Permanence of Periodic Predator-Prey System with General Nonlinear Functional Response and Stage Structure for Both Predator and Prey

Abstract

We study the permanence of periodic predator-prey system with general nonlinear functional responses and stage structure for both predator and prey and obtain that the predator and the prey species are permanent.

1. Introduction

An important and ubiquitous problem in predator-prey theory and related topics in mathematical ecology concerns the long-term coexistence of species. In the natural world there are many species whose individual members have a life history that takes them through two stages: immature and mature. In particular, we have in mind mammalian populations and some amphibious animals, which exhibit these two stages. Recently, nonautonomous systems with a stage structure have been considered in [1–16]; in particular periodic predator-prey systems with a stage structure were discussed in [3, 4, 7, 13, 14].

Already, in [3], Cui and Song proposed the following predator-prey model with stage structure for prey: 𝑥̇₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁(𝑡)−𝑝(𝑡)𝑥₁𝑥̇(𝑡)𝑦(𝑡),₂(𝑡)=𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂̇(𝑡),𝑦(𝑡)=𝑦(𝑡)(−𝑔(𝑡)+ℎ(𝑡)𝑥₁(𝑡)−𝑞(𝑡)𝑦(𝑡)). (1.1) They obtained a set of sufficient and necessary condition which guarantee the permanence of the system.

In [4], Cui and Takeuchi considered the following periodic predator-prey system with a stage structure: 𝑥̇₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(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)𝑦(𝑡)−𝑞(𝑡)𝑦(𝑡)), (1.2) where

0<𝜙(𝑡,𝑥₁𝜕)<𝐿,𝜕𝑥₁(𝜙(𝑡,𝑥₁)𝑥₁)≥0(𝑥₁>0). (1.3)

Recently, Huang et al. [7] studied the following periodic stage-structured three-species predator-prey system with Holling IV and Beddington-DeAngelis functional response:

𝑥̇₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁ℎ(𝑡)−¹(𝑡)𝑥₁(𝑡)𝑘₁(𝑡)+𝑥²₁𝑦(𝑡)¹𝑥̇(𝑡),₂(𝑡)=𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂ℎ(𝑡)−²(𝑡)𝑥₂(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥₂(𝑡)+𝑛(𝑡)𝑦₂𝑦(𝑡)²𝑦̇(𝑡),₁(𝑡)=𝑦₁(𝑡)(−𝑞₁𝑝(𝑡)+¹(𝑡)𝑥₁(𝑡)𝑘₁(𝑡)+𝑥²₁(𝑡)−𝑔₁(𝑡)𝑦₁𝑦̇(𝑡)),₂(𝑡)=𝑦₂(𝑡)(−𝑞₂𝑝(𝑡)+²(𝑡)𝑥₂(𝑡)𝑘₂(𝑡)+𝑚(𝑡)𝑥₂(𝑡)+𝑛(𝑡)𝑦₂(𝑡)−𝑔₂(𝑡)𝑦₂(𝑡)), (1.4) where 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡),𝑑(𝑡),𝑓(𝑡),𝑚(𝑡),𝑛(𝑡),ℎ_𝑖(𝑡),𝑘_𝑖(𝑡),𝑝_𝑖(𝑡),𝑞_𝑖(𝑡), and 𝑔_𝑖(𝑡)(𝑖=1,2) are all continuous positive 𝜔-periodic functions; 𝑥₁(𝑡) and 𝑥₂(𝑡) denote the density of immature and mature prey species at time 𝑡, respectively; 𝑦₁(𝑡) represents the density of the predator that preys on immature prey; 𝑦₁(𝑡) represents the density of the other predator that preys on mature prey at time 𝑡.

It is assumed in the classical predator-prey model that each individual predator admits the same ability to attack prey and each individual prey admits the same risk to be attacked by predator. This assumption seems not to be realistic for many animals. On the other hand, predator-prey systems where only immature individuals are consumed by their predator are well known in nature. One example is described in [9], where the Chinese fire-bellied newt, which is unable to prey upon the mature rana chensinensis, can only prey on its immature.

To the best of the authors' knowledge, for the nonautonomous periodic case of predator-prey systems with stage structure for both predator and prey, whether one could obtain the permanence of the system or not is still an open problem.

Motivated by the above question, we consider the following periodic predator-prey system with general nonlinear functional responses and stage structure for both predator and prey: 𝑥̇₁(𝑡)=𝑎₁(𝑡)𝑥₂(𝑡)−𝑏₁(𝑡)𝑥₁(𝑡)−𝑑₁(𝑡)𝑥²₁(𝑡)−𝑔(𝑥₁(𝑡))𝑦₂𝑥̇(𝑡),₂(𝑡)=𝑐₁(𝑡)𝑥₁(𝑡)−𝑓₁(𝑡)𝑥²₂𝑦̇(𝑡),₁(𝑡)=𝑎₂(𝑡)𝑦₂(𝑡)−𝑏₂(𝑡)𝑦₁(𝑡)−𝑑₂(𝑡)𝑦²₁(𝑡)+𝑘(𝑡)𝑔(𝑥₁(𝑡))𝑦₁𝑦̇(𝑡),₂(𝑡)=𝑐₂(𝑡)𝑦₁(𝑡)−𝑓₂(𝑡)𝑦²₂(𝑡), (1.5) where 𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐_𝑖(𝑡),𝑑_𝑖(𝑡),𝑓_𝑖(𝑡),𝑖=1,2, and 𝑘(𝑡) are all continuous positive 𝜔-periodic functions. Here 𝑥₁(𝑡) and 𝑥₂(𝑡) denote the density of immature and mature prey species, respectively, and 𝑦₁(𝑡) and 𝑦₂(𝑡) denote the density of immature and mature predator species, respectively. The function 𝑔(𝑥) is assumed to satisfy the following assumptions which has been studied in detail by Georgescu and Morosanu in [17].

(G) 𝑔(𝑥) of class 𝐶¹ is increasing on 𝑅₊,𝑔(0)=0, and such that 𝑥↦𝑔(𝑥)/𝑥 is decreasing on 𝑅₊,|𝑔^′(𝑥)|≤𝐿 for 𝑥∈𝑅₊, where 𝐿≥0.

Note that hypothesis (G) is satisfied if function 𝑔(𝑥) represents Holling type II functional response, that is, 𝑔(𝑥)=𝑎𝑥/(1+𝑏𝑥), in which 𝑎 is the search rate of the resource and of the intermediate consumer, and 𝑏 represents the corresponding clearance rate, that is, search rate multiplied by the (supposedly constant) handling time.

The aim of this paper is, by further developing the analysis technique of Cui and Song [3] and Cui and Takeuchi [4], to obtain a set of sufficient and necessary conditions which ensure the permanence of the system (1.5). 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.

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.

Lemma 2.2 (see [6]). 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}.

Theorem 2.3. System (1.5) is permanent.

3. Proof of the Main Results

We need the following propositions to prove Theorem 2.3. The hypotheses of the lemmas and theorems of the preceding section are assumed to hold in what follows.

Proposition 3.1. There exists a positive constant 𝑀_𝑥 such that limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀_𝑥,𝑖=1,2. (3.1)

Proof. Obviously, 𝑅⁺₄ is a positively invariant set of system (1.5). Given any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of (1.5), from the first and second equations of system(1.5), we have 𝑥̇₁(𝑡)≤𝑎₁(𝑡)𝑥₂(𝑡)−𝑏₁(𝑡)𝑥₁(𝑡)−𝑑₁(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)≤𝑐₁(𝑡)𝑥₁(𝑡)−𝑓₁(𝑡)𝑥²₂(𝑡). (3.2)
By Lemma 2.2, the following auxiliary equation 𝑢̇₁(𝑡)=𝑎₁(𝑡)𝑢₂(𝑡)−𝑏₁(𝑡)𝑢₁(𝑡)−𝑑₁(𝑡)𝑢²₁𝑢̇(𝑡),₂(𝑡)=𝑐₁(𝑡)𝑢₁(𝑡)−𝑓₁(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. From the global attractivity of (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)), for any positive 𝜀>0 small enough, there exists a 𝑇₀>0 such that |𝑢_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|<𝜀,𝑡≥𝑇₀ (3.5) (3.4) combined with (3.5) leads to 𝑥_𝑖(𝑡)<𝑥^∗_𝑖(𝑡)+𝜀,𝑡>𝑇₀. (3.6)
Let 𝑀_𝑥=max_{0≤𝑡≤𝜔}{𝑥^∗_𝑖(𝑡)+𝜀∶𝑖=1,2}, then we have limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀_𝑥. (3.7) This completes the proof of Proposition 3.1.

Proposition 3.2. There exists a positive constant 𝑚_𝑦 such that liminf_𝑡→∞𝑦_𝑖(𝑡)≥𝑚_𝑦,𝑖=1,2. (3.8)

Proof. Given any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of (1.5), from the third and fourth equations of system (1.5), we have 𝑦̇₁(𝑡)≥𝑎₂(𝑡)𝑦₂(𝑡)−𝑏₂(𝑡)𝑦₁(𝑡)−𝑑₂(𝑡)𝑦²₁𝑦̇(𝑡),₂(𝑡)≥𝑐₂(𝑡)𝑦₁(𝑡)−𝑓₂(𝑡)𝑦²₂(𝑡). (3.9)
By Lemma 2.2, the following auxiliary equation 𝑣̇₁(𝑡)=𝑎₂(𝑡)𝑣₂(𝑡)−𝑏₂(𝑡)𝑣₁(𝑡)−𝑑₂(𝑡)𝑣²₁𝑣̇(𝑡),₂(𝑡)=𝑐₂(𝑡)𝑣₁(𝑡)−𝑓₂(𝑡)𝑣²₂(𝑡) (3.10) has a globally asymptotically stable positive 𝜔-periodic solution (𝑦^∗₁(𝑡),𝑦^∗₂(𝑡)). Let (𝑣₁(𝑡),𝑣₂(𝑡)) be the solution of (3.10) with 𝑣_𝑖(0)=𝑦_𝑖(0). By comparison, we then have 𝑦_𝑖(𝑡)≥𝑣_𝑖(𝑡),𝑖=1,2 (3.11) for 𝑡≥0. From the global attractivity of (𝑦^∗₁(𝑡),𝑦^∗₂(𝑡)), for any positive 𝜀>0 small enough (𝜀<min_{0≤𝑡≤𝜔}{𝑦^∗_𝑖(𝑡)∶𝑖=1,2}), there exists a 𝑇₁>0 such that |𝑣_𝑖(𝑡)−𝑦^∗_𝑖(𝑡)|<𝜀,𝑡≥𝑇₁ (3.12) (3.11) combined with (3.12) leads to 𝑦_𝑖(𝑡)>𝑦^∗_𝑖(𝑡)−𝜀,𝑡>𝑇₁. (3.13)
Let 𝑚_𝑦=min_{0≤𝑡≤𝜔}{𝑦^∗_𝑖(𝑡)−𝜀∶𝑖=1,2}, then we have liminf_𝑡→∞𝑦_𝑖(𝑡)≥𝑚_𝑦. (3.14) This completes the proof of Proposition 3.2.

Proposition 3.3. There exists a positive constant 𝑀_𝑦 such that limsup_𝑡→+∞𝑦_𝑖(𝑡)≤𝑀_𝑦,𝑖=1,2. (3.15)

Proof. Given any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of (1.5), by Proposition 3.1, there exists a 𝑇₂>0 such that 0<𝑥_𝑖(𝑡)≤𝑀_𝑥,𝑡>𝑇₂. (3.16)
From the third and fourth equations of system (1.5), we have 𝑦̇₁(𝑡)≤𝑎₂(𝑡)𝑦₂(𝑡)−(𝑏₂(𝑡)−𝑘(𝑡)𝑔(𝑀_𝑥))𝑦₁(𝑡)−𝑑₂(𝑡)𝑦²₁𝑦̇(𝑡),₂(𝑡)≤𝑐₂(𝑡)𝑦₁(𝑡)−𝑓₂(𝑡)𝑦²₂(𝑡). (3.17)
By Lemma 2.2, the following auxiliary equation 𝑣̇₁(𝑡)=𝑎₂(𝑡)𝑣₂(𝑡)−(𝑏₂(𝑡)−𝑘(𝑡)𝑔(𝑀_𝑥))𝑣₁(𝑡)−𝑑₂(𝑡)𝑣²₁𝑣̇(𝑡),₂(𝑡)=𝑐₂(𝑡)𝑣₁(𝑡)−𝑓₂(𝑡)𝑣²₂(𝑡) (3.18) has a globally asymptotically stable positive 𝜔-periodic solution (𝑦^∗₁(𝑡),𝑦^∗₂(𝑡)). Let (𝑣₁(𝑡),𝑣₂(𝑡)) be the solution of (3.18) with 𝑣_𝑖(0)=𝑦_𝑖(0). By comparison, we then have 𝑦_𝑖(𝑡)≤𝑣_𝑖(𝑡),𝑖=1,2 (3.19) for 𝑡≥0. From the global attractivity of (𝑦^∗₁(𝑡),𝑦^∗₂(𝑡)), for above given positive 𝜀>0, there exists a 𝑇₃>0 such that |𝑣_𝑖(𝑡)−𝑦^∗_𝑖(𝑡)|<𝜀,𝑡≥𝑇₃ (3.20) (3.19) combined with (3.20) leads to 𝑦_𝑖(𝑡)<𝑦^∗_𝑖(𝑡)+𝜀,𝑡>𝑇₃. (3.21)
Let 𝑀_𝑦=max_{0≤𝑡≤𝜔}{𝑦^∗_𝑖(𝑡)+𝜀∶𝑖=1,2}, then we have limsup_𝑡→∞𝑦_𝑖(𝑡)≤𝑀_𝑦. (3.22) This completes the proof of Proposition 3.3.

Proposition 3.4. There exists a positive constant 𝑚_𝑥<𝑀_𝑥, such that liminf_𝑡→∞𝑥_𝑖(𝑡)≥𝑚_𝑥,𝑖=1,2. (3.23)

Proof. By Propositions 3.1 and 3.3, there exists a 𝑇₄>0 such that 0<𝑥_𝑖(𝑡)≤𝑀_𝑥;0<𝑦_𝑖(𝑡)≤𝑀_𝑦;𝑡>𝑇₄. (3.24)
From the first and second equations of system (1.5), we have 𝑥̇₁(𝑡)≥𝑎₁(𝑡)𝑥₂(𝑡)−(𝑏₁(𝑡)+𝑔(𝑥₁(𝑡))𝑥₁𝑀(𝑡)^𝑦)𝑥₁(𝑡)−𝑑₁(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)≥𝑐₁(𝑡)𝑥₁(𝑡)−𝑓₁(𝑡)𝑥²₂(𝑡). (3.25)
Since 𝑔(𝑥) is assumed to satisfy the assumptions (G), by the differential mean value theorem, we have 𝑔(𝑥₁(𝑡))𝑥₁=(𝑡)𝑔(𝑥₁(𝑡))−𝑔(0)𝑥₁(𝑡)−0=𝑔^′(𝜉)≤𝐿,0<𝜉<𝑥₁(𝑡). (3.26)
From (3.25) and (3.26), one has 𝑥̇₁(𝑡)≥𝑎₁(𝑡)𝑥₂(𝑡)−(𝑏₁(𝑡)+𝐿𝑀_𝑦)𝑥₁(𝑡)−𝑑₁(𝑡)𝑥²₁𝑥̇(𝑡),₂(𝑡)≥𝑐₁(𝑡)𝑥₁(𝑡)−𝑓₁(𝑡)𝑥²₂(𝑡). (3.27)
By Lemma 2.2, the following auxiliary equation: 𝑢̇₁(𝑡)=𝑎₁(𝑡)𝑢₂(𝑡)−(𝑏₁(𝑡)+𝐿𝑀_𝑦)𝑢₁(𝑡)−𝑑₁(𝑡)𝑢²₁𝑢̇(𝑡),₂(𝑡)=𝑐₁(𝑡)𝑢₁(𝑡)−𝑓₁(𝑡)𝑢²₂(𝑡) (3.28) has a globally asymptotically stable positive 𝜔-periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)). Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.27) with 𝑢_𝑖(0)=𝑥_𝑖(0). By comparison, we then have 𝑥_𝑖(𝑡)≥𝑢_𝑖(𝑡),𝑖=1,2 (3.29) for 𝑡≥0. From the global attractivity of (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)), for any positive 𝜀>0 small enough (𝜀<min_{0≤𝑡≤𝜔}{𝑥^∗_𝑖(𝑡)∶𝑖=1,2}), there exists a 𝑇₅>0 such that |𝑢_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|<𝜀,𝑡≥𝑇₅ (3.30) (3.29) combined with (3.30) leads to 𝑥_𝑖(𝑡)>𝑥^∗_𝑖(𝑡)−𝜀,𝑡>𝑇₅. (3.31)
Let 𝑚_𝑥=min_{0≤𝑡≤𝜔}{𝑥^∗_𝑖(𝑡)−𝜀∶𝑖=1,2}, then we have liminf_𝑡→∞𝑥_𝑖(𝑡)≥𝑚_𝑥. (3.32) This completes the proof of Proposition 3.4.

Proof of Theorem 2.3. By Propositions 3.1, 3.2, 3.3, and 3.4, system (1.5) is permanent. This completes the proof of Theorem 2.3.