Partial Extinction, Permanence, and Global Attractivity in Nonautonomous n-Species Gilpin-Ayala Competitive Systems with Impulses

Abstract

The qualitative properties of general nonautonomous n-species Gilpin-Ayala competitive systems with impulsive effects are studied. Some new criteria on the permanence, extinction, and global attractivity of partial species are established by using the methods of inequalities estimate and Liapunov functions.

1. Introduction

In [1], the general nonautonomous n-species Lotka-Volterra competitive systems with impulsive effects are investigated. By using the methods of inequalities estimate and constructing the suitable Liapunov functions, the sufficient conditions on the permanence of whole species and global attractivity of systems are established.

In [2], the authors studied the following general nonautonomous n-species Lotka-Volterra competitive systems with impulsive perturbations: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥_𝑗(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘=1,2,…,𝑖=1,2,…,𝑛, (1.1) and got a series of criteria on the extinction of a part of n-species, the permanence of other part of n-species, and the global attractivity of the systems.

In [3], a periodic n-species Gilpin-Ayala competition system with impulses is studied and obtain some useful behaviors of the system.

In this paper, we investigate the general nonautonomous n-species Gilpin-Ayala competitive systems with impulsive effects. 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥^𝜃𝑖𝑗_𝑗(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘=1,2,…,𝑖=1,2,…,𝑛, (1.2) where 𝑏_𝑖(𝑡) and 𝑎_𝑖𝑗(𝑡)(𝑖,𝑗=1,2,…,𝑛) are defined on 𝑅₊=[0,∞) and are bounded continuous functions, 𝑎_𝑖𝑗(𝑡)≥0 for all 𝑡∈𝑅₊, 𝜃_𝑖𝑗 and ℎ_𝑖𝑘>0 are constants for all 𝑘=1,2,… and 𝑖,𝑗=1,2,…,𝑛.

2. Preliminaries

Firstly, we introduce the following assumption.

Assumption H. There is a positive constant 𝜔 such that for each 𝑖=1,2,…,𝑛liminf_{𝑡→∞𝑡}^𝑡+𝜔𝑎_𝑖𝑖(𝑠)𝑑𝑠>0,liminf_{𝑡→∞𝑡}^𝑡+𝜔𝑏_𝑖(𝑠)𝑑𝑠+_{𝑡⩽𝑡𝑘⩽𝑡+𝜇}lnℎ_𝑖𝑘>0, (2.1) and functions ℎ_𝑖𝑡,𝜇=_{𝑡⩽𝑡𝑘⩽𝑡+𝜇}lnℎ_𝑖𝑘,𝑖=1,2,…,𝑛 (2.2) are bounded on 𝑡∈𝑅₊ and 0⩽𝜇⩽𝜔.

For each 𝑖∈{1,2,…,𝑛}, we consider the following logistic impulsive equation as the subsystem of system (1.2) 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑥^𝜃𝑖𝑖_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘=1,2,…. (2.3) From the above assumption, we have the following results.

Lemma 2.1. Suppose that assumption H holds. Then we have the following:(1)There exist positive constants 𝑚 and 𝑀 such that 𝑚⩽liminf_𝑡→∞𝑥_𝑖(𝑡)⩽limsup_𝑡→∞𝑥_𝑖(𝑡)⩽𝑀, (2.4) for any positive solution 𝑢_𝑖(𝑡) of (2.3).(2)lim_𝑡→∞(𝑥_𝑖⁽¹⁾(𝑡)−𝑥_𝑖⁽²⁾(𝑡))=0 for any two positive solutions 𝑥_𝑖⁽¹⁾(𝑡) and 𝑥_𝑖⁽²⁾(𝑡) of (2.3).

Proof. From assumption H, there are positive constants 𝑘₁, 𝑘₂, 𝛿 and 𝑇₀ such that for all 𝑡⩾𝑇₀ we have _𝑡^𝑡+𝜔𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑘₁𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑘<−𝛿,_{𝑡𝑡+𝜔}𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑘₂𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑘>𝛿. (2.5)(2.6) From the boundedness of function ℎ(𝑡,𝜇)=∑_{𝑡≤𝑡𝑘<𝑡+𝜇}lnℎ_𝑘, there is a positive constant 𝑃 such that for any 𝑡∈𝑅₊ and 𝜇∈[0,𝜔)ℎ𝑡,𝜇=_{𝑡≤𝑡𝑘<𝑡+𝜇}lnℎ_𝑘<𝑃. (2.7)
Firstly, we prove that there is a constant 𝑀>0 such that limsup_𝑡→∞𝑥_𝑖(𝑡)<𝑀, (2.8) for any positive solution 𝑥_𝑖(𝑡) of system (2.3). In fact, for any positive solution 𝑥_𝑖(𝑡) of system (2.3), we only need to consider the following three cases. Case I. There is a 𝑡₀≥𝑇₀ such that 𝑥(𝑡)≥𝑘^′₁=^𝜃𝑖𝑖𝑘₁ for all 𝑡≥𝑡₀. Case II. There is a 𝑡₀≥𝑇₀ such that 𝑥(𝑡)≤𝑘^′₁ for all 𝑡≥𝑡₀. Case III. 𝑥(𝑡) is oscillatory about 𝑘^′₁ for all 𝑡≥𝑇₀.
We first consider Case I. Since 𝑥_𝑖(𝑡)≥𝑘^′₁ for all 𝑡≥𝑡₀, then for 𝑡=𝑡₀+𝑙𝜔, where 𝑙≥0 is any positive integer, integrating system (2.3) from 𝑡₀ to 𝑡, from (2.5) we have 𝑥_𝑖(𝑡)=𝑥_𝑖𝑡₀exp^𝑡_𝑡0𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑥^𝜃𝑖𝑖_𝑖(𝑠)𝑑𝑠+_{𝑡0≤𝑡𝑘<𝑡}lnℎ_𝑘≤𝑥_𝑖𝑡₀exp^𝑡0_𝑡^+𝜔₀𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑘₁𝑑𝑠+_{𝑡0≤𝑡𝑘<𝑡}lnℎ_𝑘++⋯_{𝑡0𝑡+𝑙𝜔0+()𝜔𝑙−1}𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑘₁𝑑𝑠+_{𝑡0+(𝑙−1)𝜔≤𝑡𝑘<𝑡0+𝑙𝜔}lnℎ_𝑘≤𝑥_𝑖𝑡₀exp(−𝑙𝛿). (2.9) Hence, 𝑥_𝑖(𝑡)→0 as 𝑙→∞, which leads a contradiction.
Next, we consider Case III. From the oscillation of 𝑥_𝑖(𝑡) about 𝑘^′₁, we can 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≤𝑘^′₁,𝑥_𝑖𝜌⁺_𝑛≥𝑘^′₁,𝑥_𝑖𝜌^∗_𝑛≥𝑘^′₁,𝑥_𝑖𝜌^∗+_𝑛≤𝑘^′₁,𝑥_𝑖(𝑡)≥𝑘^′₁,∀𝑡∈𝜌_𝑛,𝜌^∗_𝑛,𝑥_𝑖(𝑡)≤𝑘^′₁,∀𝑡∈𝜌^∗_𝑛,𝜌_𝑛+1. (2.10) For any 𝑡≥𝑇₀, if 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛] for some integer 𝑛, then we can choose integer 𝑙≥0 and constant 0≤𝜇₁<𝜔 such that 𝑡=𝜌_𝑛+𝑙𝜔+𝜇₁. Since 𝑥̇_𝑖(𝑡)≤𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑘₁,∀𝑡∈𝜌_𝑛,𝜌^∗_𝑛,𝑡≠𝑡_𝑘, (2.11) integrating this inequality from 𝜌_𝑛 to 𝑡, by (2.5) and (2.7) we obtain 𝑥_𝑖(𝑡)=𝑥_𝑖𝜌_𝑛exp^𝑡_𝜌𝑛𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑥^𝜃𝑖𝑖(𝑠)𝑑𝑠+_{𝜌𝑛≤𝑡𝑘<𝑡}lnℎ_𝑘≤𝑘^′₁exp^𝜌𝑛_𝜌^+𝜔_𝑛𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑘₁𝑑𝑠+_{𝜌𝑛≤𝑡𝑘<𝜌𝑛+𝜔}lnℎ_𝑘++⋯_{𝜌𝑛+𝑙𝜔+𝜇1𝜌𝑛+𝑙𝜔}𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑘₁𝑑𝑠+_{𝜌𝑛+𝑙𝜔≤𝑡𝑘<𝜌𝑛+𝑙𝜔+𝜇1}lnℎ_𝑘≤𝑘^′₁exp−𝑙𝛿+^{𝜌𝑛+𝑙𝜔+𝜇1}_{𝜌𝑛+𝑙𝜔}𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑘₁𝑑𝑠+_{𝜌𝑛+𝑙𝜔≤𝑡𝑘<𝜌𝑛+𝑙𝜔+𝜇1}lnℎ_𝑘≤𝑘^′₁exp𝛼₁𝜔+𝑃, (2.12) where 𝛼₁=sup_𝑡∈𝑅+{|𝑏_𝑖(𝑡)|+𝑎_𝑖𝑖(𝑡)𝑘₁}. If there is an integer 𝑛 such that 𝑡∈(𝜌^∗_𝑛,𝜌_𝑛+1], then we obviously have 𝑥_𝑖(𝑡)≤𝑘^′₁<𝑘^′₁exp𝛼₁𝜔+𝑃. (2.13) Therefore, for Case III we always have 𝑥_𝑖(𝑡)≤𝑘^′₁exp𝛼₁𝜔+𝑃,∀𝑡≥𝑇₀. (2.14)
Lastly, if Case II holds, then we directly have 𝑥_𝑖(𝑡)≤𝑘^′₁exp𝛼₁𝜔+𝑃,∀𝑡≥𝑇₀. (2.15) Choose constant 𝑀=𝑘^′₁exp(𝛼₁𝜔+𝑃), then we see that (2.8) holds.
Secondly, a similar argument as in the proof of (2.8) we can prove that there is a constant 𝑚>0, such that liminf_𝑡→∞𝑥(𝑡)>𝑚, (2.16) for any positive solution 𝑥_𝑖(𝑡) of system (2.3). Conclusion (1.1) is proved.
Now, we prove conclusion (1.2). Let 𝑥_𝑖⁽¹⁾(𝑡) and 𝑥_𝑖⁽²⁾(𝑡) be any two positive solutions of system (2.3). From conclusion (1.1), it follows that there are positive constants 𝐴 and 𝐵 such that 𝐴⩽𝑥_𝑖⁽¹⁾(𝑡),𝑥_𝑖⁽²⁾(𝑡)⩽𝐵,∀𝑡≥0. (2.17) Choose Liapunov function as follows: 𝑉(𝑡)=ln𝑥_𝑖⁽¹⁾(𝑡)−ln𝑥_𝑖⁽²⁾(𝑡). (2.18) For any 𝑘=1,2,…, we have 𝑉𝑡⁺_𝑘=lnℎ_𝑘𝑥_𝑖⁽¹⁾𝑡_𝑘−lnℎ_𝑘𝑥_𝑖⁽²⁾𝑡_𝑘=𝑉𝑡_𝑘. (2.19) Hence, 𝑉(𝑡) is continuous for all 𝑡∈𝑅₊ and from the Mean-Value Theorem we can obtain 1𝐵𝑥_𝑖⁽¹⁾(𝑡)−𝑥_𝑖⁽²⁾(𝑡(𝑡)⩽1)⩽𝑉𝐴𝑥_𝑖⁽¹⁾(𝑡)−𝑥_𝑖⁽²⁾(𝑡). (2.20) Calculating the upper right derivative of 𝑉(𝑡), then from (2.20) we obtain 𝐷⁺𝑉=sign𝑥_𝑖⁽¹⁾(𝑡)−𝑥_𝑖⁽²⁾(𝑡)𝑥̇_𝑖⁽¹⁾(𝑡)𝑥_𝑖⁽¹⁾(𝑡)−𝑥̇_𝑖⁽²⁾(𝑡)𝑥_𝑖⁽²⁾(𝑡)=−𝑎_𝑖𝑖(𝑡)𝑥^{(1)𝜃𝑖𝑖}_𝑖(𝑡)−𝑥^{(2)𝜃𝑖𝑖}_𝑖(𝑡)≤−𝑎_𝑖𝑖(𝑡)𝜃_𝑖𝑖𝐴^𝜃_𝑖𝑖𝑥_𝑖⁽¹⁾(𝑡)−𝑥_𝑖⁽²⁾(𝑡)≤−𝑎_𝑖𝑖(𝑡)𝜃_𝑖𝑖𝐴^𝜃𝑖𝑖𝑉(𝑡),𝑡≠𝑡_𝑘,𝑘=1,2,…, (2.21) where [𝜃_𝑖𝑖]≤𝜃_𝑖𝑖 is the integer part of 𝜃_𝑖𝑖.
From this, we further have for any 𝑡>0𝑉(𝑡)⩽𝑉(0)exp−𝜃_𝑖𝑖𝐴^{𝜃𝑖𝑖𝑡}₀𝑎_𝑖𝑖(𝑠)𝑑𝑠. (2.22) From condition (2.5) we can obtain ∫^𝑡₀𝑎_𝑖𝑖(𝑡)𝑑𝑡→∞ as 𝑡→∞. Hence, 𝑉(𝑡)→0 as 𝑡→∞. Further from (2.20) we finally obtain lim_𝑡→∞(𝑥_𝑖⁽¹⁾(𝑡)−𝑥_𝑖⁽²⁾(𝑡))=0. Conclusion (1.2) is proved. This completes the proof of Lemma 2.1.

Applying Lemma 2.1 and the comparison theorem of impulsive differential equations, we easily prove the following result.

Lemma 2.2. Suppose that assumption H holds then there is a constant 𝐵>0 such that limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝐵,𝑖=1,2,…,𝑛, (2.23) for any positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.2).

3. Extinction

On the partial extinction of system (1.2), we have the following result.

Theorem 3.1. Suppose that assumption H holds. Let 𝑟 be a given integer and 1≤𝑟<𝑛. If for any 𝑙>𝑟 there is a 𝑖_𝑙<𝑙 such that for any 𝑗≤𝑙𝜃_𝑖𝑙𝑗=𝜃_𝑙𝑗,limsup_𝑡→∞∫_𝑡^𝑡+𝜔𝑏_𝑙(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑙𝑘∫_𝑡^𝑡+𝜔𝑏_𝑖𝑙(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑖𝑙𝑘<liminf_𝑡→∞𝑎_𝑙𝑗(𝑡)𝑎_𝑖𝑙𝑗(𝑡),∀𝑗≤𝑙, (3.1)(3.2) or liminf_𝑡→∞∫_𝑡^𝑡+𝜔𝑏_𝑖𝑙(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑖𝑙𝑘∫_𝑡^𝑡+𝜔𝑏_𝑙(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑙𝑘>limsup_𝑡→∞𝑎_𝑖𝑙𝑗(𝑡)𝑎_𝑙𝑗(𝑡),∀𝑗≤𝑙, (3.3) then species 𝑥_𝑖(𝑖=𝑟+1,𝑟+2,…,𝑛) are extinction, that is, for any positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.2), lim_𝑡→∞𝑥_𝑖(𝑡)=0,𝑖=𝑟+1,𝑟+2,…,𝑛. (3.4)

Proof. Firstly, from assumption H, that (2.7) still holds and there are constants 𝜂₀>0 and 𝑇₀>0 such that _𝑡^𝑡+𝜔𝑏_𝑖(𝑠)𝑑𝑠+_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑖𝑘≥𝜂₀, (3.5) for all 𝑡≥𝑇₀ and 𝑖=1,2,…,𝑛.
We first prove 𝑥_𝑛(𝑡)→0 as 𝑡→∞. Without loss of generality, we assume that condition (3.2) holds. When condition (3.3) holds, a similar argument can be given. Since limsup_𝑡→∞∫_𝑡^𝑡+𝜔𝑏_𝑛(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑛𝑘∫_𝑡^𝑡+𝜔𝑏_𝑝(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑝𝑘<liminf_𝑡→∞𝑎_𝑛𝑗(𝑡)𝑎_𝑝𝑗(𝑡),𝑗=1,2,…,𝑛, (3.6) where 𝑝=𝑖_𝑛. Hence, we can choose positive constants 𝛼, 𝛽, 𝜀 and 𝑇_𝑛≥𝑇₀ such that ∫_𝑡^𝑡+𝜔𝑏_𝑛(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑛𝑘∫_𝑡^𝑡+𝜔𝑏_𝑝(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑝𝑘<𝛼𝛽𝛼−𝜀<𝛽<𝑎_𝑛𝑗(𝑡)𝑎_𝑝𝑗(𝑡), (3.7) for all 𝑡≥𝑇_𝑛 and 𝑗=1,2,…,𝑛. Hence, from (3.5) we further obtain _𝑡^𝑡+𝜔−𝛼𝑏_𝑝(𝑠)+𝛽𝑏_𝑛(𝑠)𝑑𝑠+𝛽_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑛𝑘−𝛼_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑝𝑘<−𝛽𝜀_𝑡^𝑡+𝜔𝑏_𝑝(𝑠)𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑝𝑘≤−𝛽𝜀𝜂₀,𝛼𝑎_𝑝𝑗(𝑡)−𝛽𝑎_𝑛𝑗(𝑡)=𝛽𝑎_𝑝𝑗𝛼𝛽−𝑎_𝑛𝑗(𝑡)𝑎_𝑝𝑗(𝑡)<0, (3.8)(3.9) for all 𝑡≥𝑇_𝑛 and 𝑗=1,2,…,𝑛.
Consider the Liapunov function as follows: 𝑉_𝑛(𝑡)=𝑥_𝑝(𝑡)^−𝛼𝑥_𝑛(𝑡)^𝛽. (3.10) Calculating the derivative, and from (3.1), we can obtain for any 𝑡≥0𝑑𝑉_𝑛(𝑡)𝑑𝑡=𝑉_𝑛(𝑡)−𝛼𝑏_𝑝(𝑡)−^𝑛_𝑗=1𝑎_𝑝𝑗(𝑡)𝑥^𝜃𝑝𝑗_𝑗(𝑡)+𝛽𝑏_𝑛(𝑡)−^𝑛_𝑗=1𝑎_𝑛𝑗(𝑡)𝑥^𝜃𝑛𝑗_𝑗(𝑡)=𝑉_𝑛(𝑡)−𝛼𝑏_𝑝(𝑡)+𝛽𝑏_𝑛(𝑡)+^𝑛_𝑗=1𝛼𝑎_𝑝𝑗(𝑡)−𝛽𝑎_𝑛𝑗(𝑡)𝑥^𝜃𝑝𝑗_𝑗(𝑡), (3.11) for all 𝑡≠𝑡_𝑘 and 𝑉_𝑛𝑡⁺_𝑘=ℎ^−𝛼_𝑝𝑘ℎ^𝛽_𝑛𝑘𝑉_𝑛𝑡_𝑘, (3.12) for all 𝑘=1,2,…. From (3.9), we further have 𝑑𝑉_𝑛(𝑡)𝑑𝑡≤𝑉_𝑛(𝑡)−𝛼𝑏_𝑝(𝑡)+𝛽𝑏_𝑛(𝑡),𝑡≥𝑇_𝑛,𝑡≠𝑡_𝑘,𝑉_𝑛𝑡⁺_𝑘=ℎ^−𝛼_𝑝𝑘ℎ^𝛽_𝑛𝑘𝑉_𝑛𝑡_𝑘,𝑘=1,2,…. (3.13) For any 𝑡>𝑇_𝑛, there is an integer 𝑞_𝑡≥0 such that 𝑡∈[𝑇_𝑛+𝑞_𝑡𝜔,𝑇_𝑛+(𝑞_𝑡+1)𝜔]. Hence, by integrating (3.13) from 𝑇_𝑛 to 𝑡, we obtain 𝑉_𝑛(𝑡)≤𝑉_𝑛𝑇_𝑛exp^𝑡_𝑇𝑛−𝛼𝑏_𝑝(𝑠)+𝛽𝑏_𝑛(𝑠)𝑑𝑠+_{𝑇𝑛≤𝑡𝑘<𝑡}lnℎ^−𝛼_𝑝𝑘ℎ^𝛽_𝑛𝑘=𝑉_𝑛𝑇_𝑛exp^𝑇𝑛_𝑇^+𝜔_𝑛−𝛼𝑏_𝑝(𝑠)+𝛽𝑏_𝑛(𝑠)𝑑𝑠+_{𝑇𝑛≤𝑡𝑘<𝑇𝑛+𝜔}lnℎ^−𝛼_𝑝𝑘ℎ^𝛽_𝑛𝑘+⋯+^𝑡_{𝑇𝑛+𝑞𝑡𝜔}−𝛼𝑏_𝑝(𝑠)+𝛽𝑏_𝑛(𝑠)𝑑𝑠+_{𝑇𝑛+𝑞𝑡𝜔≤𝑡𝑘<𝑡}lnℎ^−𝛼_𝑝𝑘ℎ^𝛽_𝑛𝑘≤𝑀_𝑛exp−𝜀𝛽𝜂₀𝑞_𝑡, (3.14) where𝑀_𝑛=𝑉_𝑛𝑇_𝑛exp𝜔sup_𝑡≥0𝛼𝑏_𝑝(𝑡)+𝛽𝑏_𝑛(𝑡)+𝛼+𝛽𝑃. (3.15) Since 𝑞_𝑡→∞ as 𝑡→∞, it follows that from (3.14) 𝑉_𝑛(𝑡)⟶0as𝑡⟶∞. (3.16) Since 𝑥_𝑛(𝑡)^𝛽=𝑉_𝑛(𝑡)𝑥_𝑝(𝑡)^𝛼,ℎ_𝑛𝑘𝑥_𝑛𝑡_𝑘^𝛽=ℎ^𝛼_𝑝𝑘𝑥_𝑝𝑡_𝑘ℎ^−𝛼_𝑝𝑘ℎ^𝛽_𝑛𝑘𝑉_𝑛𝑡_𝑘, (3.17) by the boundedness of 𝑥(𝑡) on [0,∞) (see Lemma 2.2), we have 𝑥_𝑛(𝑡)⟶0as𝑡⟶∞. (3.18)
For any integer 𝑙>𝑟, assume that we have obtained 𝑥_𝑖(𝑡)→0 as 𝑡→∞ for all 𝑖>𝑙. Now, we prove that 𝑥_𝑙(𝑡)→0 as 𝑡→∞. Suppose that condition (3.3) holds. When condition (3.2) holds, the argument is similar. Let 𝑖=𝑖_𝑙, by (3.1), we have 𝜃_𝑖𝑙𝑗=𝜃_𝑙𝑗, then for 𝑗≤𝑙, we have 𝜃_𝑖𝑗=𝜃_𝑙𝑗. Then we can choose positive constants 𝜆, 𝜂, 𝛿 and 𝑇_𝑙≥𝑇₀ such that ∫_𝑡^𝑡+𝜔𝑏_𝑞(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑞𝑘∫_𝑡^𝑡+𝜔𝑏_𝑙(𝑠)𝑑𝑠+∑_{𝑡⩽𝑡𝑘<𝑡+𝜔}lnℎ_𝑙𝑘>𝜆𝜂𝜆+𝛿>𝜂>𝑎_𝑞𝑗(𝑡)𝑎_𝑙𝑗(𝑡), (3.19) for all 𝑡≥𝑇_𝑙, and 𝑗=1,2,…,𝑙, where 𝑞=𝑖_𝑙.
Consider the Liapunov function as follows: 𝑉_𝑙(𝑡)=𝑥_𝑞(𝑡)^−𝜂𝑥_𝑙(𝑡)^𝜆. (3.20) By calculating, we obtain for any 𝑡≥0𝑑𝑉_𝑙(𝑡)𝑑𝑡=𝑉_𝑙(𝑡)−𝜂𝑏_𝑞(𝑡)+𝜆𝑏_𝑙(𝑡)+^𝑙_𝑗=1𝜂𝑎_𝑞𝑗(𝑡)−𝜆𝑎_𝑙𝑗(𝑡)𝑥^𝜃𝑙𝑗_𝑗(𝑡)+^𝑛_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑡)𝑥^𝜃𝑞𝑗_𝑗(𝑡)−^𝑛_𝑗=𝑙+1𝜆𝑎_𝑙𝑗(𝑡)𝑥^𝜃𝑙𝑗_𝑗(𝑡), (3.21) for all 𝑡≠𝑡_𝑘 and 𝑉_𝑙𝑡⁺_𝑘=ℎ^−𝜂_𝑞𝑘ℎ^𝜆_𝑙𝑘𝑉_𝑙𝑡_𝑘, (3.22) for all 𝑘=1,2,…. From (3.3) and (3.19), we have _𝑡^𝑡+𝜔−𝜂𝑏_𝑞(𝑠)+𝜆𝑏_𝑙(𝑠)𝑑𝑠+𝜆_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑙𝑘−𝜂_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑞𝑘<−𝜂𝛿_𝑡^𝑡+𝜔𝑏_𝑙(𝑠)𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑙𝑘≤−𝛿𝜂𝜂₀,𝜂𝑎_𝑞𝑗(𝑡)−𝜆𝑎_𝑙𝑗(𝑡)<0, (3.23)(3.24) for all 𝑡≥𝑇_𝑙 and 𝑗=1,2,…,𝑙. Hence, from (3.21), it follows that 𝑑𝑉_𝑙(𝑡)𝑑𝑡≤𝑉_𝑙(𝑡)−𝜂𝑏_𝑞(𝑡)+𝜆𝑏_𝑙(𝑡)+^𝑙_𝑗=1𝜂𝑎_𝑞𝑗(𝑡)−𝜆𝑎_𝑙𝑗(𝑡)𝑥^𝜃𝑙𝑗_𝑗(𝑡)+^𝑛_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑡)𝑥^𝜃𝑞𝑗_𝑗(𝑡)−^𝑛_𝑗=𝑙+1𝜆𝑎_𝑙𝑗(𝑡)𝑥^𝜃𝑙𝑗_𝑗(𝑡),𝑡≥𝑇_𝑙,𝑡≠𝑡_𝑘,𝑉_𝑛𝑡⁺_𝑘=ℎ^−𝛼_𝑝𝑘ℎ^𝛽_𝑛𝑘𝑉_𝑛𝑡_𝑘,𝑘=1,2,…. (3.25)
Since 𝑥_𝑖(𝑡)→0 as 𝑡→∞ for all 𝑖>𝑙, by the boundedness of 𝑎_𝑖𝑗(𝑡)(𝑖,𝑗=1,2,…,𝑛) on [0,∞), we obtain lim_{𝑡→∞𝑡}^{𝑛𝑡+𝜔}_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑠)𝑥^𝜃𝑞𝑗_𝑗(𝑠)−𝜆𝑎_𝑙𝑗(𝑠)𝑥^𝜃𝑙𝑗_𝑗(𝑠)𝑑𝑠=0. (3.26) Hence, for any small 𝜀>0, there is a 𝑇^′_𝑙>0, such that _𝑡^{𝑛𝑡+𝜔}_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑠)𝑥^𝜃𝑞𝑗_𝑗(𝑠)−𝜆𝑎_𝑙𝑗(𝑠)𝑥^𝜃𝑙𝑗_𝑗(𝑠)𝑑𝑠<𝜀,𝑡>𝑇^′_𝑙. (3.27)
Combining (3.23), it follows that there is enough large 𝑇^∗_𝑙>max{𝑇_𝑙,𝑇^′_𝑙} such that for all 𝑡≥𝑇^∗_𝑙,_𝑡^𝑡+𝜔−𝜂𝑏_𝑞(𝑠)+𝜆𝑏_𝑙(𝑠)+^𝑛_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑠)𝑥^𝜃𝑞𝑗_𝑗(𝑠)−^𝑛_𝑗=𝑙+1𝜆𝑎_𝑙𝑗(𝑠)𝑥^𝜃𝑙𝑗_𝑗(𝑠)𝑑𝑠−𝜂_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑞𝑘+𝜆_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑙𝑘1≤−2𝛿𝜂𝜂₀,𝑥_𝑖(𝑡)≤𝛿∀𝑖>𝑙. (3.28) For any 𝑡>𝑇^∗_𝑙, we firstly choose an integer 𝑞_𝑡≥0 such that 𝑡∈(𝑇^∗_𝑙+𝑞_𝑡𝜔,𝑇^∗_𝑙+(𝑞_𝑡+1)𝜔]. Integrating (3.25) from 𝑇^∗_𝑙 to 𝑡, then from (3.3) and (3.28), we have 𝑉_𝑙(𝑡)≤𝑉_𝑙𝑇^∗_𝑙exp^𝑡_𝑇∗𝑙−𝜂𝑏_𝑞(𝑠)+𝜆𝑏_𝑙(𝑠)+^𝑛_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑠)𝑥^𝜃𝑞𝑗_𝑗(𝑠)−^𝑛_𝑗=𝑙+1𝜆𝑎_𝑙𝑗(𝑠)𝑥^𝜃𝑙𝑗_𝑗(𝑠)+𝑑𝑠_{𝑇∗𝑙≤𝑡𝑘<𝑡}lnℎ^−𝜂_𝑞𝑘ℎ^𝜆_𝑙𝑘=𝑉_𝑙𝑇^∗_𝑙exp^𝑇∗𝑙_𝑇^+𝜔_∗𝑙−𝜂𝑏_𝑞(𝑠)+𝜆𝑏_𝑙(𝑠)+^𝑛_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑠)𝑥^𝜃𝑞𝑗_𝑗(𝑠)−^𝑛_𝑗=𝑙+1𝜆𝑎_𝑙𝑗(𝑠)𝑥^𝜃𝑙𝑗_𝑗(𝑠)𝑑𝑠+_{𝑇∗𝑙≤𝑡𝑘<𝑇∗𝑙+𝜔}lnℎ^−𝜂_𝑞𝑘ℎ^𝜆_𝑙𝑘++⋯_{𝑇∗𝑙+𝑞𝑡𝜔𝑇∗𝑙+𝑞𝑡𝜔−1}−𝜂𝑏_𝑞(𝑠)+𝜆𝑏_𝑙(𝑠)+^𝑛_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑠)𝑥^𝜃𝑞𝑗_𝑗(𝑠)−^𝑛_𝑗=𝑙+1𝜆𝑎_𝑙𝑗(𝑠)𝑥^𝜃𝑙𝑗_𝑗(𝑠)𝑑𝑠+_{𝑇∗𝑙+𝑞𝑡−1𝜔≤𝑡𝑘<𝑇∗𝑙+𝑞𝑡𝜔}lnℎ^−𝜂_𝑞𝑘ℎ^𝜆_𝑙𝑘+^𝑡_{𝑇∗𝑙+𝑞𝑡𝜔}−𝜂𝑏_𝑞(𝑠)+𝜆𝑏_𝑙(𝑠)+^𝑛_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑠)𝑥^𝜃𝑞𝑗_𝑗(𝑠)−^𝑛_𝑗=𝑙+1𝜆𝑎_𝑙𝑗(𝑠)𝑥^𝜃𝑙𝑗_𝑗(𝑠)𝑑𝑠+_{𝑇∗𝑙+𝑞𝑡𝜔≤𝑡𝑘<𝑡}lnℎ^−𝜂_𝑞𝑘ℎ^𝜆_𝑙𝑘≤𝑀_𝑙−1exp2𝛿𝜂𝜂₀𝑞_𝑡, (3.29) where 𝑀_𝑙=𝑉_𝑙𝑇^∗_𝑙exp𝜔sup_𝑡≥0𝜂𝑏_𝑞(𝑡)+𝜆𝑏_𝑙(𝑡)+^𝑛_𝑗=𝑙+1𝜂𝑎_𝑞𝑗(𝑡)+𝜆𝑎_𝑙𝑗(𝑡)𝛿^𝜃𝑙𝑗+𝜆+𝜂𝑃, (3.30) Since 𝑞_𝑡→∞ as 𝑡→∞, we obtain from (3.29) 𝑉_𝑙(𝑡)⟶0as𝑡⟶∞, (3.31) Since 𝑥_𝑙(𝑡)^𝜂=𝑉_𝑙(𝑡)𝑥_𝑞(𝑡)^𝜆,ℎ_𝑙𝑘𝑥_𝑙𝑡_𝑘^𝜂=ℎ^𝜂_𝑙𝑘ℎ^−𝜆_𝑞𝑘𝑉_𝑙𝑡_𝑘ℎ_𝑞𝑘𝑥_𝑞𝑡_𝑘^𝜆, (3.32) by the boundedness of 𝑥(𝑡) on [0,∞), it follows that 𝑥_𝑙(𝑡)⟶0as𝑡⟶∞. (3.33) Finally, by the induction principle, we obtain that 𝑥_𝑖(𝑡)→0 as 𝑡→∞ for all 𝑖>𝑟. This completes the proof of Theorem 3.1.

4. Permanence

In this section, we study the permanence of partial species 𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑟) of system (1.2). We state and prove the following result.

Theorem 4.1. Suppose that all the conditions of Theorem 3.1 hold. If for each 𝑖=1,2,…,𝑟liminf_{𝑡→∞𝑡}^𝑡+𝜔𝑏_𝑖(𝑠)−^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑢^𝜃𝑖𝑗_𝑗0(𝑡)𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑖𝑘>0, (4.1) where 𝑢_𝑖0 is some fixed positive solution of (2.3), then species 𝑥_𝑖(𝑖=1,2,…,𝑟) are permanent, that is, there are positive constants 𝑚 and 𝑀 such that for any positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.2) 𝑚≤liminf_𝑡→∞𝑥_𝑖(𝑡)≤limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀,𝑖=1,2,…,𝑟. (4.2)

Proof. From (4.1) and the boundedness of functions 𝑎_𝑖𝑗(𝑡)(𝑖,𝑗=1,2,…,𝑛) on 𝑅₊, there are constants 𝜀₀>0 and 𝑇₁>0 such that for any 𝑡≥𝑇₁ and 𝑖=1,2,…,𝑟. _𝑡^𝑡+𝜔𝑏_𝑖(𝑠)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝜀₀−^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑢^𝜃𝑖𝑗_𝑗0(𝑠)𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑖𝑘>𝜀₀. (4.3) For any 𝑖≤𝑟, from system (1.2), we have 𝑑𝑥_𝑖(𝑡)𝑑𝑡≤𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝜃𝑖𝑗_𝑗(𝑡),≤𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖𝑖(𝑡)𝑥^𝜃𝑖𝑖_𝑖(𝑡),𝑡≠𝑡_𝑘𝑥,𝑡≥0,_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘=1,2,…, (4.4)
we have 𝑥_𝑖(𝑡)≤𝑢_𝑖(𝑡)∀𝑡≥0, (4.5) where 𝑢_𝑖(𝑡) is the solution of (2.3) with initial condition 𝑢_𝑖(0)≥𝑥_𝑖(0). From Lemma 2.1 and Theorem 3.1, for the above constant 𝜀₀ there is a 𝑇₂≥𝑇₁ such that for all 𝑡≥𝑇₂𝑥_𝑖(𝑡)≤𝑢_𝑖(𝑡)≤𝑢_𝑖0(𝑡)+𝜀₀𝑥,𝑖=1,2,…,𝑟,_𝑖(𝑡)<𝜀₀,𝑖=𝑟+1,𝑟+2,⋯,𝑛. (4.6)(4.7) Let 𝛾_𝑖=sup_𝑡≥0𝑏_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝜀₀+^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑡)𝑢^𝜃𝑖𝑗_𝑗0(𝑡),𝑚=min_{1≤𝑖≤𝑟}𝜀₀exp−𝛾_𝑖𝜔−𝑃, (4.8) where constant 𝑃>0 is given in (2.7). Obviously, 𝑚>0 and 𝑚 is independent of any positive solution of system (1.2).
Now, we prove that there is a 𝑇₃≥𝑇₂ such that 𝑥_𝑖(𝑡)≥𝑚∀𝑡≥𝑇₃,𝑖=1,2,…,𝑟. (4.9) We only need to consider the following three cases for each 𝑖=1,2,…,𝑟. Case I. There is a 𝑡₁≥𝑇₂ such that 𝑥_𝑖(𝑡)≤𝜀^′₀=√^𝜃𝑖𝑖𝜀₀ for all 𝑡≥𝑡₁. Case II. There is a 𝑡₂≥𝑇₂ such that 𝑥_𝑖(𝑡)≥𝜀^′₀ for all 𝑡≥𝑡₂. Case III. 𝑥_𝑖(𝑡) oscillates about 𝜀^′₀ for all 𝑡≥𝑇₂.
For Case I, let 𝑡=𝑡₁+𝑙𝜔, where 𝑙≥0 is any integer. From (4.3)–(4.7) we obtain 𝑥_𝑖(𝑡)=𝑥_𝑖𝑡₁exp^𝑡_𝑡1𝑏_𝑖(𝑠)−𝑎_𝑖𝑖(𝑠)𝑥^𝜃𝑖𝑖_𝑖(𝑠)−^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑥^𝜃𝑖𝑗_𝑗(𝑠)𝑑𝑠+_{𝑡1≤𝑡𝑘<𝑡}lnℎ_𝑖𝑘≥𝑥_𝑖𝑡₁exp^𝑡1_𝑡^+𝜔₁𝑏_𝑖(𝑠)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝜀₀−^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑢^𝜃𝑖𝑗_𝑗0(𝑠+)𝑑𝑠_{𝑡1≤𝑡𝑘<𝑡1+𝜔}lnℎ_𝑖𝑘+⋯+^𝑡1_𝑡^+𝑙𝜔_{1+()𝜔𝑙−1}𝑏_𝑖(𝑠)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝜀₀−^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑢^𝜃𝑖𝑗_𝑗0(𝑠)+𝑑𝑠_{𝑡1+(𝑙−1)𝜔≤𝑡𝑘<𝑡1+𝑙𝜔}lnℎ_𝑖𝑘≥𝑥_𝑖𝑡₁exp𝑙𝜀₀. (4.10) Therefore, 𝑥_𝑖(𝑡)→∞ as 𝑙→∞ which leads to a contradiction.
For Case III, we choose two sequences {𝜌_𝑛} and {𝜌^∗_𝑛} satisfying 𝑇₂≤𝜌₁<𝜌^∗₁<⋯<𝜌_𝑛<𝜌^∗_𝑛<⋯ and lim_𝑛→∞𝜌_𝑛=lim_𝑛→∞𝜌^∗_𝑛=∞ such that 𝑥_𝑖𝜌_𝑛≥𝜀^′₀,𝑥_𝑖𝜌⁺_𝑛≤𝜀^′₀,𝑥_𝑖𝜌^∗_𝑛≤𝜀^′₀,𝑥_𝑖𝜌^∗+_𝑛≥𝜀^′₀,𝑥_𝑖(𝑡)≤𝜀^′₀∀𝑡∈𝜌_𝑛,𝜌^∗_𝑛,𝑥_𝑖(𝑡)≥𝜀^′₀∀𝑡∈𝜌^∗_𝑛,𝜌_𝑛+1. (4.11) For any 𝑡≥𝑇₂, if 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛] for some integer 𝑛, then we can choose an integer 𝑙≥0 such that 𝑡=𝜌_𝑛+𝑙𝜔+𝜈_𝑖, where 𝜈_𝑖∈[0,𝜔) is a constant. Since for any 𝑡∈(𝜌_𝑛,𝜌^∗_𝑛) from (4.6) and (4.7) we have 𝑥̇_𝑖(𝑡)≥𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝜀₀−^𝑟_𝑗≠𝑖𝑏_𝑖𝑗(𝑡)𝑢^𝜃𝑖𝑗_𝑗0(𝑡),𝑡≠𝑡_𝑘. (4.12) Integrating this inequality from 𝜌_𝑛 to 𝑡, then from (4.7) and (3.28)-(3.29) we have 𝑥_𝑖(𝑡)≥𝑥𝜌_𝑛exp^𝑡_𝜌𝑛𝑏_𝑖(𝑠)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝜀₀−^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑢^𝜃𝑖𝑗_𝑗0(𝑠)𝑑𝑠+_{𝜌𝑛≤𝑡𝑘<𝑡}lnℎ_𝑖𝑘≥𝜀₀exp^𝜌𝑛_𝜌^+𝜔_𝑛𝑏_𝑖(𝑠)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝜀₀−^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑢^𝜃𝑖𝑗_𝑗0(𝑠)𝑑𝑠+_{𝜌𝑛≤𝑡𝑘<𝜌𝑛+𝜔}lnℎ_𝑖𝑘+⋯+^𝜌𝑛_𝜌^+𝑙𝜔_{𝑛+()𝜔𝑙−1}𝑏_𝑖(𝑠)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝜀₀−^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑢^𝜃𝑖𝑗_𝑗0(𝑠)𝑑𝑠+_{𝜌𝑛+()𝑙−1𝜔≤𝑡𝑘<𝜌𝑛+𝑙𝜔}lnℎ_𝑖𝑘+^{𝜌𝑛+𝑙𝜔+𝜈𝑖}_{𝜌𝑛+𝑙𝜔}𝑏_𝑖(𝑠)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝜀₀−^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑢^𝜃𝑖𝑗_𝑗0(𝑠)𝑑𝑠+_{𝜌𝑛+𝑙𝜔≤𝑡𝑘<𝜌𝑛+𝑙𝜔+𝜈𝑖}lnℎ_𝑖𝑘≥𝜀₀exp−𝛾_𝑖𝜔−𝑃. (4.13) If there exists an integer 𝑛 such that 𝑡∈(𝜌^∗_𝑛,𝜌_𝑛+1], then we obviously have 𝑥_𝑖(𝑡)≥𝜀₀>𝜀₀exp−𝛾_𝑖𝜔−𝑃. (4.14) This shows that for Case III we always have 𝑥_𝑖(𝑡)≥𝜀₀exp−𝛾_𝑖𝜔−𝑃,∀𝑡≥𝑇₂. (4.15)
Finally, if Case II holds, then from 𝑥_𝑖(𝑡)≥𝜀^′₀ for all 𝑡≥𝑡₁, we can directly obtain that (4.9) holds.
Therefore, from Lemma 2.2 and (4.9), it follows that species 𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑟) are permanent. This proof of Theorem 4.1 is completed.

5. Global Attractivity

In this section, we further discuss the global attractivity of species 𝑥_𝑖(𝑡)(𝑖≤𝑟). In order to obtain our results, we first consider the following subsystem which is composed of the species 𝑥_𝑖(𝑡)(𝑖≤𝑟) of system (1.2) and for convenience of statement we use the variable 𝑢_𝑖(𝑡)(𝑖≤𝑟) to denote the species of this subsystem, 𝑑𝑢_𝑖(𝑡)𝑑𝑡=𝑢_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑟_𝑗=1𝑎_𝑖𝑗(𝑡)𝑢^𝜃𝑖𝑗_𝑗(𝑡),𝑡≠𝑡_𝑘,𝑢_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑢_𝑖𝑡_𝑘,𝑖=1,2,…,𝑟,𝑘=1,2,…. (5.1)

We need the following lemma.

Lemma 5.1. Suppose that assumption H and condition (4.1) of Theorem 4.1 hold. Then subsystem (5.1) is permanent.
Lemma 5.1 can be proved by using the same method given in the proof of Theorem 4.1. We now state and prove the main result of this section.

Theorem 5.2. Suppose that all conditions of Theorem 3.1 and Theorem 4.1 hold. If there are positive constants 𝜌, 𝐷 and 𝑑_𝑖(𝑖=1,2,…,𝑟) and nonnegative integrable function 𝜇(𝑡) defined on 𝑅₊, satisfying ∫^𝑡_𝑠𝜇(𝜏)𝑑𝜏≥−𝐷+𝜌(𝑡−𝑠) for all 𝑡≥𝑠≥0, such that 𝑑_𝑖𝑎_𝑖𝑖(𝑡)−^𝑟_𝑗≠𝑖𝑑_𝑗𝑎_𝑗𝑖(𝑡)≥𝜇(𝑡),𝑖=1,2,…,𝑟, (5.2) for all 𝑡≥0, then for any positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.2) and any positive solution 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑟(𝑡)) of subsystem (5.1) lim_𝑡→∞𝑥_𝑖(𝑡)−𝑢_𝑖(𝑡)=0,𝑖=1,2,…,𝑟. (5.3)

Proof. Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) be a positive solution of system (1.2) and 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑟(𝑡)) be a positive solution of subsystem (5.1). By Theorem 3.1, we have 𝑥_𝑖(𝑡)→0 as 𝑡→∞ for all 𝑖>𝑟. From Theorem 4.1 and Lemma 5.1, there are positive constants 𝑚 and 𝑀 such that 𝑚≤𝑥_𝑖(𝑡),𝑢_𝑖(𝑡)≤𝑀,𝑖=1,2,…,𝑟, (5.4) for all 𝑡≥0. Choose the Liapunov function as follows: 𝑉_𝑟(𝑡)=^𝑟_𝑖=1𝑑_𝑖ln𝑥_𝑖(𝑡)−ln𝑢_𝑖(𝑡). (5.5) Since 𝑉_𝑟𝑡⁺_𝑘=^𝑟_𝑖=1𝑑_𝑖ln𝑥_𝑖𝑡⁺_𝑘−ln𝑢_𝑖𝑡⁺_𝑘=^𝑟_𝑖=1𝑑_𝑖lnℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘−lnℎ_𝑖𝑘𝑢_𝑖𝑡_𝑘=𝑉_𝑟𝑡_𝑘, (5.6) then 𝑉(𝑡) is continuous for all 𝑡≥0. Calculating the upper right derivative of 𝑉_𝑟(𝑡), we have 𝐷⁺𝑉_𝑟(𝑡)≤^𝑟_𝑖=1𝑑_𝑖−𝑎_𝑖𝑖𝑥^𝜃𝑖𝑖_𝑖(𝑡)−𝑢^𝜃𝑖𝑖_𝑖(𝑡)+^𝑟_𝑗≠𝑖𝑎_𝑖𝑗(𝑡)𝑥^𝜃𝑖𝑗_𝑗(𝑡)−𝑢^𝜃𝑖𝑗_𝑗(𝑡)+𝑔(𝑡)=−^𝑟_𝑖=1𝑑_𝑖𝑎_𝑖𝑖−^𝑟_𝑗≠𝑖𝑑_𝑗𝑎_𝑗𝑖(𝑡)𝑥^𝜃𝑗𝑖_𝑖(𝑡)−𝑢^𝜃𝑗𝑖_𝑖(𝑡)+𝑔(𝑡), (5.7) for all 𝑡≥0, where 𝑔(𝑡)=^𝑟_𝑖=1𝑑_𝑖^𝑛_𝑗=𝑟+1𝑎_𝑗𝑖(𝑡)𝑥^𝜃𝑗𝑖_𝑗(𝑡). (5.8) By (5.2), we have 𝐷⁺𝑉_𝑟(𝑡)≤−𝜇(𝑡)^𝑟_𝑖=1𝑥^𝜃𝑗𝑖_𝑖(𝑡)−𝑢^𝜃𝑗𝑖_𝑖(𝑡)+𝑔(𝑡),∀𝑡≥0. (5.9) By (5.4), we further obtain 𝐷⁺𝑉_𝑟(𝑡)≤−𝜆𝜇(𝑡)𝑉_𝑟(𝑡)+𝑔(𝑡),∀𝑡≥0, (5.10) where 𝜆=min_{1≤𝑖≤𝑟}𝑑_𝑖⁻¹𝑚>0. Applying the comparison theorem and the variation of constants formula of first-order linear differential equation, we have 𝑉_𝑟(𝑡)≤𝑒^{−∫𝑡0𝜆𝜇(𝑠)𝑑𝑠𝑡}₀𝑔(𝑠)𝑒^{∫𝑠0𝜆𝜇(𝜏)𝑑𝜏}𝑑𝑠+𝑉_𝑟(0), (5.11) for all 𝑡≥0. Since 𝑔(𝑡)→0 as 𝑡→∞, from the properties of function 𝜇(𝑡) and (5.11), it is not hard to obtain 𝑉_𝑟(𝑡)→0 as 𝑡→∞. That shows lim_𝑡→∞𝑥_𝑖(𝑡)−𝑢_𝑖(𝑡)=0,𝑖=1,2,…,𝑟. (5.12) This completes the proof of Theorem 5.2.