TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 137384 10.1155/2013/137384 137384 Research Article A Mathematical Model with Pulse Effect for Three Populations of the Giant Panda and Two Kinds of Bamboo Shi Xiang-yun 1,2 Song Guo-hua 1,3 Banas J. Garcia-Pacheco F. J. Mohiuddine S. A. 1 College of Forestry Beijing Forestry University Beijing 100083 China bjfu.edu.cn 2 College of Mathematics and Information Science Xinyang Normal University Xinyang Henan, 464000 China xytc.edu.cn 3 School of Science Beijing University of Civil Engineering and Architecture Beijing 100044 China bucea.edu.cn 2013 5 12 2013 2013 02 10 2013 31 10 2013 2013 Copyright © 2013 Xiang-yun Shi and Guo-hua Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

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: (1)dx1dt=x1(a10-a11x1-a13x3),t(n+l-1)T,dx2dt=x2(a20-a22x2-a23x3),tnT,dx3dt=x3(-a30-a33x3+a31x1+a32x2),Δx1(t)=-αx1(t),t=(n+l-1)T,Δx2(t)=-βx2(t),t=nT,(x1(0+),x2(0+),x3(0+))=(x10,x20,x30), where x1(t) and x2(t) are the respective densities of two kinds of bamboo at time t and x3(t) is the density of the giant panda. ai0(i=1,2) denote the birthrate of two kinds of bamboo, respectively. aii(i=1,2) denote the density restriction coefficients of the two kinds of bamboo. ai3(i=1,2) are the predation rate of giant panda feeding upon two kinds of bamboo, respectively. (a3i/ai3)(i=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 .

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.

2. 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.

Let +=[0,+), +*=(0,+), and +3={x=(x1,x2,x3)3:x1,x2,x30}. Denote as the set of all of nonnegative integers and as f=(f1,f2,f3)T the right-hand sides of the first three equations (1). Let V:+×+3+, and then V is said to belong to class V0 if

V  is continuous on ((n-1)T,(n+l-1)T]×+3((n+l-1)T,nT]×+3 and lim(t,y)(t0,x)V(t,y)=V(t0,x) exists, where t0=(n+l-1)T+ and nT+.

V is locally Lipschitzian in x.

Definition 1.

Let VV0, for (t,x)((n-1)T,(n+l-1)T]×+3, and the upper right derivative of V with respect to the impulsive differential system (1) is defined as (2)D+V(t,x)=limsuph0+1h[V(t+h,x+hf(t,x)-V(t,x))].

The solution of system (1) is piecewise continuous function X:+×+3, X(t) is continuous on ((n-1)T,(n+l-1)T]((n+l-1)T,nT] and X(t0+)=limtt0X(t) exists, where t0=(n+l-1)T+ and nT+. The smoothness properties of f guarantee the global existence and uniqueness of solution of system (1) for details, see . Given a solution X(t) of (1), defined on [t0,t0+a) with a>0, we say that a solution Y(t) of (1) is a proper continuation to the right of X(t) if Y(t) is defined on [t0,t0+b) for some b>a and X(t)=Y(t) for t[t0,t0+a). The interval [t0,t0+a) is called the maximal interval of existence of solution (saturated solution) X(t) of (1) if x(t) is well defined on [t0,t0+a) and it doesnot have any proper continuation to the right. For other results on impulsive differential equations, see [12, 13].

Lemma 2 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Suppose VV0 and (3)D+V(t,x)g(t,V(t,x)),t(n+l-1)T,tnT,V(t,x(t+))φ1(V(t,x)),t=(n+l-1)T,V(t,x(t+))φ2(V(t,x)),t=nT, where g:+×+ satisfies.

(H):  g is continuous on ((n-1)T,(n+l-1)T]×+((n+l-1)T,nT]×+ and the limit lim(t,y)(t0,x)g(t,y)=g(t,x) exists, where t0=(n+l-1)T+ and nT+, and is finite for x+ and n, and ψ1,ψ2:++ are nondecreasing for all n. Let r(t) be the maximal solution for the impulsive Cauchy problem: (4)dudt=g(t,u(t)),t(n+l-1)T,tnT,u(t+)=ψ1(u(t)),t=(n+l-1)T,u(t+)=ψ2(u(t)),t=nT,u(0+)=u00,

defined on [0,). Then V(0+,x0)u0 implies that V(t,x(t))r(t),t0, where x(t) is any solution of (3).

Note that under appropriate conditions (such that, g is locally Lipschitz continuous with x in ((n-1)T,(n+l-1)T]×+((n+l-1)T,nT]×+ 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 r(t) 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(+,)(PC1(+,)) denote the class if real piecewise continuous (real piecewise continuously differentiable) functions are defined on +.

Lemma 3 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let the function uPC1(+,) satisfy the inequalities: (5)dudt()p(t)u(t)+f(t),tτk,t>0,u(τk)()dku(τk)+hk,k0,u(0+)()u0,

where p,fPC(+,) and dk0, hk and u0 are constants, and {τk}k0 is a strictly increasing sequence of positive real number. Then, for t>0(6)u(t)()u(0)0<tk<tdkexp(0tp(s)ds)+0<tk<t(tk<tj<tdjexp(tktp(s)ds)hk)+0t0<tk<tdkexp(stp(τ)dτ)f(s)ds.

Using Lemma 3, it is possible to prove that the solution of the Cauchy problem (1) with strictly positive initial value remains strictly positive.

Lemma 4.

The positive octant (+3) is an invariant region for system (1).

Proof.

Let us consider (x1(t),x2(t),x3(t)):[0,T0]3 a saturated solution of system (1) with a strictly positive initial value (x10,x20,x30). By Lemma 2, we can obtain that, for 0<t<T0, (7)x1(t)x10(1-α)[(t+(1-l)T)/T]×exp(0t[a11x1(s)-a13x3(s)]ds),x2(t)x20(1-β)[t/T]×exp(0t[-a22x2(s)-a23x3(s)]ds),x3(t)=x30exp(0t(-a30-a33x3+a31x1+a32x2)ds), where [x] represents the largest integer not exceeding x. That is, (x1(t),x2(t),x3(t)) remains strictly positive on [0,T0].

Lemma 5.

All solutions (x1(t),x2(t),x3(t)) of (1) with initial value (x10,x20,x30)+3 are bounded.

Proof.

Let (x1(t),x2(t),x3(t)) be a solution of (1) with a positive initial value (x10,x20,x30) and let (8)V(t)=a31a13x(t)+a32a23x2(t)+x3(t),t0. Then, if tnT,t(n+l-1)T, and t>0, we obtain that (9)dVdt=a31a13x1(a10-a11x1)+a32a23x2(a20-a22x2)-a30x3-a33x32. Then (10)dVdt+a30V(t)=x1(a10a31a13+a30-a31a11a13x1)+x2(a20a32a23+a30-a32a22a23x2)-a33x32,t(n+l-1)T,tnT,t>0. As the right-hand side of (10) is bounded from above denoted by D, it follows that (11)dVdt+a30V(t)D,t(n+l-1)T,tnT,t>0 together with (12)V((n+l-1)T+)(1-α)V((n+l-1)T),V(nT+)(1-β)V(nT).

By Lemma 3, it follows that(13)V(t)V(0+)(0<(n+l-1)T<t(1-α)0<nT<t(1-β)e-a30t)+D0t[0<(n+l-1)T<t(1-α)0<nT<t(1-β)]e-a30(t-s)ds,

which yields (14)V(t)V(0+)e-a30t+D(1-e-a30t)a30,t>0 and since the limit of the right-hand side of (14) for t is (15)V(t)Da30<, it easily follows that V(t) is bounded in its domain. Consequently, (x1(t),x2(t),x3(t)) are bounded by a constant “D/a30” for sufficiently lager t.

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 x3(t) is eradicated, it is easy to see that the equations in (1) decouple, and then we consider the properties of the subsystems: (16)dx1dt=x1(a10-a11x1),t(n+l-1)T,Δx1(t)=-αx1(t),t=(n+l-1)T,x1(0+)=x10,(17)dx2dt=x2(a20-a22x2),tnT,Δx2(t)=-βx2(t),t=nT,x2(0+)=x20.

Lemma 6 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Suppose that ln(1-α)+a10T>0, then the system (16) has a periodic solution x1*(t) with this notation, and the following properties are satisfied: (18)(i)0Tx1*(t)dt=1a11[ln(1-α)+a10T],(ii)limt|x1(t)-x1*(t)|=0 for all solutions x1(t) of (16) starting with strictly positive x10.

Similarly, we have the following Lemma 7.

Lemma 7.

Suppose that ln(1-β)+a20T>0, then the system (17) has a periodic solution x2*(t) with this notation, and the following properties are satisfied: (19)(iii)0Tx1*(t)dt=1a22[ln(1-β)+a20T],(iv)limt|x2(t)-x2*(t)|=0 for all solutions x2(t) of (17) starting with strictly positive x20.

It follows from Lemmas 6 and 7 that the system (1) has a giant panda-free periodic solution (x1*(t),x2*(t),0).

Now, we study the local stability of the giant panda-free periodic solution (x1*(t),x2*(t),0) by means of the Floquent theory. (We can see the details from Page 26 to 35 of .)

Theorem 8.

Suppose that ln(1-α)+a10T>0 and ln(1-β)+a20T>0 and (20)-a30T+a31a11(ln(1-α)+a10T)+a32a22(ln(1-β)+a20T)<0 hold, and then the giant panda-free periodic solution (x1*(t),x2*(t),0) is locally stable.

Proof.

The local stability of the periodic solution (x1*(t),x2*(t),0) may be determined by considering the behavior of small-amplitude perturbations of the solution. Define (21)x1(t)=u(t)+x1*,x2(t)=v(t)+x2*,x3(t)=w(t). Substituting (21) into system (1), it is possible to obtain a linearization of the system as follows: (22)du(t)dt=(a10-2a11x1*(t))u(t)-a13x1*(t)w(t),hhhhhhhhhhhhhhhhhhhhhhht(n+l-1)T,dv(t)dt=(a20-2a22x2*(t))v(t)-a23x2*(t)w(t),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhtnT,dw(t)dt=(-a30+a31x1*(t)+a32x2*(t))w(t), which can be written as (23)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),0tT,

where Φ(t) satisfies (24)dΦ(t)dt=(a10-2a11x1*0-a13x1*0a20-2a22x2*-a23x2*00-a30+a31x1*+a32x2*),

Φ ( 0 ) = I , and I is the identity matrix, so the fundamental solution matrix is(25)Φ(t)=(exp(0t(a10-2a11x1*(s))ds)0*0exp(0t(a20-2a22x1*(s))ds)**00Δ),

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 Δ=exp(0t(-a30+a31x1*(s)+a32x2*(s))ds).

The resetting impulsive condition of system (1) becomes (3)Δu(t)=-αu(t),Δv(t)=0,Δw(t)=0,t=(n+l-1)T,Δu(t)=0,Δv(t)=-βv(t),Δw(t)=0.t=nT.

All of the eigenvalues of(27)S=(1-α00010001)(10001-β0001)×(exp(0t(a10-2a11x1*(s))ds)0*0exp(0t(a20-2a22x1*(s))ds)**00Δ)

are (28)λ1=(1-α)exp(0t(a10-2a11x1*(s))ds)=(1-α)exp[-ln(1-α)-a10T],λ2=(1-β)exp(0t(a20-2a22x2*(s))ds)=(1-β)exp[-ln(1-β)-a20T],λ3=exp(0t(-a30+a31x1*(s)+a32x2*(s))ds)=exp(a32a22-a30T+a31a11(ln(1-α)+a10T)+a32a22(ln(1-β)+a20T)). Since ln(1-α)+a10T>0, ln(1-β)+a20T>0, and condition (20) hold, it is obvious that 0<λ1<1, 0<λ2<1, and 0<λ3<1, which implies that (x1*(t),x2*(t),0) is stable.

If the reverse of (20) is satisfied, then λ3>1 and (x1*(t),x2*(t),0) is unstable.

Theorem 9.

If the conditions of Theorem 8 are satisfied, the giant panda-free periodic solution (x1*(t),x2*(t),0) is globally asymptotically stable.

Proof.

Choose ɛ1>0, ɛ2>0 small enough that if condition (20) holds, (29)ρ=exp(a31a11-a30T+a31ɛ1T+a32ɛ2Thhhhhh+a31a11[ln(1-α)+a10T]hhhhhh+a32a22[ln(1-β)+a20T])<1. It is seen that (30)dx1dtx1(a10-a11x1),t(n+l-1)T,Δx1(t)=-αx1(t),t=(n+l-1)T,x1(0+)=x10, and so by Lemmas 2 and 3, (31)x1(t)x1*(t)+ɛ1,fort0, where x1*(t) is the periodic solution of the system (32)dudt=u(t)(a10-a11u(t)),t(n+l-1)T,Δu(t)=-αu(t),t=(n+l-1)T,u(0+)=x10,u(t)x1*(t),t, and (33)0Tx1*(t)dt=1a11[ln(1-α)+a10T]. Similarly, (34)x2(t)x2*(t)+ɛ2,fort0, where x2*(t) is the periodic solution of the system (35)dvdt=v(t)(a20-a22v(t)),tnT,Δv(t)=-αv(t),t=nT,v(0+)=x20,v(t)x2*(t),t, and (36)0Tx2*(t)dt=1a22[ln(1-β)+a20T]. Therefore, (37)dx3dtx3(-a30+a31(x1*(t)+ɛ1)+a32(x2*(t)+ɛ2)). Integrating (37) over ((n+l-1)T,(n+l)T) yields (38)x3((n+l)T)x3((n+l-1)T+)×exp(n+l-1)T(n+l)T(-a30+a31(x1*(t)+ɛ1)hhhhhhhhhhhhhh+a32(x2*(t)+ɛ2))dtx3((n+l-1)T)×exp(-a30T+a31a11(ln(1-α)+a10T)hhhhhhh+a32a22(ln(1-β)+a20T)hhhhhhh+(a31ɛ1+a32ɛ2)Ta31a11).=x3((n+l-1)T)ρ. Therefore (39)x3((n+l+k)T)x3((n+l+k-1)T)ρx3((n+l-1)T)ρk.ρk0 as k. This implies that x3(t)0 as t.

Next, we prove that x1(t)x1*(t) and x2(t)x2*(t) as t if limtx3(t)=0. For ɛ3>0 there exists a T1>0 such that 0<x3(t)<ɛ3, t>T1. Without loss of generality, we may assume that 0<x3(t)<ɛ3 for all t0. Then we have (40)dx1dtx1(a10-a11x1-a13ɛ3),t(n+l-1)T,Δx1(t)=-αx1(t),t=(n+l-1)T. By Lemmas 2 and 3, we have (41)x1(t)x~1(t)-ɛ1,fort0, where x~1(t) is the periodic solution of the system (42)dudt=u(t)(a10-a11u(t)-a13ɛ3),t(n+l-1)T,Δu(t)=-αu(t),t=(n+l-1)T,u(0+)=x10,u(t)x~1(t) as t, and (43)0Tx~1(t)dt=1a11[ln(1-α)+(a10-a13ɛ3)T]. Therefore, for ɛ1>0, we have (44)x~1(t)-ɛ1x1(t)x1*(t)+ɛ1 for t large enough. Let ɛ30, and we get x1*(t)-ɛ1x1(t)x1*(t)+ɛ1 for large t, which implies x1(t)x1*(t) as t.

Similarly, we can get that x2(t)x2*(t) as t. This completes the proof.

Remark 10.

Condition (20) can be rewritten as follows: (45)T<(a31/a11)ln(1-α)+(a32/a22)ln(1-β)a30-(a31/a11)a10-(a32/a22)a20. Denote T*=((a31/a11)ln(1-α)+(a32/a22)ln(1-β))/(a30-(a31/a11)a10-(a32/a22)a20), and we find that when T<T*, 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 T*, the bamboo cannot be revived to support giant panda again, so giant panda will die by starvation.

4. Permanence

We make mention of the definition of permanence before starting the permanence of system (1).

Definition 11.

System (1) is said to be permanent if there exist two positive constants m and M such that every positive solution (x1(t),x2(t),x3(t)) of system (1) with x10,x20,x30>0 satisfies mx1(t)M, mx2(t)M, and mx3(t)M for sufficiently large t.

Theorem 12.

Suppose that ln(1-α)+(a10-a13M)T>0,ln(1-β)+(a20-a23)T>0, and (46)-a30T+a31a11[ln(1-α)+a10T]+a32a22[ln(1-β)+a20T]>0 hold, and system (1) is permanent, where M is upper bound of the solution of system (1).

Proof.

Let (x1(t),x2(t),x3(t)) be a solution of (1). From Lemma 3, there exists a constant M>0 such that x1(t)M,x2(t)M,x3(t)M for each solution X=(x1(t),x2(t),x3(t)) of (1) for all sufficiently large t. The first equation of (1) implies (47)x1(t)x1(a10-a11x1-a13M). By Lemmas 2 and 3, we have (48)x1(t)u*(t)-ɛm1,forsufficientlysmallɛ>0, where u*(t) is the periodic solution of system: (49)dudt=u(t)(a10-a11u(t)-a13M),t(n+l-1)T,Δu(t)=-αu(t),t=(n+l-1)T,u(0+)=x10,u(t)u*(t),t, and (50)0Tu*(t)dt=1a11[ln(1-α)+(a10-a13M)T]. Similarly, if ln(1-β)+(a20-a23M)T>0 holds, we can get x2(t)>v*(t)-ɛm2, where v*(t) is the periodic solution of system: (51)dvdt=v(t)(a20-a22v(t)-a23M),tnT,Δv(t)=-αv(t),t=nT,v(0+)=x10,v(t)v*(t),t, and (52)0Tv*(t)dt=1a22[ln(1-α)+(a20-a23M)T].

Therefore, it is necessary only to find an m3>0 such that x3(t)m3 for sufficiently large t. This can be done in the following two steps.

Step   1. Choose m0>0, ɛ4>0, ɛ5>0 small enough that if condition (46) holds, (53)σ=exp(a31a11-a30T-a33m0T-a31ɛ4T-a32ɛ5Thhhhhh+a31a11[ln(1-α)+(a10-a13m0)T])hhhhhh+a32a22[ln(1-β)+(a20-a23m0)T])>1.

This step will show that x3(t)m0 for some t1>0. Assuming the contrary, x3(t)<0 for all t, from system (1), we have (54)dx1dtx1(a10-a11x1-a13m0),t(n+l-1)T,dx2dtx2(a20-a22x2-a23m0),tnT,dx3dtx3(-a30-a33m0+a31x1+a32x2),Δx1(t)=-αx1(t),t=(n+l-1)T,Δx2(t)=-βx2(t),t=nT,(x1(0+),x2(0+),x3(0+))=(x10,x20x30). Consider the corresponding impulsive compare system: (55)du(t)dt=u(t)(a10-a11u(t)-a13m0),t(n+l-1)T,dv(t)dt=v(t)(a20-a22v(t)-a23m0),tnT,dwdt=w(t)(-a30-a33m0+a31u(t)+a32v(t)),Δu(t)=-αu(t),t=(n+l-1)T,Δv(t)=-βv(t),t=nT,(u(0+),v(0+),w(0+))=(x10,x20,x30). By Lemma 2, x1(t)u(t),x2(t)>v(t), and x3(t)w(t). By Lemma 3, (56)u*(t)-ɛ4u(t)u*(t)+ɛ4,forsufficientlysmallɛ4,0Tu*(t)dt=1a11[ln(1-α)+(a10-a13m0)T]. Also, (57)v*(t)-ɛ5v(t)v*(t)+ɛ5,forsufficientlysmallɛ2,0Tv*(t)dt=1a22[ln(1-β)+(a20-a23m0)T]. Thus (58)dwdt(-a30-a30m0+a31(u*(t)-ɛ4)+a32(v*(t)-ɛ5))w(t). Integrating (58) over ((n+l-1)T,(n+l)T) yields (59)w((n+l)T)w((n+l-1)T+)×exp(n+l-1)T(n+l)T(a32(v*(t)-ɛ5)-a30-a33m0+a31(u*(t)-ɛ4)hhhhhhhhhhhhhhhh+a32(v*(t)-ɛ5))dtw((n+l-1)T)σ. Therefore (60)x3((n+l+k)T)w((n+l+k)T)w((n+l+k-1)T)σw((n+l-1)T)σk.

Since σ>1, σk as k. This implies that x3(t) as t, which contradicts the boundedness of x3(t).

Step    2. If x3(t)m0 for all tt1, then the proof is complete. If not, let t*=inftt1{x3(t)<m0}, and then x3(t)m0 for t[t1,t*) and x3(t*)=m0. By Step 1, there exists a t>t* such that x(t)m0. Set t2=inft>t*{x3(t)m0}, and then x3(t)<m0 for t(t*,t2) and x3(t2)=m0. 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 [tn,tn+1],(n), such that x3(t)m0,t[tn,tn+1]. Let T=sup|tn+1-tn|,n. If T=, there must exist a subsequence tni such that tni+1-tni as ni. From Step 1, this can lead to a contradiction with the boundedness of x3(t); therefore, T<. Then (61)x3(t)x3(tn)×exptnt(-a30-a33M+a31u*(t)-a31ɛ4+a32v*(t)-a32ɛ5)dsm0exp((-a30-a33M)T)m3. Let m=min{m1,m2,m3}, and then (62)liminftx1(t)m,liminftx2(t)m,liminftx3(t)m. This completes the proof.

Next, we consider the following two subsystems: (63)dx1dt=x1(a10-a11x1-a13x3),t(n+l-1)T,dx3dt=x3(a30-a33x3+a31x1),Δx1(t)=-αx1(t),t=(n+l-1)T,(x1(0+),x3(0+))=(x10,x30),(64)dx2dt=x2(a20-a22x2-a23x3),tnT,dx3dt=x3(a30-a33x3+a32x2),Δx2(t)=-βx2(t),t=nT,(x2(0+),x3(0+))=(x20,x30).

Imitating the proof of Theorem 12, we can obtain the following theorems.

Theorem 13.

Subsystem (63) is permanent if (65)-a30T+a31a11[ln(1-α)+a10T]>0 holds.

Theorem 14.

Subsystem (64) is permanent if (66)-a30T+a31a22[ln(1-β)+a20T]>0 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.

5. 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 T*. When T<T*, 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 T*, 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.

Acknowledgments

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).

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