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

Abstract

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 [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: 𝑑𝑥₁𝑑𝑡=𝑥₁𝑎₁₀−𝑎₁₁𝑥₁−𝑎₁₃𝑥₃,𝑡≠(𝑛+𝑙−1)𝑇,𝑑𝑥₂𝑑𝑡=𝑥₂𝑎₂₀−𝑎₂₂𝑥₂−𝑎₂₃𝑥₃,𝑡≠𝑛𝑇,𝑑𝑥₃𝑑𝑡=𝑥₃−𝑎₃₀−𝑎₃₃𝑥₃+𝑎₃₁𝑥₁+𝑎₃₂𝑥₂,Δ𝑥₁(𝑡)=−𝛼𝑥₁(𝑡),𝑡=(𝑛+𝑙−1)𝑇,Δ𝑥₂(𝑡)=−𝛽𝑥₂(𝑡)𝑥,𝑡=𝑛𝑇,₁0⁺,𝑥₂0⁺,𝑥₃0⁺=𝑥₁₀,𝑥₂₀,𝑥₃₀, (1) 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. 𝑎_𝑖0(𝑖=1,2) denote the birthrate of two kinds of bamboo, respectively. 𝑎_𝑖𝑖(𝑖=1,2) denote the density restriction coefficients of the two kinds of bamboo. 𝑎_𝑖3(𝑖=1,2) are the predation rate of giant panda feeding upon two kinds of bamboo, respectively. (𝑎_3𝑖/𝑎_𝑖3)(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 [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.

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 ℝ₊=[0,+∞), ℝ^∗₊=(0,+∞), and ℝ³₊={𝑥=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³∶𝑥₁,𝑥₂,𝑥₃≥0}. Denote as ℕ 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∶ℝ₊×ℝ³₊→ℝ₊, and then 𝑉 is said to belong to class 𝑉₀ if(1)𝑉is continuous on ((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]×ℝ³₊∪((𝑛+𝑙−1)𝑇,𝑛𝑇]×ℝ³₊ and lim_{(𝑡,𝑦)→(𝑡0,𝑥)}𝑉(𝑡,𝑦)=𝑉(𝑡₀,𝑥) exists, where 𝑡₀=(𝑛+𝑙−1)𝑇⁺ and 𝑛𝑇⁺.(2)𝑉 is locally Lipschitzian in 𝑥.

Definition 1. Let 𝑉∈𝑉₀, for (𝑡,𝑥)∈((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]×ℝ³₊, and the upper right derivative of 𝑉 with respect to the impulsive differential system (1) is defined as 𝐷⁺𝑉(𝑡,𝑥)=limsup_ℎ→0+1ℎ𝑉𝑡+ℎ,𝑥+ℎ𝑓(𝑡,𝑥)−𝑉(𝑡,𝑥). (2)
The solution of system (1) is piecewise continuous function 𝑋∶ℝ₊×ℝ³₊, 𝑋(𝑡) is continuous on ((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]∪((𝑛+𝑙−1)𝑇,𝑛𝑇] and 𝑋(𝑡⁺₀)=lim_𝑡→𝑡0𝑋(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 [12]. 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 [12]). Suppose 𝑉∈𝑉₀ and 𝐷⁺𝑉()((()𝑉𝑡𝑡,𝑥≤𝑔𝑡,𝑉𝑡,𝑥)),𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑡,𝑥⁺≤𝜑₁(𝑉(()𝑉𝑡𝑡,𝑥)),𝑡=𝑛+𝑙−1𝑇,𝑡,𝑥⁺≤𝜑₂(𝑉(𝑡,𝑥)),𝑡=𝑛𝑇, (3) where 𝑔∶ℝ₊×ℝ₊→ℝ satisfies.
(H):  𝑔 is continuous on ((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]×ℝ₊∪((𝑛+𝑙−1)𝑇,𝑛𝑇]×ℝ₊ 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∈ℝ₊ and 𝑛∈ℕ, and 𝜓₁,𝜓₂∶ℝ₊→ℝ₊ are nondecreasing for all 𝑛∈ℕ. Let 𝑟(𝑡) be the maximal solution for the impulsive Cauchy problem: 𝑑𝑢((𝑡()𝑢𝑡𝑑𝑡=𝑔𝑡,𝑢)),𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,₊=𝜓₁(𝑢(𝑡()𝑢𝑡)),𝑡=𝑛+𝑙−1𝑇,₊=𝜓₂(𝑢(𝑡𝑢0)),𝑡=𝑛𝑇,₊=𝑢₀≥0, (4)
defined on [0,∞). Then 𝑉(0⁺,𝑥₀)≤𝑢₀ implies that 𝑉(𝑡,𝑥(𝑡))≤𝑟(𝑡),𝑡≥0, 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]×ℝ₊∪((𝑛+𝑙−1)𝑇,𝑛𝑇]×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 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(ℝ₊,ℝ)(PC¹(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 𝑢∈𝑃𝐶¹(ℝ₊,ℝ) satisfy the inequalities: 𝑑𝑢≤(≥)𝑝(𝑡)𝑢(𝑡)(𝑡)𝑑𝑡+𝑓,𝑡≠𝜏_𝑘𝑢𝜏,𝑡>0,_𝑘≤(≥)𝑑_𝑘𝑢𝜏_𝑘+ℎ_𝑘𝑢0,𝑘≥0,₊≤(≥)𝑢₀, (5)

where 𝑝,𝑓∈𝑃𝐶(ℝ₊,ℝ) and 𝑑_𝑘≥0, ℎ_𝑘 and 𝑢₀ are constants, and {𝜏_𝑘}_𝑘≥0 is a strictly increasing sequence of positive real number. Then, for 𝑡>0𝑢(𝑡)≤(≥)𝑢(0)_{0<𝑡𝑘<𝑡}𝑑_𝑘exp^𝑡₀𝑝(𝑠)d𝑠+_{0<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝑑_𝑗exp^𝑡_𝑡𝑘𝑝(𝑠)d𝑠ℎ_𝑘+^𝑡_{00<𝑡𝑘<𝑡}𝑑_𝑘exp^𝑡_𝑠𝑝(𝜏)d𝜏𝑓(𝑠)d𝑠. (6)

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 (ℝ³₊) is an invariant region for system (1).

Proof. Let us consider (𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))∶[0,𝑇₀]→ℝ³ 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, 𝑥₁(𝑡)≥𝑥₁₀(1−𝛼)^{[(𝑡+(1−𝑙)𝑇)/𝑇]}×exp^𝑡₀𝑎₁₁𝑥₁(𝑠)−𝑎₁₃𝑥₃(𝑠,𝑥)𝑑𝑠₂(𝑡)≥𝑥₂₀1−𝛽^[𝑡/𝑇]×exp^𝑡₀−𝑎₂₂𝑥₂(𝑠)−𝑎₂₃𝑥₃(𝑠,𝑥)𝑑𝑠₃(𝑡)=𝑥₃₀exp^𝑡₀−𝑎₃₀−𝑎₃₃𝑥₃+𝑎₃₁𝑥₁+𝑎₃₂𝑥₂𝑑𝑠, (7) 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)∈ℝ³₊ 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 𝑉(𝑡)=𝑎₃₁𝑎₁₃𝑥(𝑡)+𝑎₃₂𝑎₂₃𝑥₂(𝑡)+𝑥₃(𝑡),𝑡≥0. (8) Then, if 𝑡≠𝑛𝑇,𝑡≠(𝑛+𝑙−1)𝑇, and 𝑡>0, we obtain that 𝑑𝑉=𝑎𝑑𝑡𝑎³¹₁₃𝑥₁𝑎₁₀−𝑎₁₁𝑥₁+𝑎₃₂𝑎₂₃𝑥₂𝑎₂₀−𝑎₂₂𝑥₂−𝑎₃₀𝑥₃−𝑎₃₃𝑥²₃. (9) Then 𝑑𝑉𝑑𝑡+𝑎₃₀𝑉(𝑡)=𝑥₁𝑎₁₀𝑎₃₁𝑎₁₃+𝑎₃₀−𝑎₃₁𝑎₁₁𝑎₁₃𝑥₁+𝑥₂𝑎₂₀𝑎₃₂𝑎₂₃+𝑎₃₀−𝑎₃₂𝑎₂₂𝑎₂₃𝑥₂−𝑎₃₃𝑥²₃,𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,𝑡>0. (10) As the right-hand side of (10) is bounded from above denoted by 𝐷, it follows that 𝑑𝑉𝑑𝑡+𝑎₃₀𝑉(𝑡)≤𝐷,𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,𝑡>0 (11) together with 𝑉(𝑛+𝑙−1)𝑇⁺≤()𝑉)𝑇),𝑉1−𝛼((𝑛+𝑙−1𝑛𝑇⁺≤1−𝛽𝑉(𝑛𝑇). (12)
By Lemma 3, it follows that𝑉(𝑡)≤𝑉0⁺_{()0<𝑛+𝑙−1𝑇<𝑡}(1−𝛼)_{0<𝑛𝑇<𝑡}1−𝛽𝑒^{−𝑎30𝑡}+𝐷^𝑡_{0()0<𝑛+𝑙−1𝑇<𝑡}(1−𝛼)_{0<𝑛𝑇<𝑡}1−𝛽𝑒^{−𝑎30(𝑡−𝑠)}𝑑𝑠, (13)
which yields 𝑉(𝑡)≤𝑉0⁺𝑒^{−𝑎30𝑡}+𝐷1−𝑒^{−𝑎30𝑡}𝑎₃₀,𝑡>0 (14) and since the limit of the right-hand side of (14) for 𝑡→∞ is 𝑉(𝑡)≤𝐷𝑎₃₀<∞, (15) 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: 𝑑𝑥₁𝑑𝑡=𝑥₁𝑎₁₀−𝑎₁₁𝑥₁,𝑡≠(𝑛+𝑙−1)𝑇,Δ𝑥₁(𝑡)=−𝛼𝑥₁(𝑡)()𝑥,𝑡=𝑛+𝑙−1𝑇,₁0⁺=𝑥₁₀,𝑑𝑥₂𝑑𝑡=𝑥₂𝑎₂₀−𝑎₂₂𝑥₂,𝑡≠𝑛𝑇,Δ𝑥₂(𝑡)=−𝛽𝑥₂(𝑡)𝑥,𝑡=𝑛𝑇,₂0⁺=𝑥₂₀. (16)(17)

Lemma 6 (see [14]). Suppose that ln(1−𝛼)+𝑎₁₀𝑇>0, then the system (16) has a periodic solution 𝑥^∗₁(𝑡) with this notation, and the following properties are satisfied: (𝑖)^𝑇₀𝑥^∗₁(𝑡)1𝑑𝑡=𝑎₁₁ln(1−𝛼)+𝑎₁₀𝑇,(𝑖𝑖)lim_𝑡→∞𝑥₁(𝑡)−𝑥^∗₁(𝑡)=0 (18) 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 𝑥^∗₂(𝑡) with this notation, and the following properties are satisfied: (𝑖𝑖𝑖)^𝑇₀𝑥^∗₁(𝑡)1𝑑𝑡=𝑎₂₂ln1−𝛽+𝑎₂₀𝑇,(𝑖𝑣)lim_𝑡→∞𝑥₂(𝑡)−𝑥^∗₂(𝑡)=0 (19) 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 (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),0).

Now, we study the local stability of the giant panda-free periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),0) by means of the Floquent theory. (We can see the details from Page 26 to 35 of [13].)

Theorem 8. Suppose that ln(1−𝛼)+𝑎₁₀𝑇>0 and ln(1−𝛽)+𝑎₂₀𝑇>0 and −𝑎₃₀𝑎𝑇+³¹𝑎₁₁ln(1−𝛼)+𝑎₁₀𝑇+𝑎₃₂𝑎₂₂ln1−𝛽+𝑎₂₀𝑇<0 (20) hold, and then the giant panda-free periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),0) is locally stable.

Proof. The local stability of the periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),0) may be determined by considering the behavior of small-amplitude perturbations of the solution. Define 𝑥₁(𝑡)=𝑢(𝑡)+𝑥^∗₁,𝑥₂(𝑡)=𝑣(𝑡)+𝑥^∗₂,𝑥₃(𝑡)=𝑤(𝑡). (21) Substituting (21) into system (1), it is possible to obtain a linearization of the system as follows: 𝑑𝑢(𝑡)=𝑎𝑑𝑡¹⁰−2𝑎₁₁𝑥^∗₁(𝑡)𝑢(𝑡)−𝑎₁₃𝑥^∗₁(𝑡)𝑤(𝑡),()(𝑡)𝑡≠𝑛+𝑙−1𝑇,𝑑𝑣=𝑎𝑑𝑡²⁰−2𝑎₂₂𝑥^∗₂(𝑡)𝑣(𝑡)−𝑎₂₃𝑥^∗₂(𝑡)𝑤(𝑡),(𝑡)𝑡≠𝑛𝑇,𝑑𝑤=𝑑𝑡−𝑎₃₀+𝑎₃₁𝑥^∗₁(𝑡)+𝑎₃₂𝑥^∗₂(𝑡)𝑤(𝑡), (22) which can be written as 𝑢(𝑡)𝑣(𝑡)𝑤(𝑡)(𝑡)𝑢(0)𝑣(0)𝑤(0)=Φ,0≤𝑡≤𝑇, (23)
where Φ(𝑡) satisfies 𝑑Φ(𝑡)=𝑎𝑑𝑡₁₀−2𝑎₁₁𝑥^∗₁0−𝑎₁₃𝑥^∗₁0𝑎₂₀−2𝑎₂₂𝑥^∗₂−𝑎₂₃𝑥^∗₂00−𝑎₃₀+𝑎₃₁𝑥^∗₁+𝑎₃₂𝑥^∗₂, (24)
Φ(0)=𝐼, and 𝐼 is the identity matrix, so the fundamental solution matrix isΦ(𝑡)=exp∫^𝑡₀𝑎₁₀−2𝑎₁₁𝑥^∗₁(𝑠)∫𝑑𝑠0∗0exp^𝑡₀𝑎₂₀−2𝑎₂₂𝑥^∗₁(𝑠)𝑑𝑠∗∗00Δ, (25)
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(∫^𝑡₀(−𝑎₃₀+𝑎₃₁𝑥^∗₁(𝑠)+𝑎₃₂𝑥^∗₂(𝑠))𝑑𝑠).
The resetting impulsive condition of system (1) becomes (𝑡)(𝑡),(𝑡)(𝑡)()(𝑡)(𝑡)(𝑡),(𝑡)Δ𝑢=−𝛼𝑢Δ𝑣=0,Δ𝑤=0,𝑡=𝑛+𝑙−1𝑇,Δ𝑢=0,Δ𝑣=−𝛽𝑣Δ𝑤=0.𝑡=𝑛𝑇. (3)
All of the eigenvalues of×∫𝑆=1−𝛼0001000110001−𝛽0001exp^𝑡₀𝑎₁₀−2𝑎₁₁𝑥^∗₁(𝑠∫)𝑑𝑠0∗0exp^𝑡₀𝑎₂₀−2𝑎₂₂𝑥^∗₁(𝑠)𝑑𝑠∗∗00Δ (27)
are 𝜆₁=(1−𝛼)exp^𝑡₀𝑎₁₀−2𝑎₁₁𝑥^∗₁(𝑠)=()()𝑑𝑠1−𝛼exp−ln1−𝛼−𝑎₁₀𝑇,𝜆₂=1−𝛽exp^𝑡₀𝑎₂₀−2𝑎₂₂𝑥^∗₂(𝑠)=𝑑𝑠1−𝛽exp−ln1−𝛽−𝑎₂₀𝑇,𝜆₃=exp^𝑡₀−𝑎₃₀+𝑎₃₁𝑥^∗₁(𝑠)+𝑎₃₂𝑥^∗₂(𝑠)𝑑𝑠=exp−𝑎₃₀𝑎𝑇+³¹𝑎₁₁ln(1−𝛼)+𝑎₁₀𝑇+𝑎₃₂𝑎₂₂ln1−𝛽+𝑎₂₀𝑇. (28) 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 (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),0) is stable.
If the reverse of (20) is satisfied, then 𝜆₃>1 and (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),0) is unstable.

Theorem 9. If the conditions of Theorem 8 are satisfied, the giant panda-free periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),0) is globally asymptotically stable.

Proof. Choose 𝜀₁>0, 𝜀₂>0 small enough that if condition (20) holds, 𝜌=exp−𝑎₃₀𝑇+𝑎₃₁𝜀₁𝑇+𝑎₃₂𝜀₂𝑇+𝑎₃₁𝑎₁₁ln(1−𝛼)+𝑎₁₀𝑇+𝑎₃₂𝑎₂₂ln1−𝛽+𝑎₂₀𝑇<1. (29) It is seen that 𝑑𝑥₁𝑑𝑡≤𝑥₁𝑎₁₀−𝑎₁₁𝑥₁,𝑡≠(𝑛+𝑙−1)𝑇,Δ𝑥₁(𝑡)=−𝛼𝑥₁(𝑡)()𝑥,𝑡=𝑛+𝑙−1𝑇,₁0⁺=𝑥₁₀, (30) and so by Lemmas 2 and 3, 𝑥₁(𝑡)≤𝑥^∗₁(𝑡)+𝜀₁,for𝑡≥0, (31) where 𝑥^∗₁(𝑡) is the periodic solution of the system 𝑑𝑢(𝑡)𝑎𝑑𝑡=𝑢₁₀−𝑎₁₁𝑢(𝑡)()(𝑡)(𝑡)()𝑢0,𝑡≠𝑛+𝑙−1𝑇,Δ𝑢=−𝛼𝑢,𝑡=𝑛+𝑙−1𝑇,₊=𝑥₁₀, (32)𝑢(𝑡)→𝑥^∗₁(𝑡),𝑡→∞, and ^𝑇₀𝑥^∗₁(𝑡)1𝑑𝑡=𝑎₁₁ln(1−𝛼)+𝑎₁₀𝑇. (33) Similarly, 𝑥₂(𝑡)≤𝑥^∗₂(𝑡)+𝜀₂,for𝑡≥0, (34) where 𝑥^∗₂(𝑡) is the periodic solution of the system 𝑑𝑣(𝑡)𝑎𝑑𝑡=𝑣₂₀−𝑎₂₂𝑣(𝑡)(𝑡)(𝑡)𝑣0,𝑡≠𝑛𝑇,Δ𝑣=−𝛼𝑣,𝑡=𝑛𝑇,₊=𝑥₂₀, (35)𝑣(𝑡)→𝑥^∗₂(𝑡),𝑡→∞, and ^𝑇₀𝑥^∗₂(𝑡)1𝑑𝑡=𝑎₂₂ln1−𝛽+𝑎₂₀𝑇. (36) Therefore, 𝑑𝑥₃𝑑𝑡≤𝑥₃−𝑎₃₀+𝑎₃₁𝑥^∗₁(𝑡)+𝜀₁+𝑎₃₂𝑥^∗₂(𝑡)+𝜀₂. (37) Integrating (37) over ((𝑛+𝑙−1)𝑇,(𝑛+𝑙)𝑇) yields 𝑥₃((𝑛+𝑙)𝑇)≤𝑥₃(𝑛+𝑙−1)𝑇⁺×exp_()𝑇^{(𝑛+𝑙)𝑇}_{𝑛+𝑙−1}−𝑎₃₀+𝑎₃₁𝑥^∗₁(𝑡)+𝜀₁+𝑎₃₂𝑥^∗₂(𝑡)+𝜀₂𝑑𝑡≤𝑥₃((𝑛+𝑙−1)𝑇)×exp−𝑎₃₀𝑎𝑇+³¹𝑎₁₁ln(1−𝛼)+𝑎₁₀𝑇+𝑎₃₂𝑎₂₂ln1−𝛽+𝑎₂₀𝑇+𝑎₃₁𝜀₁+𝑎₃₂𝜀₂𝑇.=𝑥₃((𝑛+𝑙−1)𝑇)𝜌. (38) Therefore 𝑥₃((𝑛+𝑙+𝑘)𝑇)≤𝑥₃((𝑛+𝑙+𝑘−1)𝑇)𝜌≤𝑥₃((𝑛+𝑙−1)𝑇)𝜌^𝑘. (39)𝜌^𝑘→0 as 𝑘→∞. This implies that 𝑥₃(𝑡)→0 as 𝑡→∞.
Next, we prove that 𝑥₁(𝑡)→𝑥^∗₁(𝑡) and 𝑥₂(𝑡)→𝑥^∗₂(𝑡) as 𝑡→∞ if lim_𝑡→∞𝑥₃(𝑡)=0. For 𝜀₃>0 there exists a 𝑇₁>0 such that 0<𝑥₃(𝑡)<𝜀₃, 𝑡>𝑇₁. Without loss of generality, we may assume that 0<𝑥₃(𝑡)<𝜀₃ for all 𝑡≥0. Then we have 𝑑𝑥₁𝑑𝑡≥𝑥₁𝑎₁₀−𝑎₁₁𝑥₁−𝑎₁₃𝜀₃,𝑡≠(𝑛+𝑙−1)𝑇,Δ𝑥₁(𝑡)=−𝛼𝑥₁(𝑡),𝑡=(𝑛+𝑙−1)𝑇. (40) By Lemmas 2 and 3, we have 𝑥₁(𝑡)≥𝑥₁(𝑡)−𝜀₁,for𝑡≥0, (41) where 𝑥₁(𝑡) is the periodic solution of the system 𝑑𝑢(𝑡)𝑎𝑑𝑡=𝑢₁₀−𝑎₁₁𝑢(𝑡)−𝑎₁₃𝜀₃()(𝑡)(𝑡)()𝑢0,𝑡≠𝑛+𝑙−1𝑇,Δ𝑢=−𝛼𝑢,𝑡=𝑛+𝑙−1𝑇,₊=𝑥₁₀, (42)𝑢(𝑡)→𝑥₁(𝑡) as 𝑡→∞, and ^𝑇₀𝑥₁(𝑡)1𝑑𝑡=𝑎₁₁ln(1−𝛼)+𝑎₁₀−𝑎₁₃𝜀₃𝑇. (43) Therefore, for 𝜀₁>0, we have 𝑥₁(𝑡)−𝜀₁≤𝑥₁(𝑡)≤𝑥^∗₁(𝑡)+𝜀₁ (44) for 𝑡 large enough. Let 𝜀₃→0, and we get 𝑥^∗₁(𝑡)−𝜀₁≤𝑥₁(𝑡)≤𝑥^∗₁(𝑡)+𝜀₁ for large 𝑡, which implies 𝑥₁(𝑡)→𝑥^∗₁(𝑡) as 𝑡→∞.
Similarly, we can get that 𝑥₂(𝑡)→𝑥^∗₂(𝑡) as 𝑡→∞. This completes the proof.

Remark 10. Condition (20) can be rewritten as follows: 𝑎𝑇<³¹/𝑎₁₁ln(1−𝛼)+𝑎₃₂/𝑎₂₂ln1−𝛽𝑎₃₀−𝑎₃₁/𝑎₁₁𝑎₁₀−𝑎₃₂/𝑎₂₂𝑎₂₀. (45) 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 𝑚≤𝑥₁(𝑡)≤𝑀, 𝑚≤𝑥₂(𝑡)≤𝑀, and 𝑚≤𝑥₃(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 −𝑎₃₀𝑎𝑇+³¹𝑎₁₁ln(1−𝛼)+𝑎₁₀𝑇+𝑎₃₂𝑎₂₂ln1−𝛽+𝑎₂₀𝑇>0 (46) 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 𝑥₁(𝑡)≥𝑥₁𝑎₁₀−𝑎₁₁𝑥₁−𝑎₁₃𝑀. (47) By Lemmas 2 and 3, we have 𝑥₁(𝑡)≥𝑢^∗(𝑡)−𝜀≡𝑚₁,forsufficientlysmall𝜀>0, (48) where 𝑢^∗(𝑡) is the periodic solution of system: 𝑑𝑢(𝑡)𝑎𝑑𝑡=𝑢₁₀−𝑎₁₁𝑢(𝑡)−𝑎₁₃𝑀()(𝑡)(𝑡)()𝑢0,𝑡≠𝑛+𝑙−1𝑇,Δ𝑢=−𝛼𝑢,𝑡=𝑛+𝑙−1𝑇,₊=𝑥₁₀, (49)𝑢(𝑡)→𝑢^∗(𝑡),𝑡→∞, and ^𝑇₀𝑢^∗(𝑡)1𝑑𝑡=𝑎₁₁ln(1−𝛼)+𝑎₁₀−𝑎₁₃𝑀𝑇. (50) Similarly, if ln(1−𝛽)+(𝑎₂₀−𝑎₂₃𝑀)𝑇>0 holds, we can get 𝑥₂(𝑡)>𝑣^∗(𝑡)−𝜀≡𝑚₂, where 𝑣^∗(𝑡) is the periodic solution of system: 𝑑𝑣(𝑡)𝑎𝑑𝑡=𝑣₂₀−𝑎₂₂𝑣(𝑡)−𝑎₂₃𝑀(𝑡)(𝑡)𝑣0,𝑡≠𝑛𝑇,Δ𝑣=−𝛼𝑣,𝑡=𝑛𝑇,₊=𝑥₁₀, (51)𝑣(𝑡)→𝑣^∗(𝑡),𝑡→∞, and ^𝑇₀𝑣^∗(𝑡)1𝑑𝑡=𝑎₂₂ln(1−𝛼)+𝑎₂₀−𝑎₂₃𝑀𝑇. (52)
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, 𝜀₄>0, 𝜀₅>0 small enough that if condition (46) holds, 𝜎=exp−𝑎₃₀𝑇−𝑎₃₃𝑚₀𝑇−𝑎₃₁𝜀₄𝑇−𝑎₃₂𝜀₅𝑇+𝑎₃₁𝑎₁₁ln(1−𝛼)+𝑎₁₀−𝑎₁₃𝑚₀𝑇+𝑎₃₂)𝑎₂₂ln1−𝛽+𝑎₂₀−𝑎₂₃𝑚₀𝑇>1. (53)
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 𝑑𝑥₁𝑑𝑡≥𝑥₁𝑎₁₀−𝑎₁₁𝑥₁−𝑎₁₃𝑚₀,𝑡≠(𝑛+𝑙−1)𝑇,𝑑𝑥₂𝑑𝑡≥𝑥₂𝑎₂₀−𝑎₂₂𝑥₂−𝑎₂₃𝑚₀,𝑡≠𝑛𝑇,𝑑𝑥₃𝑑𝑡≥𝑥₃−𝑎₃₀−𝑎₃₃𝑚₀+𝑎₃₁𝑥₁+𝑎₃₂𝑥₂,Δ𝑥₁(𝑡)=−𝛼𝑥₁(𝑡),𝑡=(𝑛+𝑙−1)𝑇,Δ𝑥₂(𝑡)=−𝛽𝑥₂(𝑡)𝑥,𝑡=𝑛𝑇,₁0⁺,𝑥₂0⁺,𝑥₃0⁺=𝑥₁₀,𝑥₂₀𝑥₃₀. (54) Consider the corresponding impulsive compare system: 𝑑𝑢(𝑡)(𝑡)𝑎𝑑𝑡=𝑢₁₀−𝑎₁₁𝑢(𝑡)−𝑎₁₃𝑚₀()(𝑡),𝑡≠𝑛+𝑙−1𝑇,𝑑𝑣(𝑡𝑎𝑑𝑡=𝑣)₂₀−𝑎₂₂𝑣(𝑡)−𝑎₂₃𝑚₀,𝑡≠𝑛𝑇,𝑑𝑤(𝑡)𝑑𝑡=𝑤−𝑎₃₀−𝑎₃₃𝑚₀+𝑎₃₁𝑢(𝑡)+𝑎₃₂𝑣(𝑡),(𝑡)(𝑡)()(𝑡)(𝑡)𝑢0Δ𝑢=−𝛼𝑢,𝑡=𝑛+𝑙−1𝑇,Δ𝑣=−𝛽𝑣,𝑡=𝑛𝑇,₊,𝑣0⁺,𝑤0⁺=𝑥₁₀,𝑥₂₀,𝑥₃₀. (55) By Lemma 2, 𝑥₁(𝑡)≥𝑢(𝑡),𝑥₂(𝑡)>𝑣(𝑡), and 𝑥₃(𝑡)≥𝑤(𝑡). By Lemma 3, 𝑢^∗(𝑡)−𝜀₄≤𝑢(𝑡)≤𝑢^∗(𝑡)+𝜀₄,forsufficientlysmall𝜀₄,_𝑇0𝑢^∗(𝑡)1𝑑𝑡=𝑎₁₁ln(1−𝛼)+𝑎₁₀−𝑎₁₃𝑚₀𝑇. (56) Also, 𝑣^∗(𝑡)−𝜀₅≤𝑣(𝑡)≤𝑣^∗(𝑡)+𝜀₅,forsufficientlysmall𝜀₂,_𝑇0𝑣^∗(𝑡)1𝑑𝑡=𝑎₂₂ln1−𝛽+𝑎₂₀−𝑎₂₃𝑚₀𝑇. (57) Thus 𝑑𝑤≥𝑑𝑡−𝑎₃₀−𝑎₃₀𝑚₀+𝑎₃₁𝑢^∗(𝑡)−𝜀₄+𝑎₃₂𝑣^∗(𝑡)−𝜀₅𝑤(𝑡). (58) Integrating (58) over ((𝑛+𝑙−1)𝑇,(𝑛+𝑙)𝑇) yields 𝑤)𝑇)()𝑇((𝑛+𝑙≥𝑤𝑛+𝑙−1⁺×exp_()𝑇^{(𝑛+𝑙)𝑇}_{𝑛+𝑙−1}−𝑎₃₀−𝑎₃₃𝑚₀+𝑎₃₁𝑢^∗(𝑡)−𝜀₄+𝑎₃₂𝑣^∗(𝑡)−𝜀₅)𝑇)𝑑𝑡≥𝑤((𝑛+𝑙−1𝜎. (59) Therefore 𝑥₃)𝑇))𝑇))𝑇)𝜎)𝑇)𝜎((𝑛+𝑙+𝑘≥𝑤((𝑛+𝑙+𝑘≥𝑤((𝑛+𝑙+𝑘−1≥𝑤((𝑛+𝑙−1^𝑘. (60)
Since 𝜎>1, 𝜎^𝑘→∞ as 𝑘→∞. This implies that 𝑥₃(𝑡)→∞ as 𝑡→∞, which contradicts the boundedness of 𝑥₃(𝑡).
Step    2. If 𝑥₃(𝑡)≥𝑚₀ for all 𝑡≥𝑡₁, then the proof is complete. If not, let 𝑡^∗=inf_𝑡≥𝑡1{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=inf_𝑡>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 [𝑡_𝑛,𝑡_𝑛+1],(𝑛∈ℕ), such that 𝑥₃(𝑡)≥𝑚₀,𝑡∈[𝑡_𝑛,𝑡_𝑛+1]. Let 𝑇^′=sup|𝑡_𝑛+1−𝑡_𝑛|,𝑛∈ℕ. If 𝑇^′=∞, there must exist a subsequence 𝑡_𝑛𝑖 such that 𝑡_𝑛𝑖+1−𝑡_𝑛𝑖→∞ as 𝑛_𝑖→∞. From Step 1, this can lead to a contradiction with the boundedness of 𝑥₃(𝑡); therefore, 𝑇^′<∞. Then 𝑥₃(𝑡)≥𝑥₃𝑡_𝑛×exp^𝑡_𝑡𝑛−𝑎₃₀−𝑎₃₃𝑀+𝑎₃₁𝑢^∗(𝑡)−𝑎₃₁𝜀₄+𝑎₃₂𝑣^∗(𝑡)−𝑎₃₂𝜀₅𝑑𝑠≥𝑚₀exp−𝑎₃₀−𝑎₃₃𝑀𝑇^′≡𝑚₃. (61) Let 𝑚=min{𝑚₁,𝑚₂,𝑚₃}, and then liminf_𝑡→∞𝑥₁(𝑡)≥𝑚,liminf_𝑡→∞𝑥₂(𝑡)≥𝑚,liminf_𝑡→∞𝑥₃(𝑡)≥𝑚. (62) This completes the proof.

Next, we consider the following two subsystems: 𝑑𝑥₁𝑑𝑡=𝑥₁𝑎₁₀−𝑎₁₁𝑥₁−𝑎₁₃𝑥₃,𝑡≠(𝑛+𝑙−1)𝑇,𝑑𝑥₃𝑑𝑡=𝑥₃𝑎₃₀−𝑎₃₃𝑥₃+𝑎₃₁𝑥₁,Δ𝑥₁(𝑡)=−𝛼𝑥₁(𝑡)()𝑥,𝑡=𝑛+𝑙−1𝑇,₁0⁺,𝑥₃0⁺=𝑥₁₀,𝑥₃₀,𝑑𝑥₂𝑑𝑡=𝑥₂𝑎₂₀−𝑎₂₂𝑥₂−𝑎₂₃𝑥₃,𝑡≠𝑛𝑇,𝑑𝑥₃𝑑𝑡=𝑥₃𝑎₃₀−𝑎₃₃𝑥₃+𝑎₃₂𝑥₂,Δ𝑥₂(𝑡)=−𝛽𝑥₂(𝑡)𝑥,𝑡=𝑛𝑇,₂0⁺,𝑥₃0⁺=𝑥₂₀,𝑥₃₀. (63)(64)

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

Theorem 13. Subsystem (63) is permanent if −𝑎₃₀𝑎𝑇+³¹𝑎₁₁ln(1−𝛼)+𝑎₁₀𝑇>0 (65) holds.

Theorem 14. Subsystem (64) is permanent if −𝑎₃₀𝑎𝑇+³¹𝑎₂₂ln1−𝛽+𝑎₂₀𝑇>0 (66) 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).