An Impulsive Periodic Single-Species Logistic System with Diffusion

Abstract

We study a single-species periodic logistic type dispersal system in a patchy environment with impulses. On the basis of inequality estimation technique, sufficient conditions of integrable form for the permanence and extinction of the system are obtained. By constructing an appropriate Lyapunov function, conditions for the existence of a unique globally attractively positive periodic solution are also established. Numerical examples are shown to verify the validity of our results and to further discuss the model.

1. Introduction

In the practical world, on the one hand, owing to natural enemy, severe competition, or deterioration of the patch environment, species dispersal in two or more patches becomes one of the most prevalent phenomena of nature. Many empirical works and monographies on population dynamics in a spatial heterogeneous environment have been done (see [1–9] and the references cited therein). On the other hand, many natural and man-made factors (e.g., fire, drought, flooding deforestation, hunting, harvesting, breeding, etc.) always lead to rapid decrease or increase of population number at fixed moment. Such sudden changes can often be characterized mathematically in the form of impulses. With the development of the theory of impulsive differential equations [10], various population dynamical models of impulsive differential equations have been proposed and studied extensively. For example, many important and interesting results on the permanence, persistence, extinction, global stability, the existence of positive periodic solutions, bifurcation and dynamical complexity, and so forth can be found in [11–17] and the references cited therein.

Although considerable researches on the dispersal and impulses of species have been reported in the literature, there are few papers that investigate the dynamical behavior of population systems under the circumstances in which both dispersal and impulse exist. However, dispersal species which undergoes impulses is also one of the most prevalent phenomena of nature. In our previous paper [18], an impulsive periodic predator-prey system with diffusion is studied, and some conditions for the permanence, extinction, and existence of a unique globally stable periodic solution are established. In this paper, we will present and study a single-species periodic logistic system with impulses and dispersal in 𝑛 different patches. Our model takes the form 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…, (1) where 𝑟_𝑖(𝑡) and 𝑎_𝑖(𝑡)(𝑖∈𝐼={1,2,…,𝑛}) represent the intrinsic growth rates and the density-dependent coefficients in patch 𝑖, respectively. 𝐷_𝑖𝑗(𝑡)⩾0(𝑖,𝑗∈𝐼) denotes the dispersal rate of the species from patch 𝑗 to patch 𝑖. ℎ_𝑖𝑘𝑥_𝑖(𝑡_𝑘)(𝑖∈𝐼) is the regular pulse at time 𝑡_𝑘 of species 𝑥 in patch 𝑖. Throughout this paper, we always assume the following.(𝐶₁)𝑟_𝑖(𝑡),𝑎_𝑖(𝑡), and 𝐷_𝑖𝑗(𝑡)(𝑖,𝑗∈𝐼) are continuously periodic functions with common period 𝑇, defined on 𝑅₊=[0,∞) and 𝑎_𝑖(𝑡)⩾0,𝐷₂⩾𝐷_𝑖𝑗(𝑡)⩾𝐷₁>0(𝑖≠𝑗),𝐷_𝑖𝑖(𝑡)≡0 for all 𝑖,𝑗∈𝐼 and 𝑡∈𝑅₊.(𝐶₂)ℎ_𝑖𝑘>0 for all 𝑖∈𝐼,𝑘=1,2,… and there exists a positive integer 𝑞 such that 𝑡_𝑘+𝑞=𝑡_𝑘+𝑇 and ℎ_{𝑖(𝑘+𝑞)}=ℎ_𝑖𝑘 for any 𝑖∈𝐼.

The organization of this paper is as follows. In the next section, some sufficient conditions for the permanence and extinction of system (1) are obtained. In Section 3, conditions for the existence of a unique globally attractively positive periodic solution are also established. Finally, some numerical simulations are proposed to illustrate the feasibility of our results and discuss the model further.

2. Permanence and Extinction

In this section, applying inequality estimation technique, we get some sufficient conditions on the permanence and extinction of system (1).

Theorem 1. There exists a positive constant 𝑀 such that limsup_𝑡→∞𝑥_𝑖(𝑡)<𝑀 for all 𝑖∈𝐼 if ^𝑇₀𝑎(𝑡)d𝑡>0, (2) where 𝑎(𝑡)=min_𝑖∈𝐼{𝑎_𝑖(𝑡)}.

Proof. Let ℎ_𝑘=max_𝑖∈𝐼{ℎ_𝑖𝑘} for any 𝑘=1,2,…, then we have ℎ_𝑘+𝑞=ℎ_𝑘 and there exists a positive constant 𝐻 such that function ℎ𝑡,𝜇=_{𝑡⩽𝑡𝑘<𝑡+𝜇}lnℎ_𝑘⩽^𝑞_𝑘=1lnℎ_𝑘⩽𝐻 (3) for all 𝑡∈𝑅₊ and 𝜇∈[0,𝑇]. Choose 𝑟(𝑡)=max_𝑖∈𝐼{𝑟_𝑖(𝑡)}, and 𝑟(𝑡) is bounded for all 𝑡∈𝑅₊. Then from conditions (2) and (3), we have two positive constants 𝜏 and 𝛿 such that ^𝑇₀(𝑟(𝑡)−𝑎(𝑡)𝜏)d𝑡+^𝑞_𝑘=1lnℎ_𝑘<−𝛿. (4)
Define the function 𝑉(𝑡)=max_𝑖∈𝐼{𝑥_𝑖(𝑡)}. For any 𝑡∈𝑅₊, there is an 𝑖=𝑖(𝑡)∈𝐼 such that 𝑉(𝑡)=𝑥_𝑖(𝑡). Calculating the upper-right derivative of 𝑉(𝑡), we obtain 𝐷⁺𝑉(𝑡)⩽𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡(𝑡𝑟(𝑡)(𝑡)𝑉(𝑡.)⩽𝑉)(−𝑎)) (5) When 𝑡=𝑡_𝑘, we have 𝑉(𝑡⁺_𝑘)=max{𝑥_𝑖(𝑡⁺_𝑘)}=max{ℎ_𝑖𝑘𝑥_𝑖(𝑡_𝑘)}⩽max{ℎ_𝑖𝑘}max{𝑥_𝑖(𝑡_𝑘)}=ℎ_𝑘𝑉(𝑡_𝑘).
Consider the following auxiliary system: 𝐷⁺𝑤(𝑡)(𝑡𝑟(𝑡)(𝑡)𝑤(𝑡,𝑤𝑡=𝑤)(−𝑎))_+𝑘=ℎ_𝑘𝑤𝑡_𝑘,𝑘=1,2,… (6) with the initial condition 𝑉(0)⩽𝑤(0). If there is a constant 𝑀₀>0 such that limsup_𝑡→∞𝑤(𝑡)<𝑀₀ (7) for any positive solution 𝑤(𝑡) of system (6), then, according to the comparison theorem of impulsive differential equations [10], we have 𝑉(𝑡)⩽𝑤(𝑡) for all 𝑡⩾0. Therefore, choose 𝑀=𝑀₀, and we will finally have limsup_𝑡→∞𝑥_𝑖(𝑡)⩽limsup_𝑡→∞𝑉(𝑡)⩽limsup_𝑡→∞𝑤(𝑡)<𝑀 for all 𝑖∈𝐼.
Next, we will prove that (7) holds. In fact, for any positive solution 𝑤(𝑡) of system (6), we only need to consider the following three cases.
Case 1. There is a 𝑡₀⩾0 such that 𝑤(𝑡)⩾𝜏 for all 𝑡⩾𝑡₀.
Case 2. There is a 𝑡₀⩾0 such that 𝑤(𝑡)⩽𝜏 for all 𝑡⩾𝑡₀.
Case 3. 𝑤(𝑡) is oscillatory about 𝜏 for all 𝑡⩾0.
We first consider Case 1. Since 𝑤(𝑡)⩾𝜏 for all 𝑡⩾𝑡₀, then for 𝑡=𝑡₀+𝑙𝑇, where 𝑙 is any positive integer, integrating system (6) from 𝑡₀ to 𝑡, from (4) we have 𝑤(𝑡)=𝑤𝑡₀exp^𝑡_𝑡0(𝑟(𝑠)−𝑎(𝑠)𝑤(𝑠))d𝑠+_{𝑡0⩽𝑡𝑘<𝑡}lnℎ_𝑘⩽𝑤𝑡₀exp^𝑡0_𝑡^+𝑇₀(𝑟(𝑠)(𝑠)𝜏)+−𝑎d𝑠+⋯_{𝑡0𝑡+𝑙𝑇0+()𝑇𝑙−1}(𝑟(𝑠)−𝑎(𝑠)𝜏)d𝑠+𝑙^𝑞_𝑘=1lnℎ_𝑘⩽𝑤𝑡₀exp(−𝑙𝛿). (8) Hence, 𝑤(𝑡)→0 as 𝑙→∞, which leads to a contradiction.
Then, we consider Case 3. From the oscillation of 𝑤(𝑡) about 𝜏, we can choose two sequences {𝜌_𝑛} and {𝜌^∗_𝑛} satisfying 0<𝜌₁<𝜌^∗₁<⋯<𝜌_𝑛<𝜌^∗_𝑛<⋯ and lim_𝑛→∞𝜌_𝑛=lim_𝑛→∞𝜌^∗_𝑛=∞ such that 𝑤𝜌_𝑛⩽𝜏,𝑤𝜌⁺_𝑛𝑤𝜌⩾𝜏,_∗𝑛⩾𝜏,𝑤𝜌_𝑛^∗+𝑤(𝑡)𝜌⩽𝜏,⩾𝜏∀𝑡∈_𝑛,𝜌^∗_𝑛,𝑤(𝑡)⩽𝜏∀𝑡∈𝜌^∗_𝑛,𝜌_𝑛+1. (9) For any 𝑡⩾𝜌₁, if 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛] for some integer 𝑛, then we can choose integer 𝑙⩾0 and constant 0⩽𝜈<𝑇 such that 𝑡=𝜌_𝑛+𝑙𝑇+𝜈. Since 𝐷⁺𝑤(𝑡)⩽𝑤(𝑡)(𝑟(𝑡)−𝑎(𝑡)𝜏) for all 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛) and 𝑡≠𝑡_𝑘, integrating this inequality from 𝜌_𝑛 to 𝑡, by (3) and (4) we obtain 𝑤(𝑡)⩽𝑤𝜌_𝑛exp^𝑡_𝜌𝑛(𝑟(𝑠)−𝑎(𝑠)𝜏)d𝑠+_{𝜌𝑛⩽𝑡𝑘<𝑡}lnℎ_𝑘⩽𝜏exp𝑙^𝑇₀(𝑟(𝑡)−𝑎(𝑡)𝜏)d𝑡+^𝑞_𝑘=1lnℎ_𝑘+^𝜌𝑛_𝜌^+𝜈_𝑛(𝑟(𝑡)−𝑎(𝑡)𝜏)d𝑡+_{𝜌𝑛⩽𝑡𝑘<𝜌𝑛+𝜈}lnℎ_𝑘⩽𝜏exp(−𝑙𝛿+𝑟𝑇+𝐻)⩽𝜏exp(𝑟𝑇+𝐻), (10) where 𝑟=sup_{𝑡∈[0,𝑇]}{|𝑟(𝑡)|+𝑎(𝑡)𝜏}. If there is an integer 𝑛 such that 𝑡∈(𝜌^∗_𝑛,𝜌_𝑛+1], then we have 𝑤(𝑡)⩽𝜏⩽𝜏exp(𝑟𝑇+𝐻). Therefore, for Case 3 we always have 𝑤(𝑡)⩽𝜏exp(𝑟𝑇+𝐻) for all 𝑡⩾𝜌₁.
Lastly, if Case 2 holds, then we directly have 𝑤(𝑡)⩽𝜏exp(𝑟𝑇+𝐻) for all 𝑡⩾0.
Choose constant 𝑀₀=𝜏exp(𝑟𝑇+𝐻)+1, then we see that (7) holds. This completes the proof.

Remark 2. It can be seen from Theorem 1 that, in one time period 𝑇, if the density-dependent coefficient in patch 𝑖 (𝑖∈𝐼) is strictly greater than zero and the impulsive coefficient ℎ_𝑖𝑘 is bounded in the same time period, the dispersal species 𝑥 is always ultimately bounded.

Theorem 3. Assume that all conditions of Theorem 1 hold. In addition, there is an 𝑖₀∈𝐼 such that ^𝑇₀𝑟_𝑖0(𝑡)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_𝑖0𝑘>0. (11) Then system (1) is permanent.

Proof. The ultimate boundedness of system (1) has been proved in Theorem 1. In the following, we mainly prove the permanence of the system; that is, there is a constant 𝑚>0 such that liminf_𝑡→∞𝑥_𝑖(𝑡)>𝑚 (12) for each 𝑖∈𝐼 and any positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1).
From assumption ℎ_{𝑖(𝑘+𝑞)}=ℎ_𝑖𝑘, we obtain that there exists a constant 𝐻>0 such that function ℎ_𝑖𝑡,𝜇=_{𝑡⩽𝑡𝑘<𝑡+𝜇}lnℎ_𝑖𝑘⩽^𝑞_𝑘=1lnℎ_𝑖𝑘⩽𝐻 (13) for any 𝑖∈𝐼,𝑡∈𝑅₊ and 𝜇∈[0,𝑇].
For 𝑖=𝑖₀, by condition (11) and the boundedness of 𝑎_𝑖0(𝑡), there are two positive constants 𝜏 and 𝛿 such that ^𝑇₀𝑟_𝑖0(𝑡)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑡)−𝑎_𝑖0(𝑡)𝜏d𝑡+^𝑞_𝑘=1lnℎ_𝑖0𝑘>𝛿. (14)
Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) be any positive solution of system (1). Since 𝑥̇_𝑖0(𝑡)⩾𝑥_𝑖0(𝑡)𝑟_𝑖0(𝑡)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑡)−𝑎_𝑖0(𝑡)𝑥_𝑖0(𝑡),𝑡≠𝑡_𝑘, (15) by the comparison theorem of impulsive differential equations, we obtain 𝑥_𝑖0(𝑡)⩾𝑢(𝑡) for all 𝑡⩾0, where 𝑢(𝑡) is the positive solution of system 𝑢̇(𝑡)=𝑢(𝑡)𝑟_𝑖0(𝑡)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑡)−𝑎_𝑖0(𝑡)𝑢(𝑡),𝑢𝑡⁺_𝑘=ℎ_𝑖0𝑘𝑢𝑡_𝑘,𝑘=1,2,… (16) with initial condition 𝑢(0)=𝑥_𝑖0(0).
In the following, we first prove that there is a constant 𝑚>0 such that liminf_𝑡→∞𝑢(𝑡)>𝑚 (17) for any positive solution 𝑢(𝑡) of system (16). We only need to consider the following three cases.
Case 1. There is a 𝑡⩾0 such that 𝑢(𝑡)⩽𝜏 for all 𝑡⩾𝑡.
Case 2. There is a 𝑡⩾0 such that 𝑢(𝑡)⩾𝜏 for all 𝑡⩾𝑡.
Case 3. 𝑢(𝑡) is oscillatory about 𝜏 for all 𝑡⩾0.
For Case 1, let 𝑡=𝑡+𝑙𝑇, where 𝑙⩾0 is any integer. From (14), we obtain 𝑢(𝑡)=𝑢𝑡×exp^𝑡_𝑡𝑟_𝑖0(𝑠)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑠)−𝑎_𝑖0(𝑠)𝑢(𝑠)d𝑠+_{𝑡⩽𝑡𝑘<𝑡}lnℎ_𝑖0𝑘⩾𝑢𝑡exp^𝑡+𝑇_𝑡𝑟_𝑖0(𝑠)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑠)−𝑎_𝑖0(𝑠)𝜏d𝑠+⋯+^{𝑡+𝑙𝑇}_{()𝑇𝑡+𝑙−1}𝑟_𝑖0(𝑠)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑠)−𝑎_𝑖0(𝑠)𝜏d𝑠+𝑙^𝑞_𝑘=1lnℎ_𝑖0𝑘⩾𝑢𝑡exp𝑙𝛿. (18) Hence, 𝑢(𝑡)→∞ as 𝑙→∞, which leads to a contradiction.
For Case 3, we choose two sequences {𝜌_𝑛} and {𝜌^∗_𝑛} satisfying 0<𝜌₁<𝜌^∗₁<⋯<𝜌_𝑛<𝜌^∗_𝑛<⋯ and lim_𝑛→∞𝜌_𝑛=lim_𝑛→∞𝜌^∗_𝑛=∞ such that 𝑢𝜌_𝑛⩾𝜏,𝑢𝜌⁺_𝑛⩽𝜏,𝑢𝜌^∗_𝑛⩽𝜏,𝑢𝜌_𝑛^∗+⩾𝑢(𝑡)⩽𝜏,𝜏∀𝑡∈𝜌_𝑛,𝜌^∗_𝑛,𝑢(𝑡)⩾𝜏∀𝑡∈𝜌^∗_𝑛,𝜌_𝑛+1. (19) For any 𝑡⩾𝜌₁, if 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛] for some integer 𝑛, then we can choose integer 𝑙⩾0 and constant 0⩽𝜈<𝑇 such that 𝑡=𝜌_𝑛+𝑙𝑇+𝜈. Since, for all 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛) and 𝑡≠𝑡_𝑘, we have 𝑢̇(𝑡)⩾𝑢(𝑡)(𝑟_𝑖0(𝑡)−∑^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑡)−𝑎_𝑖0(𝑡)𝜏), integrating this inequality from 𝜌_𝑛 to 𝑡, then from (13) and (14) we obtain 𝑢(𝑡)⩾𝑢𝜌_𝑛×exp^𝑡_𝜌𝑛𝑟_𝑖0(𝑠)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑠)−𝑎_𝑖0(𝑠)𝜏+d𝑠_{𝜌𝑛⩽𝑡𝑘<𝑡}lnℎ_𝑖0𝑘⩾𝜏exp𝑙^𝑇₀𝑟_𝑖0(𝑡)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑡)−𝑎_𝑖0(𝑡)𝜏+d𝑡_𝑞𝑘=1lnℎ_𝑖0𝑘+^{𝜌𝑛+𝜈}_𝜌𝑛𝑟_𝑖0(𝑡)−^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑡)−𝑎_𝑖0(𝑡)𝜏d𝑡+_{𝜌𝑛⩽𝑡𝑘<𝜌𝑛+𝜈}lnℎ_𝑖0𝑘⩾𝜏exp𝑙𝛿−𝛽_𝑖0𝑇−𝐻⩾𝜏exp−𝛽_𝑖0𝑇−𝐻, (20) where 𝛽_𝑖0=sup_{𝑡∈[0,𝑇]}{|𝑟_𝑖0(𝑡)|+∑^𝑛_𝑗=1𝐷_𝑖0𝑗(𝑡)+𝑎_𝑖0(𝑡)𝜏}. If there is an integer 𝑛 such that 𝑡∈(𝜌^∗_𝑛,𝜌_𝑛+1], obviously we have 𝑢(𝑡)⩾𝜏⩾𝜏exp(−𝛽_𝑖0𝑇−𝐻). Therefore, for Case 3 we always have 𝑢(𝑡)⩾𝜏exp(−𝛽_𝑖0𝑇−𝐻) for all 𝑡⩾𝜌₁. Let constant 𝑚=𝜏exp(−𝛽_𝑖0𝑇−𝐻−1). Then 𝑚 is independent of any positive solution of system (16) and we finally have that (17) holds.
Lastly, if Case 2 holds, then from 𝑢(𝑡)⩾𝜏 for all 𝑡⩾0, we directly have that (17) holds.
From the fact that 𝑥_𝑖0(𝑡)⩾𝑢(𝑡) for all 𝑡⩾0, then we have liminf_𝑡→∞𝑥_𝑖0(𝑡)⩾liminf_𝑡→∞𝑢(𝑡)>𝑚. (21)
It follows immediately from (21) that there is a 𝑇₀>0 such that ∑^𝑛_𝑖=1𝑥_𝑖(𝑡)⩾𝑚 for all 𝑡⩾𝑇₀. Then for any 𝑖∈𝐼, when 𝑡⩾𝑇₀ and 𝑡≠𝑡_𝑘 we have 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)⩾𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)−𝐷₁+𝐷₁𝑚. (22) Obviously, there is a constant 𝛽>0 and 𝛽 is independent of any positive solution of system (1), such that for any 𝑖∈𝐼, 0⩽𝑥_𝑖(𝑡)⩽𝛽, and all 𝑡∈𝑅₊ the following inequality holds 𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)−𝐷₁+𝐷₁𝑚>𝛽. (23) Hence, for any 𝑖∈𝐼, by (22) we have 𝑥̇_𝑖(𝑡)>𝛽,𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘⩾ℎ_𝑖𝑥_𝑖𝑡_𝑘,𝑘=1,2,… (24) for all 0⩽𝑥_𝑖(𝑡)⩽𝛽 and 𝑡⩾𝑇₀, where ℎ_𝑖=min_{1⩽𝑘⩽𝑞}{ℎ_𝑖𝑘}>0.
In order to prove that (12) holds, we only need to consider the following three cases.
Case 1. There is a 𝑇₁⩾𝑇₀ such that 𝑥_𝑖(𝑡)⩾𝛽 for all 𝑡⩾𝑇₁.
Case 2. There is a 𝑇₁⩾𝑇₀ such that 𝑥_𝑖(𝑡)⩽𝛽 for all 𝑡⩾𝑇₁.
Case 3. 𝑥_𝑖(𝑡) is oscillatory about 𝛽 for all 𝑡⩾𝑇₀.
Equation (12) is obviously true if Case 1 holds.
For Case 2, there exists an impulsive time 𝑡_𝑞∗⩾𝑇₁ for some integer 𝑞^∗>0. For any 𝑡>𝑡_𝑞∗+1>𝑡_𝑞∗⩾𝑇₁, there is an integer 𝑝⩾𝑞^∗+1 such that 𝑡∈(𝑡_𝑝,𝑡_𝑝+1], and from system (24) we have 𝑥_𝑖(𝑡)>𝑥_𝑖𝑡⁺_𝑝+𝛽𝑡−𝑡_𝑝⩾ℎ_𝑖𝑥_𝑖𝑡_𝑝+𝛽𝑡−𝑡_𝑝⩾ℎ_𝑖𝑥_𝑖𝑡⁺_𝑝−1+𝛽𝑡_𝑝−𝑡_𝑝−1+𝛽𝑡−𝑡_𝑝>ℎ_𝑖𝛽𝜏₀, (25) where 𝜏₀=min_{1⩽𝑘⩽𝑞}{𝑡_𝑘−𝑡_𝑘−1}>0. Therefore, we obtain liminf_𝑡→∞𝑥_𝑖(𝑡)>ℎ_𝑖𝛽𝜏₀ (26) for any 𝑖∈𝐼.
Then, we consider Case 3. We choose two sequences {𝜌_𝑛} and {𝜌^∗_𝑛} satisfying 𝑇₀<𝜌₁<𝜌^∗₁<⋯<𝜌_𝑛<𝜌^∗_𝑛<⋯ and lim_𝑛→∞𝜌_𝑛=lim_𝑛→∞𝜌^∗_𝑛=∞ such that 𝑥_𝑖𝜌_𝑛⩾𝛽,𝑥_𝑖𝜌⁺_𝑛𝑥⩽𝛽,_𝑖𝜌^∗_𝑛⩽𝛽,𝑥_𝑖𝜌_𝑛^∗+𝑥⩾𝛽,_𝑖(𝑡)⩽𝛽∀𝑡∈𝜌_𝑛,𝜌^∗_𝑛,𝑥_𝑖(𝑡)⩾𝛽∀𝑡∈𝜌^∗_𝑛,𝜌_𝑛+1. (27)
For any 𝑡⩾𝜌₁, if 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛] for some integer 𝑛, we first of all show that 𝜌_𝑛 must be an impulsive time. Otherwise, there exists a positive constant 𝛿 such that there is no pulse in the interval (𝜌_𝑛,𝜌_𝑛+𝛿)⊂(𝜌_𝑛,𝜌^∗_𝑛]. Then for any 𝑡∈(𝜌_𝑛,𝜌_𝑛+𝛿), from system (24) we have 𝑥_𝑖(𝑡)>𝑥_𝑖(𝜌⁺_𝑛)+𝛽(𝑡−𝜌_𝑛)>𝑥_𝑖(𝜌_𝑛)⩾𝛽, which leads to a contradiction.
If there is only one impulsive time in the interval (𝜌_𝑛,𝜌^∗_𝑛](which must be 𝜌_𝑛), then from system (24) we get 𝑥_𝑖(𝑡)>𝑥_𝑖𝜌⁺_𝑛+𝛽𝑡−𝜌_𝑛⩾ℎ_𝑖𝑥_𝑖𝜌_𝑛+𝛽𝑡−𝜌_𝑛>ℎ_𝑖𝛽. (28)
If there are at least twice pulses in (𝜌_𝑛,𝜌^∗_𝑛], then we can denote (𝜌_𝑛,𝜌^∗_𝑛]=⋃^𝑞∗−1_𝑝=𝑝0(𝑡_𝑝,𝑡_𝑝+1]⋃(𝑡_𝑞∗,𝜌^∗_𝑛], where 𝑡_𝑝0=𝜌_𝑛, 𝑞^∗>𝑝₀, 𝑝₀,𝑝 and 𝑞^∗ are some positive integers. Hence, for any 𝑡∈(𝑡_𝑝,𝑡_𝑝+1], there is 𝑥_𝑖(𝑡)>𝑥_𝑖(𝑡⁺_𝑝)+𝛽(𝑡−𝑡_𝑝) from system (24). Moreover, if 𝑝=𝑝₀, we have 𝑥_𝑖(𝑡)>𝑥_𝑖𝑡⁺_𝑝+𝛽𝑡−𝑡_𝑝⩾ℎ_𝑖𝑥_𝑖𝜌_𝑛+𝛽𝑡−𝜌_𝑛>ℎ_𝑖𝛽. (29) If 𝑝>𝑝₀, then 𝑥_𝑖(𝑡)>𝑥_𝑖𝑡⁺_𝑝+𝛽𝑡−𝑡_𝑝⩾ℎ_𝑖𝑥_𝑖𝑡⁺_𝑝−1+𝛽𝑡_𝑝−𝑡_𝑝−1+𝛽𝑡−𝑡_𝑝>ℎ_𝑖𝛽𝜏₀. (30) However, when 𝑡∈(𝑡_𝑞∗,𝜌^∗_𝑛], then 𝑥_𝑖(𝑡)>𝑥_𝑖𝑡⁺_𝑞∗+𝛽𝑡−𝑡_𝑞∗⩾ℎ_𝑖𝑥_𝑖𝑡⁺_𝑞∗−1+𝛽𝑡_𝑞∗−𝑡_𝑞∗−1+𝛽𝑡−𝑡_𝑞∗>ℎ_𝑖𝛽𝜏₀. (31)
It follows from (28)–(31) that 𝑥_𝑖(𝑡)⩾minℎ_𝑖𝛽,ℎ_𝑖𝛽𝜏₀ (32) for all 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛] and any 𝑖∈𝐼.
If there is an integer 𝑛 such that 𝑡∈(𝜌^∗_𝑛,𝜌_𝑛+1], obviously we have 𝑥_𝑖(𝑡)⩾𝛽. Therefore, for Case 3 we always have 𝑥_𝑖(𝑡)⩾minℎ_𝑖𝛽,ℎ_𝑖𝛽𝜏₀,𝛽 (33) for all 𝑡⩾𝜌₁ and any 𝑖∈𝐼.
From (26) and (33), choose 𝑚=min_𝑖∈𝐼{ℎ_𝑖𝛽,ℎ_𝑖𝛽𝜏₀,𝛽}/2. Then we finally have that (12) holds. System (1) is permanent. The proof of Theorem 3 is completed.

Remark 4. It follows from Theorem 3 that system (1) is permanent if there is a positive average growth rate (which does not include the dispersal entrance) in one time period 𝑇 in any patch 𝑖₀(𝑖₀∈𝐼). In paper [4], the authors showed an interesting result that the dispersal species without impulses is permanent in all other patches if it is permanent in a patch. However, we extend this result to a periodic case with impulses.

Remark 5. In paper [18], we studied an impulsive periodic predator-prey system with Holling type III functional response and diffusion, and the paper mainly considers the influences of Holling type functional response and impulses. The conditions of the main result Theorem 3 require the minimum of the coefficients. However, in this paper, we consider a single-species logistic system. Although the predator is not involved in the model, which is simple than the model in [18], a more accurate and reasonable condition is established in the present paper; that is, in dispersal system, species with impulses is permanent in all patches if it is permanent in a patch, which improves the minimum conditions in the previous paper.

Theorem 6. System (1) is extinct if conditions ^𝑇₀𝑎(𝑡)d𝑡>0,^𝑇₀𝛾(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_𝑘⩽0 (34) hold, where 𝛾(𝑡)=max_𝑖∈𝐼{𝑟_𝑖(𝑡)−∑^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)+∑^𝑛_𝑗=1𝐷_𝑗𝑖(𝑡)} for all 𝑡∈[0,𝑇], and 𝑎(𝑡) and ℎ_𝑘 are defined in Theorem 1.

Proof. In fact, from (34), for any constant 𝜀>0, there is a positive constant 𝛿 such that ^𝑇₀𝛾(𝑡)−𝜀𝑎(𝑡)𝑛d𝑡+^𝑞_𝑘=1lnℎ_𝑘<−𝛿. (35) Define 𝑉(𝑡)=∑^𝑛_𝑖=1𝑥_𝑖(𝑡). When 𝑡≠𝑡_𝑘, calculating the right-upper derivative of 𝑉(𝑡), we have 𝐷⁺𝑉(𝑡)=^𝑛_𝑖=1𝑥̇_𝑖(𝑡)=^𝑛_𝑖=1𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑗𝑖(𝑡)−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)⩽𝛾(𝑡)𝑉(𝑡)−𝑎(𝑡)^𝑛_𝑖=1𝑥²_𝑖(𝑡)(𝑡)𝛾(𝑡)−𝑎(𝑡)⩽𝑉(𝑡).𝑛𝑉 (36) When 𝑡=𝑡_𝑘, we obtain 𝑉(𝑡⁺_𝑘)=∑^𝑛_𝑖=1𝑥_𝑖(𝑡⁺_𝑘)=∑^𝑛_𝑖=1ℎ_𝑖𝑘𝑥_𝑖(𝑡_𝑘)⩽max_𝑖∈𝐼{ℎ_𝑖𝑘}𝑉(𝑡_𝑘)=ℎ_𝑘𝑉(𝑡_𝑘). From this and (35), a similar argument as in the proof of (7), we can obtain 𝑉(𝑡)⩽𝜀exp(𝛾(𝑡)𝑇+𝐻) for 𝑡⩾0. Then from the arbitrariness of 𝜀, we obtain 𝑉(𝑡)→0 as 𝑡→∞. Finally, we have 𝑥_𝑖(𝑡)→0 as 𝑡→∞ for all 𝑖∈𝐼. This completes the proof of Theorem 6.

Remark 7. It can be seen from Theorem 6 that system (1) is always extinct if, in one time period 𝑇, there are a positive density-dependent coefficient and a nonpositive average growth rate (which includes the dispersal entrance) in the time period in each patch 𝑖(𝑖∈𝐼).

3. Periodic Solutions

In this section, by constructing an appropriate Lyapunov function, sufficient conditions for the existence of the unique globally attractively positive 𝑇-periodic solution of system (1) are established.

Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) and 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡)) be any two positive solutions of system (1). From Theorem 3, we can obtain that there are constants 𝐴>0 and 𝐵>0 such that 𝐴⩽𝑥_𝑖(𝑡),𝑥^∗_𝑖(𝑡)⩽𝐵∀𝑡⩾0,𝑖∈𝐼. (37)

Theorem 8. Suppose all the conditions of Theorem 3 hold. Moreover, if ^𝑇₀𝛽(𝑡)d𝑡>0, (38) where 𝛽(𝑡)=min_𝑖∈𝐼{𝑎_𝑖(𝑡)−∑^𝑛_𝑗=1𝐷_𝑗𝑖(𝑡)/𝐴}⩾0, 𝑡∈[0,𝑇], then system (1) has a unique globally attractively positive 𝑇-periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡)); that is, any positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1) satisfies lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)=0,𝑖∈𝐼. (39)

Proof. Choose Lyapunov function 𝑉(𝑡)=∑^𝑛_𝑖=1|ln𝑥_𝑖(𝑡)−ln𝑥^∗_𝑖(𝑡)|. Since for any impulsive time 𝑡_𝑘 we have 𝑉𝑡⁺_𝑘=^𝑛_𝑖=1ln𝑥_𝑖𝑡⁺_𝑘−ln𝑥^∗_𝑖𝑡⁺_𝑘=^𝑛_𝑖=1lnℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘−lnℎ_𝑖𝑘𝑥^∗_𝑖𝑡_𝑘=𝑉𝑡_𝑘, (40) then 𝑉(𝑡) is continuous for all 𝑡⩾0. On the other hand, from (37) we can obtain that for any 𝑡∈𝑅₊ and 𝑡≠𝑡_𝑘1𝐵𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)⩽ln𝑥_𝑖(𝑡)−ln𝑥^∗_𝑖(𝑡⩽1)𝐴𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡). (41) For any 𝑡∈𝑅₊ and 𝑡≠𝑡_𝑘, calculating the derivative of 𝑉(𝑡), we obtain 𝐷⁺𝑉(𝑡)=^𝑛_𝑖=1sgn𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡𝑥̇)^𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥̇^∗_𝑖(𝑡)𝑥^∗_𝑖(𝑡)⩽^𝑛_𝑖=1−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝐷_𝑖𝑗(𝑡), (42) where 𝐷_𝑖𝑗(𝑡)=𝐷_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)𝑥_𝑖(𝑡)−𝑥^∗_𝑗(𝑡)𝑥^∗_𝑖(𝑡),𝑥_𝑖(𝑡)>𝑥^∗_𝑖(𝑡),𝐷_𝑖𝑗(𝑡)𝑥^∗_𝑗(𝑡)𝑥^∗_𝑖(𝑡)−𝑥_𝑗(𝑡)𝑥_𝑖(𝑡),𝑥_𝑖(𝑡)<𝑥^∗_𝑖(𝑡). (43) For all 𝑡⩾0, we estimate 𝐷_𝑖𝑗(𝑡) under the following two cases.(i)If 𝑥_𝑖(𝑡)⩾𝑥^∗_𝑖(𝑡), then 𝐷_𝑖𝑗(𝑡)⩽𝐷_𝑖𝑗(𝑡)/𝑥_𝑖(𝑡)(𝑥_𝑗(𝑡)−𝑥^∗_𝑗(𝑡))⩽𝐷_𝑖𝑗(𝑡)/𝐴|𝑥_𝑗(𝑡)−𝑥^∗_𝑗(𝑡)|.(ii)If 𝑥_𝑖(𝑡)<𝑦_𝑖(𝑡), then 𝐷_𝑖𝑗(𝑡)⩽𝐷_𝑖𝑗(𝑡)/𝑥^∗_𝑖(𝑡)(𝑥_𝑗(𝑡)−𝑥^∗_𝑗(𝑡))⩽𝐷_𝑖𝑗(𝑡)/𝐴|𝑥_𝑗(𝑡)−𝑥^∗_𝑗(𝑡)|.
It follows from the estimation of 𝐷_𝑖𝑗(𝑡) and (41) that 𝐷⁺𝑉(𝑡)⩽^𝑛_𝑖=1−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡+_{𝑛𝑛𝑖=1𝑗=1})𝐷_𝑖𝑗(𝑡)𝐴𝑥_𝑗(𝑡)−𝑥^∗_𝑗(𝑡)⩽−^𝑛_𝑖=1𝑎_𝑖(𝑡)−^𝑛_𝑗=1𝐷_𝑗𝑖(𝑡)𝐴𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡(𝑡)(𝑡).)⩽−𝛽𝐴𝑉 (44) From this and condition (38), we have 𝑉(𝑡)⩽𝑉(0)exp(−𝐴∫^𝑡₀𝛽(𝑠)d𝑠)→0 as 𝑡→∞. Further more, from (41) we have that (39) holds.
Now let us consider the sequence (𝑥^∗₁(𝑚𝑇,𝑧₀),𝑥^∗₂(𝑚𝑇,𝑧₀),…,𝑥^∗_𝑛(𝑚𝑇,𝑧₀))=𝑧(𝑚𝑇,𝑧₀), where 𝑚=1,2,… and 𝑧₀=(𝑥^∗₁(0),𝑥^∗₂(0),…,𝑥^∗_𝑛(0)). It is compact in the domain [𝐴,𝐵]^𝑛 since 𝐴⩽𝑥^∗_𝑖(𝑡)⩽𝐵 for all 𝑡⩾0 and 𝑖=1,2,…,𝑛. Let 𝑧 be a limit point of this sequence, with 𝑧=lim_𝑛→∞𝑧(𝑚_𝑛𝑇,𝑧₀). Then 𝑧(𝑇,𝑧)=𝑧. Indeed, since 𝑧(𝑇,𝑧(𝑚_𝑛𝑇,𝑧₀))=𝑧(𝑚_𝑛𝑇,𝑧(𝑇,𝑧₀)) and 𝑧(𝑚_𝑛𝑇,𝑧(𝑇,𝑧₀))−𝑧(𝑚_𝑛𝑇,𝑧₀)→0 as 𝑚_𝑛→∞, we get ‖𝑧(𝑇,𝑧)−𝑧‖_[𝐴,𝐵]^𝑛⩽𝑧(𝑇,𝑧)−𝑧𝑇,𝑧𝑚_𝑛𝑇,𝑧_{0[𝐴,𝐵]𝑛}+𝑧𝑇,𝑧𝑚_𝑛𝑇,𝑧₀−𝑧𝑚_𝑛𝑇,𝑧_{0[𝐴,𝐵]𝑛}+𝑧(𝑚_𝑛𝑇,𝑧₀)−𝑧_[𝐴,𝐵]^𝑛⟶0,𝑛⟶∞. (45) The sequence 𝑧(𝑚𝑇,𝑧₀),𝑚=1,2,… has a unique limit point. On the contrary, let the sequence have two limit points 𝑧=lim_𝑛→∞𝑧(𝑚_𝑛𝑇,𝑧₀) and 𝑧=lim_𝑛→∞𝑧(𝑚_𝑛𝑇,𝑧₀). Then, taking into account (39) and 𝑧=𝑧(𝑚_𝑛𝑇,𝑧), we have ‖𝑧−𝑧‖_[𝐴,𝐵]^𝑛⩽𝑧−𝑧𝑚_𝑛𝑇,𝑧_{0[𝐴,𝐵]𝑛}+𝑧𝑚_𝑛𝑇,𝑧₀−𝑧_{[𝐴,𝐵]𝑛}⟶0,𝑛⟶∞, (46) and hence 𝑧=𝑧. The solution (𝑥^∗₁(𝑡,𝑧),𝑥^∗₂(𝑡,𝑧),…,𝑥^∗_𝑛(𝑡,𝑧)) is the unique periodic solution of system (1). By (39), it is globally attractive. This completes the proof of Theorem 8.

4. Numerical Simulation and Discussion

In this paper, we have investigated a class of single-species periodic logistic system with impulses and dispersal in 𝑛 different patches. By means of inequality estimation technique and Lyapunov function, we gave the criteria for the permanence, extinction, and existence of a unique globally stable positive periodic solution of system (1).

In order to testify the validity of our results and present a more in-depth problem for further discussion, we discuss the following two patches 𝑇-periodic dispersal system: 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)+𝐷₁₂(𝑡)𝑥₂(𝑡)−𝑥₁(𝑡),𝑥̇₂(𝑡)=𝑥₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)+𝐷₂₁(𝑡)𝑥₁(𝑡)−𝑥₂(𝑡),𝑡≠𝑡_𝑘,𝑥₁𝑡⁺_𝑘=ℎ_1𝑘𝑥₁𝑡_𝑘,𝑥₂𝑡⁺_𝑘=ℎ_2𝑘𝑥₂𝑡_𝑘,𝑘=1,2,…. (47)

We take 𝑟₁(𝑡)=5−|sin(𝜋/2)𝑡|, 𝑟₂(𝑡)=2.5+0.5cos𝜋𝑡, 𝑎₁(𝑡)=1.5, 𝑎₂(𝑡)=0.8, 𝐷₁₂(𝑡)=1.2, 𝐷₂₁(𝑡)=0.7, ℎ_1𝑘=1.2, ℎ_2𝑘=0.8, and 𝑡_𝑘=0.1𝑘, 𝑘=1,2,…. Obviously, 𝑟(𝑡)=max{𝑟₁(𝑡),𝑟₂(𝑡)}=5−|sin(𝜋/2)𝑡|, 𝑎(𝑡)=min{𝑎₁(𝑡),𝑎₂(𝑡)}=0.8, ℎ_𝑘=max{ℎ_1𝑘,ℎ_2𝑘}=1.2, and system (47) is periodic with period 𝑇=2. For 𝑞=20, we have 𝑡_𝑘+𝑞=𝑡_𝑘+𝑇, ℎ_1(𝑘+𝑞)=ℎ_1𝑘, and ℎ_2(𝑘+𝑞)=ℎ_2𝑘 for all 𝑘=1,2,…. It is easy to verify that ℎ_𝑘(𝑡,𝜇) and ℎ_𝑖𝑘(𝑡,𝜇)(𝑖=1,2) are bounded for all 𝑡∈𝑅₊ and 𝜇∈[0,𝑇]. Further more, since ^𝑇₀𝑎(𝑡)d𝑡=1.6>0,_𝑇0𝑟₁(𝑡)−𝐷₁₂(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_1𝑘=9.9732>0𝑖₀=1, (48) all the conditions of Theorem 3 are satisfied. Hence, system (47) is permanent. See Figures 1 and 2.

Figure 1 

The time series of the permanence of species 𝑥.

Figure 2 

The phase of the permanence of species 𝑥.

However, if the survival environment of the two patches is austere, the intrinsic growth rates will be negative. Hence, if we take 𝑟₁(𝑡)=−1−|sin(𝜋/2)𝑡|, 𝑟₂(𝑡)=−4+0.5cos𝜋𝑡 and all other parameters are retained, then we obtain 𝛾(𝑡)=max{𝑟₁(𝑡)−𝐷₁₂(𝑡)+𝐷₂₁(𝑡),𝑟₂(𝑡)−𝐷₂₁(𝑡)+𝐷₁₂(𝑡)}=−1.5−|sin(𝜋/2)𝑡| and ^𝑇₀𝛾(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_𝑘=−0.6268<0, (49) hence conditions of Theorem 6 are satisfied. From Theorem 6 we find that any positive solution of system (47) will be extinct. See Figure 3.

Figure 3 

The time series and phase of the extinction of species 𝑥.

From the illustrations of the theorems, we note that there is a great difference on the choice of the intrinsic growth rates 𝑟_𝑖(𝑡)(𝑖=1,2), which guarantee that the system is permanent or extinct. These differences make us want to know what results will be if all the parameters satisfy ^𝑇₀𝑟_𝑖(𝑡)−²_𝑗=1𝐷_𝑖𝑗(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_𝑖𝑘⩽0𝑖=1,2,_𝑇0𝛾(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_𝑘>0. (50) For this aim, we choose 𝑟₁(𝑡)=2.55−5|sin(𝜋/2)𝑡|, 𝑟₂(𝑡)=2+0.5cos𝜋𝑡 and all other parameters are retained; then ^𝑇₀𝑟₁(𝑡)−𝐷₁₂(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_1𝑘=−0.0198<0,_𝑇0𝑟₂(𝑡)−𝐷₂₁(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_2𝑘=−1.8629<0,_𝑇0𝛾(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_𝑘=11.3732>0, (51) and all the conditions of Theorems 3 and 6 are not satisfied. But from Figures 4 and 5 we find that the system is permanent.

Figure 4 

The time series of the permanence of species 𝑥.

Figure 5 

The phase of the permanence of species 𝑥.

Furthermore, if we choose 𝑟₁(𝑡)=−0.5−|sin(𝜋/2)𝑡|, 𝑟₂(𝑡)=−4+0.5cos𝜋𝑡 and keep all other parameters, then we have ^𝑇₀𝑟₁(𝑡)−𝐷₁₂(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_1𝑘=−1.0268<0,_𝑇0𝑟₂(𝑡)−𝐷₂₁(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_2𝑘=−12.4629<0,_𝑇0𝛾(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_𝑘=1.3732>0, (52) which do not satisfy conditions of any theorem. But, from Figure 6 we see that any positive solution of system (1) is extinct.

Figure 6 

The time series and phase of the extinction of species 𝑥.

Remark 9. Through the above analysis, we realize that there is a little flaw of the finding conditions of the theorems. A challenging problem is to find some sufficient and necessary conditions (if the conditions hold, then the system will be permanent, otherwise, it will be extinct) to guarantee the permanence and extinction of the system.

Throughout Figures 1–6, we always take the initial condition 𝑥(0)=(𝑥₁(0),𝑥₂(0))=(2,3).

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (60702071), Program for New Century Excellent Talents in University (NCET-06-0811), 973 National Basic Research Program of China (2010CB732501), Foundation of Sichuan Excellent Young Talents (09ZQ026-035), and Open Project of State Key Laboratory for Novel Software Technology of Nanjing University and Zhejiang Provincial Natural Science Foundation of China (Q13A010080).