On a Periodic Predator-Prey System with Holling III Functional Response and Stage Structure for Prey

Abstract

We propose and study the permanence of the following periodic Holling III predator-prey system with stage structure for prey and both two predators which consume immature prey. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

1. Introduction

The aim of this paper is to investigate the permanence of the following periodic stage-structure predator-prey system with Holling III functional response:

𝑥̇₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑝(𝑡)−¹(𝑡)𝑥²₁(𝑡)𝑘₁(𝑡)+𝑥²₁𝑦(𝑡)¹−𝑝(𝑡)₂(𝑡)𝑥₁(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥₁(𝑡)+𝑛(𝑡)𝑦₂𝑦(𝑡)²𝑥̇(𝑡),₂(𝑡)=𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂𝑝(𝑡)−³(𝑡)𝑥²₂(𝑡)𝑘₃(𝑡)+𝑥²₂𝑦(𝑡)²𝑦̇(𝑡),₁(𝑡)=𝑦₁(𝑡)(−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥²₁(𝑡)𝑘₁(𝑡)+𝑥²₁(𝑡)−𝑞₁(𝑡)𝑦₁𝑦̇(𝑡)),₂(𝑡)=𝑦₂(𝑡)(−𝑔₂ℎ(𝑡)+²(𝑡)𝑥₁(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥₁(𝑡)+𝑛(𝑡)𝑦₂+ℎ(𝑡)³(𝑡)𝑥²₂(𝑡)𝑘₃(𝑡)+𝑥²₂(𝑡)−𝑞₂(𝑡)𝑦₂(𝑡)), (1.1) where 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑓(𝑡), 𝑝_𝑖(𝑡), 𝑔_𝑖(𝑡), 𝑞_𝑖(𝑡), 𝑖=1,2 and ℎ_𝑖(𝑡), 𝑘_𝑖(𝑡), 𝑖=1,2,3, are all continuous positive 𝜔-periodic functions. Here 𝑥₁(𝑡) and 𝑥₂(𝑡) denote the density of immature and mature prey species, respectively, and 𝑦_𝑖 is the density of the predators.

The periodic functions in (1.1) have the following biological meanings. The birth rate into the immature population is given by 𝑎(𝑡)𝑥₂; that is, it is assumed to be proportional to the existing mature population, with a proportionality coefficient 𝑎(𝑡). The death rate of the immature population is proportional to the existing immature population and to its square with coefficients 𝑏(𝑡) and 𝑑(𝑡), respectively. The death rate of the mature population is of a logistic nature, with proportionality coefficient 𝑓(𝑡). The transition rate from the immature individuals to the mature individuals is assumed to be proportional to the existing immature population, with proportionality coefficient 𝑐(𝑡). Similarly, −𝑔_𝑖(𝑡)𝑦_𝑖−𝑞_𝑖(𝑡)𝑦_𝑖 gives the density-dependent death rate of the predators. 𝑝_𝑖(𝑡) and ℎ_𝑖(𝑡) give the coefficients that relate to the conversion rate of the immature prey biomass into predator biomass. More details about the biological background for (1.1) can be found in [1–10].

The function 𝑝_𝑖(𝑡)𝑥²₁(𝑡)/(𝑘_𝑖(𝑡)+𝑥²₁(𝑡)) represents the functional response of predator to immature prey. Let 𝜑_𝑖(𝑡,𝑥₁)=𝑝_𝑖(𝑡)𝑥²₁(𝑡)/(𝑘_𝑖(𝑡)+𝑥²₁(𝑡)), 𝑖=1,2, then we have

𝜕𝜕𝑥₁𝜑_𝑖(𝑡,𝑥₁)≥0,𝑥₁(𝑡)>0,𝑖=1,2. (1.2)

The functional response of predator species 𝑦_𝑖(𝑡) to immature prey species takes the Holling type III, that is, 𝑝_𝑖(𝑖)𝑥²₁(𝑡)/(𝑘_𝑖(𝑡)+𝑥²₁(𝑡)). Holling type III is the third function in which Holling proposed three kinds of functional response of the predator to prey based on numerous experiments for different species. The Holling type form of functional response is intituled prey-dependent model form. It is applied to almost invertebrate that is one of the most extensive applied functional responses.

In [2], Cui and Takeuchi considered the following periodic predator-prey system with a stage structure:

𝑥̇₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₂(𝑡)−𝑑(𝑡)𝑥²₁(𝑡)−𝑝(𝑡)𝜙(𝑡,𝑥₁)𝑥₁𝑥̇𝑦(𝑡),₂(𝑡)=𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂̇(𝑡),𝑦(𝑡)=𝑦(𝑡)(−𝑔(𝑡)+ℎ(𝑡))𝜙(𝑡,𝑥₁)𝑥₁𝑦(𝑡)−𝑞(𝑡)𝑦(𝑡), (1.3) where

0<𝜙(𝑡,𝑥₁𝜕)<𝐿,𝜕𝑥₁(𝜙(𝑡,𝑥₁)𝑥₁)≥0(𝑥₁>0). (1.4)

Different predators usually consume prey in different stage structures. Some predators only prey on immature prey, and some predators only prey on mature prey [5]. Based on system (1.3), we also consider another predator species which also consumes immature prey. Assuming that the predator consumes immature prey according to Holling III functional response while the other predator consumes mature prey also according to the Holling III functional response, we get model (1.1).

To the best of the authors' knowledge, for the nonautonomous case of predator-prey systems with two predators which consume immature prey and stage structure for prey, whether one could obtain the sufficient and necessary conditions which insure the permanence of the system or not is still an open problem.

The aim of this paper is, by further developing the analysis technique of Cui and Takeuchi [2], to derive a set of sufficient and necessary conditions which ensure the permanence of the system (1.1). The rest of the paper is arranged as follows. In Section 2, we introduce some lemmas and then state the main result of this paper. The result is proved in Section 3.

Throughout this paper, for a continuous 𝜔-periodic function 𝑓(𝑡), we set

𝐴_𝜔(𝑓)=𝜔^−1𝜔₀𝑓(𝑡)𝑑𝑡. (1.5)

2. Main Results

Definition 2.1. The system 𝑥̇=𝐹(𝑡,𝑥),𝑥∈𝑅^𝑛 (2.1) is said to be permanent if there exists a compact set 𝐾 in the interior of 𝑅^𝑛₊={(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑅^𝑛∣𝑥_𝑖≥0,𝑖=1,2,…,𝑛}, such that all solutions starting in the interior of 𝑅^𝑛₊ ultimately enter 𝐾 and remain in 𝐾.

The following lemma can be found in [4].

Lemma 2.2. If 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), and 𝑓(𝑡) are all 𝜔-periodic, then system 𝑥̇₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)=𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂(𝑡) (2.2) has a positive 𝜔-periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) which is globally asymptotically stable with respect to 𝑅²₊={(𝑥₁,𝑥₂)∶𝑥₁>0,𝑥₂>0}.

Lemma 2.3 (see [11]). If 𝑏(𝑡) and 𝑎(𝑡) are all 𝜔-periodic, and if 𝐴_𝜔(𝑏)>0 and 𝐴_𝜔(𝑎)>0 for all 𝑡∈𝑅, then the system 𝑥̇=𝑥[𝑏(𝑡)−𝑎(𝑡)𝑥] (2.3) has a positive 𝜔-periodic solution which is globally asymptotically stable.

Theorem 2.4. Suppose that 𝐴_𝜔(−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥^∗₁(𝑡))²𝑘₁(𝑡)+(𝑥^∗₁(𝑡))²𝐴)>0,_𝜔(−𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₁(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥^∗₁+ℎ(𝑡)³(𝑡)(𝑥^∗₂(𝑡))²𝑘₃(𝑡)+(𝑥^∗₂(𝑡))²)>0 (2.4) holds then system (1.1) is permanent, where (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) is the unique positive periodic solution of system (2.2) given by Lemma 2.2.

Theorem 2.5. System (1.1) is permanent if and only if (2.4) holds.

3. Proof of the Main Results

We need the following propositions to prove Theorems 2.4 and 2.5. The hypothesis of the lemmas and theorems of the preceding section are assumed to hold in what follows.

Proposition 3.1. There exist positive constants 𝑀_𝑥 and 𝑀_𝑦 such that limsup_𝑡→+∞𝑥_𝑖(𝑡)≤𝑀_𝑥,limsup_𝑡→+∞𝑦_𝑖(𝑡)≤𝑀_𝑦,𝑖=1,2. (3.1)

Proof. Obviously, 𝑅⁴₊ is a positively invariant set of system (1.1). Given any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of (1.1), we have 𝑥̇₁(𝑡)≤𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)≤𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂(𝑡). (3.2)
By Lemma 2.2, the following auxiliary equation 𝑢̇₁(𝑡)=𝑎(𝑡)𝑢₂(𝑡)−𝑏(𝑡)𝑢₁(𝑡)−𝑑(𝑡)𝑢²₁𝑢̇(𝑡),₂(𝑡)=𝑐(𝑡)𝑢₁(𝑡)−𝑓(𝑡)𝑢²₂(𝑡) (3.3) has a globally asymptotically stable positive 𝜔-periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)). Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.3) with 𝑢_𝑖(0)=𝑥_𝑖(0). By comparison, we then have 𝑥_𝑖(𝑡)≤𝑢_𝑖(𝑡),𝑖=1,2 (3.4) for 𝑡≥0. By (2.4), we can choose a positive 𝜀>0 small enough such that 𝐴_𝜔(−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥^∗₁(𝑡)+𝜀)²𝑘₁𝐴(𝑡))>0,_𝜔(−𝑔₂ℎ(𝑡)+²(𝑡)(𝑥^∗₁(𝑡)+𝜀)𝑘₂+ℎ(𝑡)³(𝑡)(𝑥^∗₂(𝑡)+𝜀)²𝑘₃(𝑡))>0. (3.5) Thus, from the global attractivity of (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)), for the above given 𝜀>0, there exists a 𝑇₀>0 such that |𝑢_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|<𝜀,𝑡≥𝑇₀, (3.6) Equation (3.4) combined with (3.6) leads to 𝑥_𝑖(𝑡)<𝑥^∗_𝑖(𝑡)+𝜀,𝑡>𝑇₀. (3.7) In addition, for 𝑡≥𝑇₀, from the third and fourth equations of (1.1) and (3.7) we get 𝑦̇₁(𝑡)≤𝑦₁(𝑡)[−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥²₁(𝑡)𝑘₁(𝑡)−𝑞₁(𝑡)𝑦₁(𝑡)]≤𝑦₁(𝑡)[−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥^∗₁(𝑡)+𝜀)²𝑘₁(𝑡)−𝑞₁(𝑡)𝑦₁𝑦̇(𝑡)],₂(𝑡)≤𝑦₂(𝑡)[−𝑔₂ℎ(𝑡)+²(𝑡)𝑥₁(𝑡)𝑘₂+ℎ(𝑡)³(𝑡)𝑥²₂(𝑡)𝑘₃(𝑡)−𝑞₂(𝑡)𝑦₂(𝑡)]≤𝑦₂(𝑡)[−𝑔₂ℎ(𝑡)+²(𝑡)(𝑥^∗₁(𝑡)+𝜀)𝑘₂+ℎ(𝑡)³(𝑡)(𝑥^∗₂(𝑡)+𝜀)²𝑘₃(𝑡)−𝑞₂(𝑡)𝑦₂(𝑡)]. (3.8) Consider the following auxiliary equation: 𝑣̇₁(𝑡)=𝑣₁(𝑡)[−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥^∗₁(𝑡)+𝜀)²𝑘₁(𝑡)−𝑞₁(𝑡)𝑣₁𝑣̇(𝑡)],₂(𝑡)=𝑣₂(𝑡)[−𝑔₂ℎ(𝑡)+²(𝑡)(𝑥^∗₁(𝑡)+𝜀)𝑘₂+ℎ(𝑡)³(𝑡)(𝑥^∗₂(𝑡)+𝜀)²𝑘₃(𝑡)−𝑞₂(𝑡)𝑣₂(𝑡)]. (3.9) It follows from (3.5) and Lemma 2.3 that (3.9) has a unique positive 𝜔-periodic solution 𝑦^∗_𝑖(𝑡)>0 which is globally asymptotically stable. Similarly to the above analysis, there exists a 𝑇₁>𝑇₀ such that for the above 𝜀, one has 𝑦_𝑖(𝑡)<𝑦^∗_𝑖(𝑡)+𝜀,𝑡≥𝑇₁. (3.10) Let 𝑀_𝑥=max_{0≤𝑡≤𝜔}{𝑥^∗_𝑖(𝑡)+𝜀∶𝑖=1,2}, 𝑀_𝑦=max_{0≤𝑡≤𝜔}{𝑦^∗_𝑖(𝑡)+𝜀∶𝑖=1,2}, then we have limsup_𝑡→+∞𝑥_𝑖(𝑡)≤𝑀_𝑥,limsup_𝑡→+∞𝑦_𝑖(𝑡)≤𝑀_𝑦. (3.11) This completes the proof of Proposition 3.1.

Proposition 3.2. There exist positive constants 𝛿_𝑖<𝑀_𝑥, 𝑖=1,2, such that liminf_𝑡→+∞𝑥_𝑖(𝑡)≥𝛿_𝑖,𝑖=1,2. (3.12)

Proof. By Proposition 3.1, there exists 𝑇₁>0 such that 0<𝑥_𝑖(𝑡)≤𝑀_𝑥;0<𝑦_𝑖(𝑡)≤𝑀_𝑦;𝑡>𝑇₁. (3.13) Hence, from the first and second equations of system (1.1), we have 𝑥̇₁(𝑡)≥𝑎(𝑡)𝑥₂𝑝(𝑡)−(𝑏(𝑡)+(¹(𝑡)𝑘₁+𝑝(𝑡)²(𝑡)𝑘₂(𝑡))𝑀_𝑦)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)≥𝑐(𝑡)𝑥₁𝑝(𝑡)−(𝑓(𝑡)+³(𝑡)𝑘₃𝑀(𝑡)^𝑦)𝑥²₂ (3.14) for 𝑡>𝑇₁. By Lemma 2.2, the following auxiliary equation 𝑢̇₁(𝑡)=𝑎(𝑡)𝑢₂𝑝(𝑡)−(𝑏(𝑡)+(¹(𝑡)𝑘₁+𝑝(𝑡)²(𝑡)𝑘₂(𝑡))𝑀_𝑦)𝑢₁(𝑡)−𝑑(𝑡)𝑢²₁𝑢̇(𝑡),₂(𝑡)≥𝑐(𝑡)𝑢₁𝑝(𝑡)−(𝑓(𝑡)+³(𝑡)𝑘₃𝑀(𝑡)^𝑦)𝑢²₂ (3.15) has a globally asymptotically stable positive 𝜔-periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)). Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.15) with (𝑢₁(𝑇₁),𝑢₂(𝑇₁))=(𝑥₁(𝑇₁),𝑥₂(𝑇₁)); by comparison, we have 𝑥_𝑖(𝑡)≥𝑢_𝑖(𝑡)(𝑖=1,2),𝑡>𝑇₁. (3.16) Thus, from the global attractivity of (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)), there exists a 𝑇₂>𝑇₁, such that |𝑢_𝑖(𝑡)−𝑥^∗_𝑖𝑥(𝑡)|<^∗𝑖(𝑡)2(𝑖=1,2),𝑡>𝑇₂. (3.17) Equation (3.17) combined with (3.16) leads to 𝑥_𝑖(𝑡)>𝛿_𝑖=min_{0≤𝑡≤𝜔}{𝑥^∗_𝑖(𝑡)2},𝑖=1,2,𝑡>𝑇₂. (3.18) That is, we have liminf_𝑡→+∞𝑥_𝑖(𝑡)≥𝛿_𝑖,𝑖=1,2. (3.19) This completes the proof of Proposition 3.2.

Proposition 3.3. There exists a positive constant 𝛿_𝑦 such that limsup_𝑡→+∞𝑦_𝑖(𝑡)≥𝛿_𝑦,𝑖=1,2. (3.20)

Proof. By assumption (2.4), we can choose a constant 𝜀₀>0 and the same constant 𝜀 as in Proposition 3.1 such that 𝐴_𝜔(𝜓_𝑖𝜀0(𝑡))>0,𝑖=1,2, (3.21) where 𝜓_1𝜀0(𝑡)=−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥^∗₁(𝑡)−𝜀₀)²𝑘₁(𝑡)+(𝑥^∗₁(𝑡)−𝜀₀)²−𝑞₁(𝑡)𝜀₀,𝜓_2𝜀0(𝑡)=−𝑔₂ℎ(𝑡)+²(𝑡)(𝑥^∗₁(𝑡)−𝜀₀)𝑘₂(𝑡)+𝑚(𝑡)(𝑥^∗₁(𝑡)+𝜀)+𝑛(𝑡)𝜀₀+ℎ₃(𝑡)(𝑥^∗₂(𝑡)−𝜀₀)²𝑘₃(𝑡)+(𝑥^∗₂(𝑡)−𝜀₀)²−𝑞₂(𝑡)𝜀₀. (3.22) Consider the following equation with a parameter 𝛽>0: 𝑥̇₁(𝑡)=𝑎(𝑡)𝑥₂𝑝(𝑡)−(𝑏(t)+2𝛽(¹(𝑡)𝑘₁+𝑝(𝑡)²(𝑡)𝑘₂(𝑡))𝑀_𝑦)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)=𝑐(𝑡)𝑥₁𝑝(𝑡)−(𝑓(𝑡)+³(𝑡)𝑘₃(𝑡)2𝛽)𝑥²₂. (3.23) By Lemma 2.2, (3.23) has a unique positive 𝜔-periodic solution (𝑥_1𝛽(𝑡),𝑥_2𝛽(𝑡)), which is globally asymptotically stable. Let (𝑥_1𝛽(𝑡),𝑥_2𝛽(𝑡)) be the solution of (3.23) with initial condition 𝑥_𝑖𝛽(0)=𝑥^∗_𝑖(0), 𝑖=1,2; then, for the above 𝜀₀, there exists a 𝑇₃>𝑇₂, such that |𝑥_𝑖𝛽(𝑡)−𝑥_𝑖𝛽𝜀(𝑡)|<⁰4,𝑖=1,2,𝑡>𝑇₃. (3.24) By continuity of the solution in the parameter, we have (𝑥_1𝛽(𝑡),𝑥_2𝛽(𝑡))→(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) uniformly in [𝑇₃,𝑇₃+𝜔] as 𝛽→0. Hence, for 𝜀₀>0 there exists 𝛽₀=𝛽₀(𝜀₀)>0 such that |𝑥_𝑖𝛽(𝑡)−𝑥^∗_𝑖𝜀(𝑡)|<⁰4,𝑡∈[𝑇₃,𝑇₃+𝜔],0<𝛽<𝛽₀. (3.25) So we have |𝑥_𝑖𝛽(𝑡)−𝑥^∗_𝑖(𝑡)|≤|𝑥_𝑖𝛽(𝑡)−𝑥_𝑖𝛽(𝑡)|+|𝑥_𝑖𝛽(𝑡)−𝑥^∗_𝑖𝜀(𝑡)|<⁰2,𝑡∈[𝑇₃,𝑇₃+𝜔]. (3.26) Since 𝑥_𝑖𝛽(𝑡) and 𝑥^∗_𝑖(𝑡) are all 𝜔-periodic, we have |𝑥_𝑖𝛽(𝑡)−𝑥^∗_𝑖𝜀(𝑡)|<⁰2,𝑡≥0,0<𝛽<𝛽₀. (3.27) Choose a constant 𝛽₁(0<𝛽₁<𝛽₀,2𝛽₁<𝜀₀) and 𝑥_𝑖𝛽1(𝑡)≥𝑥^∗_𝑖𝜀(𝑡)−⁰2,𝑡≥0. (3.28) Suppose that the conclusion (3.20) is not true. Then there exists a 𝑍∈𝑅⁴₊ such that, for the positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of (1.1) with an initial condition (𝑥₁(0),𝑥₂(0),𝑦₁(0),𝑦₂(0))=𝑍, we have limsup_𝑡→+∞𝑦_𝑖(𝑡)<𝛽₁. (3.29) So there exists a 𝑇₄>𝑇₃ such that 𝑦_𝑖(𝑡)<2𝛽₁,𝑡≥𝑇₄. (3.30) By applying (3.30), from the first and second equations of system (1.1) it follows that for all 𝑡≥𝑇₄, 𝑥̇₁(𝑡)≥𝑎(𝑡)𝑥₂(𝑡)−(𝑏(𝑡)+2𝛽₁𝑝₁(𝑡)𝑘₁(𝑡)+2𝛽₁𝑝₂(𝑡)𝑘₂(𝑡))𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)≥𝑐(𝑡)𝑥₁𝑝(𝑡)−(𝑓(𝑡)+³(𝑡)𝑘₃(𝑡)2𝛽₁)𝑥²₂(𝑡). (3.31) Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.23) with 𝛽=𝛽₁ and 𝑢_𝑖(𝑇₄)=𝑥_𝑖(𝑇₄), 𝑖=1,2; we know that 𝑥_𝑖(𝑡)≥𝑢_𝑖(𝑡), 𝑡≥𝑇₄, 𝑖=1,2.
By the global asymptotic stability of (𝑥_1𝛽1(𝑡),𝑥_2𝛽1(𝑡)), for the given 𝜀=𝜀₀/2, there exists 𝑇₅≥𝑇₄ such that |𝑢_𝑖(𝑡)−𝑥_𝑖𝛽1𝜀(𝑡)|<⁰2,𝑡≥𝑇₅. (3.32) So we have 𝑥_𝑖(𝑡)≥𝑢_𝑖(𝑡)>𝑥_𝑖𝛽1𝜀(𝑡)−⁰2,𝑡≥𝑇₅ (3.33) and hence 𝑥_𝑖(𝑡)>𝑥^∗_𝑖(𝑡)−𝜀₀,𝑡≥𝑇₅. (3.34) From (3.7) and (3.34), we have 𝑥^∗_𝑖(𝑡)−𝜀₀<𝑥_𝑖(𝑡)<𝑥^∗_𝑖(𝑡)+𝜀,𝑡≥𝑇₅. (3.35)
By (3.35) and (1.2), from the third and fourth equations of system (1.1) we have 𝑦̇_𝑖(𝑡)≥𝜓_𝑖𝜀0(𝑡)𝑦_𝑖(𝑡),𝑡≥𝑇₅. (3.36) Integrating (3.36) from 𝑇₅ to 𝑡 yields 𝑦_𝑖(𝑡)≥𝑦_𝑖(𝑇₅)exp{^𝑡_𝑇5𝜓_𝑖𝜀0(𝑡)𝑑𝑡},𝑡≥𝑇₅. (3.37) By (3.21), we know that 𝑦_𝑖(𝑡)→+∞ as 𝑡→+∞, 𝑖=1,2, which is a contradiction. This completes the proof.

Proof of Theorem 2.4. By Propositions 3.2 and 3.3, system (1.1) is uniform weak persistent [2]. From [12, Propositions 3.1 and Theorem 1.3.3], system (1.1) is permanent. This completes the proof of Theorem 2.4.

Proof of Theorem 2.5. The sufficiency of Theorem 2.5 now follows from Theorem 2.4. We thus only need to prove the necessity of Theorem 2.5. Suppose that 𝐴_𝜔(−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥^∗₁(𝑡))²𝑘₁(𝑡)+(𝑥^∗₁(𝑡))²𝐴)≤0,_𝜔(−𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₁(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥^∗₁+ℎ(𝑡)³(𝑡)(𝑥^∗₂(𝑡))²𝑘₃(𝑡)+(𝑥^∗₂(𝑡))²)≤0. (3.38) We will show that lim_𝑡→+∞𝑦_𝑖(𝑡)=0,𝑖=1,2. (3.39) In fact, by (3.38), we know that, for any given 0<𝜀<1, there exist 𝜀₁>0 and 𝜀₀>0 such that 𝐴_𝜔(−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥^∗₁(𝑡)+𝜀₁)²𝑘₁(𝑡)+(𝑥^∗₁(𝑡)+𝜀₁)²−𝑞₁𝜀(𝑡)𝜀)≤−2𝐴_𝜔(𝑞₁(𝑡))≤−𝜀₀,𝐴_𝜔(−𝑔₂ℎ(𝑡)+²(𝑡)(𝑥^∗₁(𝑡)+𝜀₁)𝑘₂(𝑡)+𝑚(𝑡)(𝑥^∗₁(𝑡)+𝜀₁+ℎ)+𝑛(𝑡)𝜀³(𝑡)(𝑥^∗₂(𝑡)+𝜀₁)²𝑘₃(𝑡)+(𝑥^∗₂(𝑡)+𝜀₁)²−𝑞₂𝜀(𝑡)𝜀)≤−2𝐴_𝜔(𝑞₂(𝑡))≤−𝜀₀. (3.40) Note that 𝑞_𝑖(𝑡)>0 for 𝑡≥0. Since 𝑥̇₁(𝑡)≤𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)≤𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₁(𝑡), (3.41) we know that, for the given 𝜀₁, there exists 𝑇⁽¹⁾>0 such that 𝑥_𝑖(𝑡)≤𝑥^∗_𝑖(𝑡)+𝜀₁,𝑡≥𝑇⁽¹⁾,𝑖=1,2. (3.42) By (3.40), (1.2), and (3.42), we have 𝐴_𝜔(−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥²₁(𝑡)𝑘₁(𝑡)+𝑥²₁(𝑡)−𝑞₁(𝑡)𝜀)≤−𝜀₀,𝑡≥𝑇⁽¹⁾,𝐴_𝜔(−𝑔₂ℎ(𝑡)+²(𝑡)𝑥₁(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥₁+ℎ(𝑡)+𝑛(𝑡)𝜀³(𝑡)𝑥²₂(𝑡)𝑘₃(𝑡)+𝑥²₂(𝑡)−𝑞₂(𝑡)𝜀)≤−𝜀₀,𝑡≥𝑇⁽¹⁾. (3.43) We now show that there must exist 𝑇⁽²⁾(>𝑇⁽¹⁾) such that 𝑦_𝑖(𝑇⁽²⁾)<𝜀. Otherwise, by (3.43), we have 𝜀≤𝑦₁(𝑡)≤𝑦₁(𝑇⁽¹⁾)exp{^𝑡_𝑇(1)[−𝑔₁ℎ(𝑠)+¹(𝑠)(𝑥₁(𝑠))²𝑘₁(𝑠)+(𝑥₁(𝑠))²−𝑞₁(𝑠)𝜀]𝑑𝑠}→0,𝑡→+∞,𝜀≤𝑦₂(𝑡)≤𝑦₂(𝑇⁽¹⁾)exp{^𝑡_𝑇(1)[−𝑔₂ℎ(𝑠)+²(𝑠)𝑥₁(𝑠)𝑘₂(𝑠)+𝑚(𝑠)𝑥₁+ℎ(𝑠)+𝑛(𝑠)𝜀³(𝑠)𝑥²₂(𝑠)𝑘₃(𝑠)+𝑥²₂(𝑠)−𝑞₂(𝑠)𝜀]𝑑𝑠}→0,𝑡→+∞. (3.44) This implies 𝜀≤0, which is a contradiction.
Let 𝑀₁(𝜀)=max_{0≤𝑠≤𝜔}{−𝑔₁ℎ(𝑠)+¹(𝑠)(𝑥₁(𝑠))²𝑘₁(𝑠)+(𝑥₁(𝑠))²−𝑞₁𝑀(𝑠)𝜀},₂(𝜀)=max_{0≤𝑠≤𝜔}{−𝑔₂ℎ(𝑠)+²(𝑠)𝑥₁(𝑠)𝑘₂(𝑠)+𝑚(𝑠)𝑥₁+ℎ(𝑠)+𝑛(𝑠)𝜀³(𝑠)𝑥²₂(𝑠)𝑘₃(𝑠)+𝑥²₂(𝑠)−𝑞₂(𝑠)𝜀}. (3.45) We know that 𝑀_𝑖(𝜀) is bounded (Proposition 3.1 given). We now show that 𝑦_𝑖(𝑡)≤𝜀exp{𝑀_𝑖(𝜀)𝜔},𝑡≥𝑇⁽²⁾,𝑖=1,2. (3.46) Otherwise, there exists 𝑇⁽³⁾>𝑇⁽²⁾ such that 𝑦_𝑖(𝑇⁽³⁾)>𝜀exp{𝑀_𝑖(𝜀)𝜔}. (3.47) By the continuity of 𝑦_𝑖(𝑡), there must exist 𝑇⁽⁴⁾∈(𝑇⁽²⁾,𝑇⁽³⁾) such that 𝑦_𝑖(𝑇⁽⁴⁾)=𝜀 and 𝑦_𝑖(𝑡)>𝜀 for 𝑡∈(𝑇⁽⁴⁾,𝑇⁽³⁾]. Let 𝑃₁ be the nonnegative integer such that 𝑇⁽³⁾∈(𝑇⁽⁴⁾+𝑃₁𝜔,𝑇⁽⁴⁾+(𝑃₁+1)𝜔]; by (3.43) we have 𝜀exp{𝑀₁(𝜀)𝜔}<𝑦₁(𝑇⁽³⁾)<𝑦₁(𝑇⁽⁴⁾)exp{^𝑇(3)_𝑇(4)[−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥₁(𝑡))²𝑘₁(𝑡)+(𝑥₁(𝑡))²−𝑞₁(𝑡)𝜀]𝑑𝑡}=𝜀exp{^{𝑇(4)+𝑃1𝜔}_𝑇(4)+^𝑇(3)_{𝑇(4)+𝑃1𝜔}}[−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥₁(𝑡))²𝑘₁(𝑡)+(𝑥₁(𝑡))²−𝑞₁(𝑡)𝜀]𝑑𝑡≤𝜀exp{^𝑇(3)_{𝑇(4)+𝑃1𝜔}[−𝑔₁ℎ(𝑡)+¹(𝑡)(𝑥₁(𝑡))²𝑘₁(𝑡)+(𝑥₁(𝑡))²−𝑞₁(𝑡)𝜀]𝑑𝑡}<𝜀exp{𝑀₁(𝜀)𝜔},𝜀exp{𝑀₂(𝜀)𝜔}<𝑦₂(𝑇⁽³⁾)<𝑦₂(𝑇⁽⁴⁾)exp{^𝑇(3)_𝑇(4)[−𝑔₂ℎ(𝑡)+²(𝑡)𝑥₁(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥₁+ℎ(𝑡)+𝑛(𝑡)𝜀₃(𝑡)𝑥²₂(𝑡)𝑘₃(𝑡)+𝑥²₂(𝑡)−𝑞₂(𝑡)𝜀]𝑑𝑡}=𝜀exp{^{𝑇(4)+𝑃1𝜔}_𝑇(4)+^𝑇(3)_{𝑇(4)+𝑃1𝜔}}[−𝑔₂ℎ(𝑡)+²(𝑡)𝑥₁(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥₁+ℎ(𝑡)+𝑛(𝑡)𝜀₃(𝑡)𝑥²₂(𝑡)𝑘₃(𝑡)+𝑥²₂(𝑡)−𝑞₂(𝑡)𝜀]𝑑𝑡≤𝜀exp{^𝑇(3)_{𝑇(4)+𝑃1𝜔}[−𝑔₂ℎ(𝑡)+²(𝑡)𝑥₁(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥₁+ℎ(𝑡)+𝑛(𝑡)𝜀³(𝑡)𝑥²₂(𝑡)𝑘₃(𝑡)+𝑥²₂(𝑡)−𝑞₂(𝑡)𝜀]𝑑𝑡}<𝜀exp{𝑀₂(𝜀)𝜔}, (3.48) which is a contradiction. This implies that (3.46) holds. We then conclude, by the arbitrariness of 𝜀, that 𝑦_𝑖(𝑡)→0 as 𝑡→+∞, 𝑖=1,2. This completes the proof of Theorem 2.5.