, Eds., Sichuan Publishing House of Science and Technology, Sichuan, China, 1990.View at: Google Scholar
J. Hu, “Studies on the food habits of the Giant Panda,” in Research and Progress in Biology of the Giant Panda, J. Hu, F. Wei, C. Yuan, and Y. Wu, Eds., Sichuan Publishing House of Science and Technology, Sichuan, China, 1990.View at: Google Scholar
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory or Impulsive Differential Equations, World Scientific, Singapore, 1989.
D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, John Wiley & Sons, New York, NY, USA, 1993.