Existence of Periodic Solutions in a Discrete Predator-Prey System with Beddington-DeAngelis Functional Responses

Abstract

A discrete predator-prey model with Holling II and Beddington-DeAngelis functional responses is investigated. With the aid of differential equations with piecewise constant arguments, a discrete version of continuous nonautonomous delayed predator-prey model with Beddington-DeAngelis functional responses is proposed. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive solutions of the model are established.

1. Introduction

In population dynamics, the functional response refers to the number of prey eaten per predator per unit time as a function of prey density. Based on a lot of experiments on mammals, Holling [1] proposed three kinds of functional responses as follows:(1)𝑝₁(𝑥)=𝑎𝑥, (2)𝑝₂(𝑥)=𝑎𝑥/(𝑎+𝑥), (3)𝑝₃(𝑥)=𝑎𝑥²/(𝑎+𝑥²),

where 𝑥 represents the density of prey. Functions 𝑝₁(𝑥), 𝑝₂(𝑥), and 𝑝₃(𝑥) refer to Holling types I, II, and III, respectively, and 𝑎>0 is the predation rate of the predator. After that, the dynamical properties of predator-prey systems with functional response have received great attention from both theoretical and mathematical biologists; for example, Liu et al. [2] investigated the coexistence of predators and preys of the following predator-prey system with Holling II functional response:𝑑𝑥(𝑡𝑥(𝑡)𝑑𝑡=𝑟𝑥)1−𝐾−𝑎𝑥(𝑡)𝑦(𝑡)(𝑡)−(𝑡)𝑧(𝑡)1+𝑏𝑥𝐴𝑥(𝑡),1+𝐵𝑥𝑑𝑦(𝑡(𝑡)𝑑𝑡=𝑦)−𝑑+𝑒𝑥(𝑡),1+𝑏𝑥𝑑𝑧(𝑡(𝑡)𝑑𝑡=𝑧)−𝐷+𝐸𝑥(𝑡),1+𝐵𝑥 (1.1) where 𝑥(𝑡) stands for the density of the prey, 𝑦(𝑡) and 𝑧(𝑡) are the densities of the predators, respectively, and 𝑟, 𝑎, 𝐴, 𝑑, 𝑒, 𝐷, 𝐸, 𝐾, 𝑏, and 𝐵 are positive constants. Assuming that one predator consumes prey according to the Holling II functional response and the other predator consumes prey according to the Beddington-DeAngelis functional response, Cantrell et al. [3] proposed the revised version of system (1.1) as follows:𝑑𝑥(𝑡𝑥(𝑡)𝑑𝑡=𝑟𝑥)1−𝐾−𝑎𝑥(𝑡)𝑦(𝑡)(𝑡)−(𝑡)𝑧(𝑡)1+𝑏𝑥𝐴𝑥(𝑡)(𝑡),1+𝐵𝑥+𝐶𝑧𝑑𝑦(𝑡(𝑡)𝑑𝑡=𝑦)−𝑑+𝑒𝑥(𝑡),1+𝑏𝑥𝑑𝑧(𝑡(𝑡)𝑑𝑡=𝑧)−𝐷+𝐸𝑥(𝑡)(𝑡).1+𝐵𝑥+𝐶𝑧 (1.2) By using dynamical system technique and the geometrical singular perturbation theory, Cantrell et al. [3] made a discussion on the coexistence of predators and prey of system (1.2) which occurred along a stable positive equilibrium. Ko and Ryu [4] discussed the existence, stability, and uniqueness of coexistence states and the extinction and permanence of a diffusive two-competing-prey and one-predator system with Beddington-DeAngelis functional response. Chen et al. [5] analyzed the extinction of the predator and the global asymptotic stability of the boundary solution of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response. Huo et al. [6] studied the global existence of positive periodic solutions for a delayed predator-prey model with the Beddington-DeAngelis functional response. For more knowledge about this topic, one can see [7, 8].

Recently, Song and Li [9] investigated the following two-prey one-predator model, where two prey are competitive and the predator has Holling functional II functional response:𝑑𝑥₁𝑑𝑡=𝑥₁(𝑡)𝑏₁−𝑥₁(𝑡)−𝛼𝑥₂(𝑡)−𝜂𝑥₃(𝑡)1+𝜔₁𝑥₁(𝑡),𝑑𝑥₂𝑑𝑡=𝑥₂(𝑡)𝑏₂−𝑥₂(𝑡)−𝛽𝑥₁(𝑡)−𝜇𝑥₃(𝑡)1+𝜔₂𝑥₂(𝑡),𝑑𝑥₃𝑑𝑡=𝑥₃(𝑡)−𝑏₃+𝑑𝜂𝑥₁(𝑡)1+𝜔₁𝑥₁(𝑡)+𝑑𝜇𝑥₂(𝑡)1+𝜔₂𝑥₂(𝑡), (1.3) where 𝑥₁(𝑡) and 𝑥₂(𝑡) stand for the population size of prey (pest) species and 𝑥₃(𝑡) is the population size of the predator (natural species) species; 𝑏_𝑖>0(𝑖=1,2,3) are intrinsic rates of increase or decrease, 𝛼>0 and 𝛽>0 are parameters representing competitive effects between two prey, 𝜂>0, 𝜇>0 are positive constants, 𝜂𝑥₁(𝑡)/(1+𝜔₁𝑥₁(𝑡)) and 𝜇𝑥₂(𝑡)/(1+𝜔₂𝑥₂(𝑡)) are the Holling II functional responses. 𝑑>0 is the rate of conversing prey into predator.

To model mutual interference among predators, Beddington [10] andDeAngelis et al. [11] argued that the well-known Holling type II functional response will be replaced by the Beddington-DeAngelis functional responses which is similar to the Holling type II functional response but has an extra term 𝜈_𝑖𝑥₃(𝑖=1,2) in the denominator, then system (1.3) may be modified as the following predator-prey system with Beddington-DeAngelis functional responses:𝑑𝑥₁𝑑𝑡=𝑥₁(𝑡)𝑏₁−𝑥₁(𝑡)−𝛼𝑥₂(𝑡)−𝜂𝑥₃(𝑡)1+𝜔₁𝑥₁(𝑡)+𝜈₁𝑥₃(𝑡),𝑑𝑥₂𝑑𝑡=𝑥₂(𝑡)𝑏₂−𝑥₂(𝑡)−𝛽𝑥₁(𝑡)−𝜇𝑥₃(𝑡)1+𝜔₂𝑥₂(𝑡)+𝜈₂𝑥₃(𝑡),𝑑𝑥₃𝑑𝑡=𝑥₃(𝑡)−𝑏₃+𝑑𝜂𝑥₁(𝑡)1+𝜔₁𝑥₁(𝑡)+𝜈₁𝑥₃(𝑡)+𝑑𝜇𝑥₂(𝑡)1+𝜔₂𝑥₂(𝑡)+𝜈₂𝑥₃(𝑡), (1.4) where 𝑥₁(𝑡) and 𝑥₂(𝑡) stand for the population size of prey (pest) species and 𝑥₃(𝑡) is the population size of the predator  (natural species) species; 𝑏_𝑖>0(𝑖=1,2,3) are intrinsic rates of increase or decrease, 𝛼>0 and 𝛽>0 are parameters representing competitive effects between two prey, 𝜂>0, 𝜇>0 are positive constants, 𝜂𝑥₃(𝑡)/(1+𝜔₁𝑥₁(𝑡)+𝜈₁𝑥₃(𝑡)), 𝜇𝑥₃(𝑡)/(1+𝜔₂𝑥₂(𝑡)+𝜈₂𝑥₃(𝑡)), 𝜂𝑥₁(𝑡)/(1+𝜔₁𝑥₁(𝑡)+𝜈₁𝑥₃(𝑡)), and 𝜇𝑥₂(𝑡)/(1+𝜔₂𝑥₂(𝑡)+𝜈₂𝑥₃(𝑡)) are the Beddington-DeAngelis functional responses. 𝑑>0 is the rate of conversing prey into predator.

In the natural word, any biological and environmental parameters are naturally subject to fluctuation in time. The effect of a periodically varying environment is important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, assumptions of periodicity of parameters are a way of incorporating periodicity of the environment, such as seasonal effect of weather, food supplies, and mating habits [12]. Based on the point of view, the modification of (1.4) according to the environmental variation is the nonautonomous differential equations𝑑𝑥₁𝑑𝑡=𝑥₁(𝑡)𝑏₁(𝑡)−𝑥₁(𝑡)−𝛼(𝑡)𝑥₂(𝑡)−𝜂(𝑡)𝑥₃(𝑡)1+𝜔₁(𝑡)𝑥₁(𝑡)+𝜈₁(𝑡)𝑥₃(𝑡),𝑑𝑥₂𝑑𝑡=𝑥₂(𝑡)𝑏₂(𝑡)−𝑥₂(𝑡)−𝛽(𝑡)𝑥₁(𝑡)−𝜇(𝑡)𝑥₃(𝑡)1+𝜔₂(𝑡)𝑥₂(𝑡)+𝜈₂(𝑡)𝑥₃(𝑡),𝑑𝑥₃𝑑𝑡=𝑥₃(𝑡)−𝑏₃(𝑡)+𝑑(𝑡)𝜂(𝑡)𝑥₁(𝑡)1+𝜔₁(𝑡)𝑥₁(𝑡)+𝜈₁(𝑡)𝑥₃(𝑡)+𝑑(𝑡)𝜇(𝑡)𝑥₂(𝑡)1+𝜔₂(𝑡)𝑥₂(𝑡)+𝜈₂(𝑡)𝑥₃(𝑡). (1.5) As is known to us, discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have nonoverlapping generations. Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulations. Therefore, it is reasonable and interesting to study discrete time systems governed by difference equations. Recently, there are some papers which deal with these topics see [13–22]. The principle object of this article is to propose a discrete analogue system (1.5) and explore its dynamics.

The remainder of the paper is organized as follows: in Section 2, with the help of differential equations with piecewise constant arguments, we first propose a discrete analogue of system (1.5), modelling the dynamics of time nonautonomous predator-prey system with Beddington-DeAngelis functional responses, where populations have nonoverlapping generations. In Section 3, based on the coincidence degree and the related continuation theorem, sufficient conditions for the existence of positive solutions of the model are obtained.

2. A Discrete Version of Model (1.5)

There are several different ways of deriving discrete time version of dynamical systems corresponding to continuous time formulations. One of the ways of deriving difference equations modelling the dynamics of populations with nonoverlapping generations that we will use in the following is based on appropriate modifications of models with overlapping generations. For more details about the approach, we refer to [15, 18].

In the following, we will discrete the system (1.5). Assume that the average growth rates in system (1.5) change at regular intervals of time, then we can obtain the following modified system:1𝑥₁(𝑡)𝑥̇₁(𝑡)=𝑏₁([𝑡])−𝑥₁([𝑡])−𝛼([𝑡])𝑥₂𝑡−𝜂𝑡𝑥([])([])₃([𝑡])1+𝜔₁([𝑡])𝑥₁([𝑡])+𝜈₁([𝑡])𝑥₃𝑡,1([])𝑥₂(𝑡)𝑥̇₂(𝑡)=𝑏₂([𝑡])−𝑥₂([𝑡])−𝛽([𝑡])𝑥₁𝑡−𝜇𝑡𝑥([])([])₃([𝑡])1+𝜔₂([𝑡])𝑥₂([𝑡])+𝜈₂([𝑡])𝑥₃𝑡,1([])𝑥₃(𝑡)𝑥̇₃(𝑡)=−𝑏₃𝑡+𝑑𝑡𝜂𝑡𝑥([])([])([])₁([𝑡])1+𝜔₁([𝑡])𝑥₁([𝑡])+𝜈₁([𝑡])𝑥₃𝑡+𝑑𝑡𝜇𝑡𝑥([])([])([])₂([𝑡])1+𝜔₂([𝑡])𝑥₂([𝑡])+𝜈₂([𝑡])𝑥₃𝑡,([]) (2.1) where [𝑡] denotes the integer part of 𝑡, 𝑡∈(0,+∞) and 𝑡≠0,1,2,…. Equations of type (2.1) are known as differential equations with piecewise constant arguments, and these equations occupy a position midway between differential equations and difference equations. By a solution of (2.1), we mean a function 𝑥=(𝑥₁,𝑥₂,𝑥₃)^𝑇, which is defined for 𝑡∈[0,+∞) and has the following properties:(1)𝑥 is continuous on [0,+∞);(2)the derivative 𝑑𝑥/𝑑𝑡=(𝑑𝑥₁(𝑡)/𝑑𝑡,𝑑𝑥₂(𝑡)/𝑑𝑡,𝑑𝑥₃(𝑡)/𝑑𝑡)^𝑇 exists at each point 𝑡∈[0,+∞) with the possible exception of the points 𝑡∈{0,1,2,…}, where left-sided derivative exists;(3)The equations in (2.1) are satisfied on each interval [𝑘,𝑘+1) with 𝑘=0,1,2,….

We integrate (2.1) on any interval of the form [𝑘,𝑘+1), 𝑘=0,1,2,…, and obtain for 𝑘≤𝑡<𝑘+1, 𝑘=0,1,2,…,𝑥(𝑡)=𝑥(𝑘)exp𝑏₁(𝑘)−𝑥₁(𝑘)−𝛼(𝑘)𝑥₂(𝑘)−𝜂(𝑘)𝑥₃(𝑘)1+𝜔₁(𝑘)𝑥₁(𝑘)+𝜈₁(𝑘)𝑥₃(𝑘),𝑦(𝑡)(𝑘)𝑏(𝑡−𝑘)=𝑦exp₂(𝑘)−𝑥₂(𝑘)−𝛽(𝑘)𝑥₁(𝑘)−𝜇(𝑘)𝑥₃(𝑘)1+𝜔₂(𝑘)𝑥₂(𝑘)+𝜈₂(𝑘)𝑥₃(𝑘),𝑧(𝑡)(𝑘)(𝑡−𝑘)=𝑧exp−𝑏₃(𝑘)+𝑑(𝑘)𝜂(𝑘)𝑥₁(𝑘)1+𝜔₁(𝑘)𝑥₁(𝑘)+𝜈₁(𝑘)𝑥₃(𝑘)+𝑑(𝑘)𝜇(𝑘)𝑥₂(𝑘)1+𝜔₂(𝑘)𝑥₂(𝑘)+𝜈₂(𝑘)𝑥₃(𝑘)(𝑡−𝑘). (2.2)

Let 𝑡→𝑘+1, then (2.2) takes the following form:𝑥(𝑘+1)=𝑥(𝑘)exp𝑏₁(𝑘)−𝑥₁(𝑘)−𝛼(𝑘)𝑥₂(𝑘)−𝜂(𝑘)𝑥₃(𝑘)1+𝜔₁(𝑘)𝑥₁(𝑘)+𝜈₁(𝑘)𝑥₃(𝑘),𝑦(𝑘+1)=𝑦(𝑘)exp𝑏₂(𝑘)−𝑥₂(𝑘)−𝛽(𝑘)𝑥₁(𝑘)−𝜇(𝑘)𝑥₃(𝑘)1+𝜔₂(𝑘)𝑥₂(𝑘)+𝜈₂(𝑘)𝑥₃(𝑘),𝑧(𝑘+1)=𝑧(𝑘)exp−𝑏₃(𝑘)+𝑑(𝑘)𝜂(𝑘)𝑥₁(𝑘)1+𝜔₁(𝑘)𝑥₁(𝑘)+𝜈₁(𝑘)𝑥₃(𝑘)+𝑑(𝑘)𝜇(𝑘)𝑥₂(𝑘)1+𝜔₂(𝑘)𝑥₂(𝑘)+𝜈₂(𝑘)𝑥₃(𝑘), (2.3) which is a discrete time analogue of system (1.5), where 𝑘=0,1,2,….

3. Existence of Positive Periodic Solutions

For convenience and simplicity in the following discussion, we always use the notations below throughout the paper:𝐼_𝜔∶={0,1,2,…,𝜔−1},1𝑓∶=𝜔^𝜔−1_𝑘=0𝑓(𝑘),𝑓^𝐿=min_{𝑘∈𝐼𝜔}𝑓(𝑘),𝑓^𝑀=max_{𝑘∈𝐼𝜔}𝑓(𝑘), (3.1) where 𝑓(𝑘) is an 𝜔-periodic sequence of real numbers defined for 𝑘∈𝑍. For system (2.3), we always assume that(H1)𝑏_𝑖(𝑖=1,2,3), 𝑑, 𝛼,𝛽, 𝜂, 𝜇, 𝜔_𝑗, 𝜈_𝑗(𝑗=1,2)∶𝑍→𝑅⁺ are 𝜔-periodic.

In order to explore the existence of positive periodic solutions of (2.3) and for the reader’s convenience, we will first summarize below a few concepts and results without proof, borrowing from [23].

Let 𝑋,𝑌 be normed vector spaces, 𝐿∶Dom𝐿⊂𝑋→𝑌 is a linear mapping, 𝑁∶𝑋→𝑌 is a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=codimIm𝐿<+∞ and Im𝐿 is closed in 𝑌. If 𝐿 is a Fredholm mapping of index zero and there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑌→𝑌 such that Im𝑃=Ker𝐿,Im𝐿=Ker𝑄=Im(𝐼−𝑄), it follows that 𝐿|Dom𝐿∩Ker𝑃∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. We denote the inverse of that map by 𝐾_𝑃. If Ω is an open bounded subset of 𝑋, the mapping 𝑁 will be called 𝐿 compact on Ω if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact. Since Im𝑄 is isomorphic to Ker𝐿, there exist isomorphisms 𝐽∶Im𝑄→Ker𝐿.

Lemma 3.1 (see Continuation Theorem [23]). Let 𝐿 be a Fredholm mapping of index zero, and let 𝑁 be 𝐿 compact on Ω. Suppose that, (a)for each 𝜆∈(0,1), every solution 𝑥 of 𝐿𝑥=𝜆𝑁𝑥 is such that 𝑥∉𝜕Ω;(b)𝑄𝑁𝑥≠0 for each 𝑥∈Ker𝐿∩𝜕Ω, and deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0,then the equation 𝐿𝑥=𝑁𝑥 has at least one solution lying in Dom𝐿∩Ω.

Lemma 3.2 (see [15]). Let 𝑔∶𝑍→𝑅 be 𝜔-periodic, that is, 𝑔(𝑘+𝜔)=𝑔(𝑘). Then for any fixed 𝑘₁,𝑘₂∈𝐼_𝜔 and any 𝑘∈𝑍, one has 𝑔(𝑘)≤𝑔𝑘₁+^𝜔−1_𝑠=0𝑔()(𝑠,𝑔(𝑘)𝑘𝑠+1−𝑔)≥𝑔₂−^𝜔−1_𝑠=0𝑔(𝑠+1)−𝑔(𝑠). (3.2)

Define𝑙₃=𝑢={𝑢(𝑘)}∶𝑢(𝑘)∈𝑅³,𝑘∈𝑍. (3.3)

For 𝑎=(𝑎₁,𝑎₂,𝑎₃)^𝑇∈𝑅³, define |𝑎|=max{|𝑎₁|,|𝑎₂|,|𝑎₃|}. Let 𝑙^𝜔⊂𝑙₃ denote the subspace of all 𝜔-periodic sequences equipped with the usual supremum norm ‖⋅‖, that is, ‖𝑢‖=max_{𝑘∈𝐼𝜔}|𝑢(𝑘)|, for any 𝑢={𝑢(𝑘)∶𝑘∈𝑍}∈𝑙^𝜔. It is easy to show that 𝑙_𝜔 is a finite-dimensional Banach space.

Let𝑙^𝜔₀=𝑢={𝑢(𝑘)}∈𝑙^𝜔∶^𝜔−1_𝑘=0𝑢(𝑘),𝑙=0_𝜔𝑐=𝑢={𝑢(𝑘)}∈𝑙^𝜔∶𝑢(𝑘)=ℎ∈𝑅³,𝑘∈𝑍. (3.4)

Then it follows that 𝑙^𝜔₀ and 𝑙^𝜔_𝑐 are both closed linear subspaces of 𝑙^𝜔 and𝑙^𝜔=𝑙^𝜔₀+𝑙^𝜔_𝑐,dim𝑙^𝜔_𝑐=3. (3.5)

In the following, we will be ready to establish our result.

Theorem 3.3. Let 𝑆₁, 𝑆^′₁, 𝑆₂, 𝑆^′₂, 𝑘₀, and ℎ₀ be defined by (3.30), (3.50), (3.34), (3.46), (3.39), and (3.55), respectively. In addition to the condition (H1), assume further that the following conditions:(𝐻2)𝑏₃>𝑚𝑎𝑥{𝑑𝜂exp{𝑆₁},𝑑𝜇exp{𝑆^′₂}},𝜂^𝐿>𝑏^𝐿₁𝜈^𝑀₁, (𝐻3)𝑘₀<0,ℎ₀<0hold. Then system (2.3) has at least an 𝜔-periodic solution.

Proof. Let 𝑢₁(𝑘)=ln[𝑥₁(𝑘)], 𝑢₂(𝑘)=ln[𝑥₂(𝑘)], and 𝑢₃(𝑘)=ln[𝑥₃(𝑘)]. Then (2.3) takes the form 𝑢₁(𝑘+1)−𝑢₁(𝑘)=𝑓₁(𝑘),𝑢₂(𝑘+1)−𝑢₂(𝑘)=𝑓₂(𝑘),𝑢₃(𝑘+1)−𝑢₃(𝑘)=𝑓₃(𝑘), (3.6) where 𝑓₁(𝑘)=𝑏₁(𝑘)−exp𝑢₁(𝑘)−𝛼(𝑘)exp𝑢₂(𝑘−𝜂(𝑘)𝑢)exp₃(𝑘)1+𝜔₁(𝑘)exp𝑢₁(𝑘)+𝜈₁(𝑘)exp𝑢₃(𝑘,𝑓₂)(𝑘)=𝑏₂(𝑘)−exp𝑢₂(𝑘)−𝛽(𝑘)exp𝑢₁(𝑘−𝜇(𝑘)𝑢)exp₃(𝑘)1+𝜔₂(𝑘)exp𝑢₂(𝑘)+𝜈₂(𝑘)exp𝑢₃(𝑘,𝑓₃)(𝑘)=−𝑏₃(𝑘)+𝑑(𝑘)𝜂(𝑘)exp𝑢₁(𝑘)1+𝜔₁(𝑘)exp𝑢₁(𝑘)+𝜈₁(𝑘)exp𝑢₃(𝑘+𝑑(𝑘)𝜇(𝑘)𝑢)exp₂(𝑘)1+𝜔₂(𝑘)exp𝑢₂(𝑘)+𝜈₂(𝑘)exp𝑢₃(𝑘.) (3.7) Let 𝑋=𝑌=𝑙^𝜔, (𝑘)()(𝑘)=𝑢𝐿𝑢)(=𝑢𝑘+1−𝑢¹(𝑘+1)−𝑢₁(𝑘)𝑢₂(𝑘+1)−𝑢₂(𝑘)𝑢₃(𝑘+1)−𝑢₃(𝑘)(𝑘)=𝑓,𝑁𝑢)(¹(𝑘)𝑓₂(𝑘)𝑓₃(𝑘), (3.8)(3.9)^𝜔−1_𝑘=0𝑢₃(𝑘+1)−𝑢₃(𝑘)≤2𝑏₃𝜔. (3.18) In view of the hypothesis that 𝑢={𝑢(𝑘)}∈𝑋, there exist 𝜉_𝑖,𝜂_𝑖∈𝐼_𝜔 such that 𝑢_𝑖𝜉_𝑖=min_{𝑘∈𝐼𝜔}𝑢_𝑖(𝑘),𝑢_𝑖𝜂_𝑖=max_{𝑘∈𝐼𝜔}𝑢_𝑖(𝑘)(𝑖=1,2,3). (3.19) Then it is obvious that ∇𝑢_𝑖𝜂_𝑖≤0,∇𝑢_𝑖𝜉_𝑖≥0,𝑖=1,2,3, (3.20) where ∇ denotes the forward difference operator ∇𝑢(𝑘)=𝑢(𝑘+1)−𝑢(𝑘).
In view of (3.8), we get𝑏₁𝜉₁−exp𝑢₁𝜉₁−𝛼(𝑘)exp𝑢₂𝜉₁−𝜂𝜉₁exp𝑢₃𝜉₁1+𝜔₁𝜉₁exp𝑢₁𝜉₁+𝜈₁𝜉₁exp𝑢₃𝜉₁𝑏≥0₂𝜉₂−exp𝑢₂𝜉₂−𝛽𝜉₂exp𝑢₁𝜉₂−𝜇𝜉₂exp𝑢₃𝜉₂1+𝜔₂𝜉₂exp𝑢₂𝜉₂+𝜈₂𝜉₂exp𝑢₃𝜉₂≥0 (3.21)(3.22) Then we have exp𝑢₁𝜉₁<𝑏₁𝜉₁,exp𝑢₂𝜉₂<𝑏₂𝜉₂, (3.23) which leads to 𝑢₁𝜉₁<ln𝑏₁𝜉₁=ln𝑏^𝐿₁,𝑢₂𝜉₂<ln𝑏₂𝜉₂=ln𝑏^𝐿₂ (3.24) From (3.16), we have ^𝜔−1_𝑘=0𝑑(𝑘)𝜂(𝑘)exp𝑢₁𝜂₁+𝑑(𝑘)𝜇(𝑘)exp𝑢₂𝜂₂>𝑏₃𝜔, (3.25) that is, 𝑑𝜂exp𝑢₁𝜂₁+𝑑𝜇exp𝑢₂𝜂₂>𝑏₃ (3.26) In the sequel, we consider two cases.Case 1. If 𝑢₁(𝜂₁)≥𝑢₂(𝜂₂), then it follows from (3.26) that 𝑑𝜂exp𝑢₁𝜂₁+𝑑𝜇exp𝑢₁𝜂₁>𝑏₃, (3.27) which leads to 𝑢₁𝜂₁>ln𝑏₃𝑑(𝜂+𝜇). (3.28) In view of Lemma 3.2, (3.16), (3.24), and (3.28), we have 𝑢₁(𝑘)≤𝑢₁𝜉₁+^𝜔−1_𝑠=0𝑢₁(𝑠+1)−𝑢₁(𝑠𝑏)≤ln^𝐿₁+2𝑏₁𝜔∶=𝑏₁,𝑢₁(𝑘)≥𝑢₁𝜂₁−^𝜔−1_𝑠=0𝑢₁(𝑠+1)−𝑢₁(𝑠)≥ln𝑏₃𝑑(𝜂+𝜇)−2𝑏₁𝜔∶=𝐵₁. (3.29) It follows from (3.29) that max_{𝑘∈𝐼𝜔}𝑢₁(𝑘)<𝑚𝑎𝑥𝑏₁,𝐵₁∶=𝑆₁. (3.30) By (3.26), it is easy to obtain that 𝑑𝜂exp𝑆₁+𝑑𝜇exp𝑢₂𝜂₂>𝑏₃. (3.31) Hence, 𝑢₂𝜂₂>ln𝑏₃−𝑑𝜂exp𝑆₁.𝑑𝜇 (3.32) In view of Lemma 3.2, (3.18), (3.24), and (3.32), we have 𝑢₂(𝑘)≤𝑢₂𝜉₂+^𝜔−1_𝑠=0𝑢₂(𝑠+1)−𝑢₂(𝑠𝑏)≤ln^𝐿₂+2𝑏₂𝜔∶=𝑏₂,𝑢₂(𝑘)≥𝑢₂𝜂₁−^𝜔−1_𝑠=0𝑢₂(𝑠+1)−𝑢₂(𝑠)≥ln𝑏₃−𝑑𝜂exp𝑆₁𝑑𝜇−2𝑏₂𝜔∶=𝐵₂. (3.33) It follows from (3.33) that max_{𝑘∈𝐼𝜔}𝑢₂(𝑘)<𝑚𝑎𝑥𝑏₂,𝐵₂∶=𝑆₂. (3.34) Noticing that 𝜂(𝜉₁)exp{𝑢₃}/(1+𝜔₁(𝜉₁)exp{𝑢₁(𝜉₁)}+𝜈₁(𝜉₁)exp{𝑢₃}) is an increasing function with respect to 𝑢₃, from (3.21), we have 𝑏^𝐿₁=𝑏₁𝜉₁>𝜂𝜉₁exp𝑢₃𝜉₁1+𝜔₁𝜉₁exp𝑢₁𝜉₁+𝜈₁𝜉₁exp𝑢₃𝜉₁≥𝜂^𝐿exp𝑢₃𝜉₃1+𝜔^𝑀₁exp𝑆₁+𝜈^𝑀₁exp𝑢₃𝜉₃. (3.35) Hence, 𝑢₃𝜉₃𝑏<ln^𝐿11+𝜔^𝑀₁exp𝑆₁𝜂^𝐿−𝑏^𝐿₁𝜈^𝑀₁. (3.36) From (3.16), we get ^𝜔−1_𝑘=0𝑑(𝑘)𝜂(𝑘)exp−𝑆₁1+𝜔^𝑀₁exp𝑆₁+𝜈^𝑀₁exp𝑢₃𝜂₃+𝑑(𝑘)𝜇(𝑘)exp−𝑆₂1+𝜔^𝑀₂exp𝑆₂+𝜈^𝑀₂exp𝑢₃𝜂₃≤𝑏₃𝜔. (3.37) Then, 𝑘₂exp2𝑢₃𝜂₃+𝑘₁exp𝑢₃𝜂₃+𝑘₀≥0, (3.38) where 𝑘₀=𝑏₃1+𝜔^𝑀₁exp𝑆₁1+𝜔^𝑀₂exp𝑆₂−𝑑𝜂exp−𝑆₁1+𝜔^𝑀₂exp𝑆₂−𝑑𝜇exp−𝑆₂1+𝜔^𝑀₁exp𝑆₁,𝑘₁=𝑏₃𝜈^𝑀₂1+𝜔^𝑀₁exp𝑆₁+𝜈^𝑀₁1+𝜔^𝑀₂exp𝑆₂−𝑑𝜂𝜈^𝑀₂exp−𝑆₁+𝑑𝜇𝜈^𝑀₁exp−𝑆₂,𝑘₂=𝑏₃𝜈^𝑀₁𝜈^𝑀₂. (3.39) Thus, 𝑢₃𝜂₃≥ln−𝑘₁+𝑘²₁−𝑘₂𝑘₀2𝑘₂. (3.40) In view of Lemma 3.2, (3.18), (3.36), and (3.40), we have 𝑢₃(𝑘)≤𝑢₃𝜉₃+^𝜔−1_𝑠=0𝑢₃(𝑠+1)−𝑢₃(𝑠𝑏)≤ln^𝐿11+𝜔^𝑀₁exp𝑆₁𝜂^𝐿−𝑏^𝐿₁𝜈^𝑀₁+2𝑟𝜔∶=𝑏₃,𝑢₃(𝑘)≥𝑢₃𝜂₃−^𝜔−1_𝑠=0𝑢₃(𝑠+1)−𝑢₃(𝑠)≥ln−𝑘₁+𝑘²₁−𝑘₂𝑘₀2𝑘₂−2𝑏₃𝜔∶=𝐵₃. (3.41) It follows from (3.41) that max_{𝑘∈𝐼𝜔}𝑢₃(𝑘)<𝑚𝑎𝑥𝑏₃,𝐵₃∶=𝑆₃. (3.42)Case 2. If 𝑢₁(𝜂₁)<𝑢₂(𝜂₂), then it follows from (3.26) that 𝑑𝜂exp𝑢₂𝜂₂+𝑑𝜇exp𝑢₂𝜂₂>𝑏₃, (3.43) which leads to 𝑢₂𝜂₂>ln𝑏₃𝑑(𝜂+𝜇). (3.44) In view of Lemma 3.2, (3.17), (3.24), and (3.44), we have 𝑢₂(𝑘)≤𝑢₂𝜉₂+^𝜔−1_𝑠=0𝑢₂(𝑠+1)−𝑢₂(𝑠𝑏)≤ln^𝐿₂+2𝑏₂𝜔∶=𝑏^′₂,𝑢₂(𝑘)≥𝑢₂𝜂₂−^𝜔−1_𝑠=0𝑢₂(𝑠+1)−𝑢₂(𝑠)≥ln𝑏₃𝑑(𝜂+𝜇)−2𝑏₂𝜔∶=𝐵^′₂. (3.45) Combining both equation of (3.45), one obtains max_{𝑘∈𝐼𝜔}𝑢₂(𝑘)<𝑚𝑎𝑥𝑏^′₂,𝐵^′₂∶=𝑆^′₂. (3.46) By (3.26), it is easy to obtain that 𝑑𝜂exp𝑢₁𝜂₁+𝑑𝜇exp𝑆^′₂>𝑏₃. (3.47) Hence, 𝑢₁𝜂₁>ln𝑏₃−𝑑𝜇exp𝑆^′₂.𝑑𝜂 (3.48) By virtue of Lemma 3.2, (3.16), (3.24), and (3.48), we have 𝑢₁(𝑘)≤𝑢₁𝜉₁+^𝜔−1_𝑠=0𝑢₁(𝑠+1)−𝑢₁(𝑠𝑏)≤ln^𝐿₁+2𝑏₂𝜔∶=𝑏^′₁,𝑢₁(𝑘)≥𝑢₁𝜂₁−^𝜔−1_𝑠=0𝑢₁(𝑠+1)−𝑢₁(𝑠)≥ln𝑏₃−𝑑𝜇exp𝑆^′₂𝑑𝜂−2𝑏₁𝜔∶=𝐵^′₁. (3.49) It follows from (3.49) that max_{𝑘∈𝐼𝜔}𝑢₁(𝑘)<𝑚𝑎𝑥𝑏^′₁,𝐵^′₁∶=𝑆^′₁. (3.50) Considering that 𝜂(𝜉₁)exp{𝑢₃}/(1+𝜔₁(𝜉₁)exp{𝑢₁(𝜉₁)}+𝜈₁(𝜉₁)exp{𝑢₃}) is an increasing function with respect to 𝑢₃, from (3.21), we have 𝑏^𝐿₁=𝑏₁𝜉₁>𝜂𝜉₁exp𝑢₃𝜉₁1+𝜔₁𝜉₁exp𝑢₁𝜉₁+𝜈₁𝜉₁exp𝑢₃𝜉₁≥𝜂^𝐿exp𝑢₃𝜉₃1+𝜔^𝑀₁exp𝑆^′₁+𝜈^𝑀₁exp𝑢₃𝜉₃. (3.51) Therefore, 𝑢₃𝜉₃𝑏<ln^𝐿11+𝜔^𝑀₁exp𝑆^′₁𝜂^𝐿−𝑏^𝐿₁𝜈^𝑀₁. (3.52) From (3.16), we know that ^𝜔−1_𝑠=0𝑑(𝑘)𝜂(𝑘)exp−𝑆^′₁1+𝜔^𝑀₁exp𝑆^′₁+𝜈^𝑀₁exp𝑢₃𝜂₃+𝑑(𝑘)𝜇(𝑘)exp−𝑆^′₂1+𝜔^𝑀₂exp𝑆^′₂+𝜈^𝑀₂exp𝑢₃𝜂₃≤𝑏₃𝜔. (3.53) Then, ℎ₂exp2𝑢₃𝜂₃+ℎ₁exp𝑢₃𝜂₃+ℎ₀≥0, (3.54) where ℎ₀=𝑏₃1+𝜔^𝑀₁exp𝑆^′₁1+𝜔^𝑀₂exp𝑆^′₂−𝑑𝜂exp−𝑆^′₁1+𝜔^𝑀₂exp𝑆^′₂−𝑑𝜇exp−𝑆^′₂1+𝜔^𝑀₁exp𝑆^′₁,ℎ₁=𝑏₃𝜈^𝑀₂1+𝜔^𝑀₁exp𝑆^′₁+𝜈^𝑀₁1+𝜔^𝑀₂exp𝑆^′₂−𝑑𝜂𝜈^𝑀₂exp−𝑆^′₁+𝑑𝜇𝜈^𝑀₁exp−𝑆^′₂,ℎ₂=𝑏₃𝜈^𝑀₁𝜈^𝑀₂. (3.55) Thus, 𝑢₃𝜂₃≥ln−ℎ₁+ℎ²₁−ℎ₂ℎ₀2ℎ₂. (3.56) In view of Lemma 3.2, (3.18), (3.52), and (3.56), we have 𝑢₃(𝑘)≤𝑢₃𝜉₃+^𝜔−1_𝑠=0𝑢₃(𝑠+1)−𝑢₃(𝑠𝑏)≤ln^𝐿11+𝜔^𝑀₁exp𝑆₁𝜂^𝐿−𝑏^𝐿₁𝜈^𝑀₁+2𝑟𝜔∶=𝑏^′₃,𝑢₃(𝑘)≥𝑢₃𝜂₃−^𝜔−1_𝑠=0𝑢₃(𝑠+1)−𝑢₃(𝑠)≥ln−𝑘₁+𝑘²₁−𝑘₂𝑘₀2𝑘₂−2𝑏₃𝜔∶=𝐵^′₃. (3.57) It follows from (3.57) that max_{𝑘∈𝐼𝜔}𝑢₃(𝑘)<𝑚𝑎𝑥𝑏^′₃,𝐵^′₃∶=𝑆^′₃. (3.58) Obviously, 𝑆₁, 𝑆₂, 𝑆^′₂, 𝑆₃, and 𝑆^′₃ are independent of 𝜆∈(0,1). Take 𝑀=𝑆₁+max{𝑆₂,𝑆^′₂}+max{𝑆₃,𝑆^′₃}+𝑀₀, where 𝑀₀ is taken sufficiently large such that each solution (𝑢^∗₁,𝑢^∗₂,𝑢^∗₃)^𝑇 of the following algebraic equations: 𝑏₁−exp𝑢₁−𝛼exp𝑢₂−1𝜔^𝜔−1_𝑘=0𝜂(𝑘)exp𝑢₃1+𝜔₁(𝑘)exp𝑢₁+𝜈₁(𝑘)exp𝑢₃=0,𝑏₂−exp𝑢₂−𝛽exp𝑢₁−1𝜔^𝜔−1_𝑘=0𝜇(𝑘)exp𝑢₃1+𝜔₂(𝑘)exp𝑢₂+𝜈₂(𝑘)exp𝑢₃−=0,𝑏₃+1𝜔^𝜔−1_𝑘=0𝑑(𝑘)𝜂(𝑘)exp𝑢₁1+𝜔₁(𝑘)exp𝑢₁+𝜈₁(𝑘)exp𝑢₃+1𝜔^𝜔−1_𝑘=0𝑑(𝑘)𝜇(𝑘)exp𝑢₂1+𝜔₂(𝑘)exp𝑢₂+𝜈₂(𝑘)exp𝑢₃=0 (3.59) satisfies max{|𝑢^∗₁|,|𝑢^∗₂|,|𝑢^∗₃|}<𝑀₀.Now we have proved that any solution 𝑢={𝑢(𝑘)}={(𝑢₁(𝑘),𝑢₂(𝑘),𝑢₃(𝑘))^𝑇} of (3.6) in 𝑋 satisfies ‖𝑢‖<𝑀,𝑘∈𝑍.
Let Ω∶={𝑢={𝑢(𝑘)}∈𝑋∶‖𝑢‖<𝑀}. Then it is easy to see that Ω is an open, bounded set in 𝑋 and verifies requirement (a) of Lemma 3.1. When 𝑢∈𝜕Ω∩Ker𝐿, 𝑢={(𝑢₁,𝑢₂,𝑢₃)^𝑇} is a constant vector in 𝑅³ with ‖𝑢‖=max{|𝑢₁|,|𝑢₂|,|𝑢₃|}=𝑀. ThenQ𝑁𝑢=𝑏₁−exp𝑢₁−𝛼exp𝑢₂−1𝜔^𝜔−1_𝑘=0𝜂(𝑘)exp𝑢₃1+𝜔₁(𝑘)exp𝑢₁+𝜈₁(𝑘)exp𝑢₃𝑏₂−exp𝑢₂−𝛽exp𝑢₁−1𝜔^𝜔−1_𝑘=0𝜇(𝑘)exp(𝑧)1+𝜔₂(𝑘)exp𝑢₂+𝜈₂(𝑘)exp𝑢₃−𝑏₃+1𝜔^𝜔−1_𝑘=0𝑑(𝑘)𝜂(𝑘)exp𝑢₁1+𝜔₁(𝑘)exp𝑢₁+𝜈₁(𝑘)exp𝑢₃+1𝜔^𝜔−1_𝑘=0𝑑(𝑘)𝜇(𝑘)exp𝑢₂1+𝜔₂(𝑘)exp𝑢₂+𝜈₂(𝑘)exp𝑢₃≠0. (3.60) Define the homotopy 𝜙∶Dom𝐿×[0,1]→𝑋 by 𝜙(𝑢₁,𝑢₂,𝑢₃,𝜇)=𝜇𝑄𝑁𝑢+(1−𝜇)𝐺𝑢, 𝜇∈[0,1], where 1𝐺𝑢=𝜔^𝜔−1_𝑘=0𝑏₁−exp𝑢₁𝑏₂−exp𝑢₂−𝑏₃+1𝜔^𝜔−1_𝑘=0𝑑(𝑘)𝜂(𝑘)exp𝑢₁1+𝜔₁(𝑘)exp𝑢₁+𝜈₁(𝑘)exp𝑢₃. (3.61) Let 𝐽 be the identity mapping. According to the definition of topology, direct calculation yields deg𝐽𝑄𝑁𝑢₁,𝑢₂,𝑢₃^𝑇𝑢;Ω∩ker𝐿;0=deg𝑄𝑁₁,𝑢₂,𝑢₃^𝑇𝑢;Ω∩Ker𝐿;0=deg𝑄𝑁₁,𝑢₂,𝑢₃,1^𝑇𝑢;Ω∩Ker𝐿;0=deg𝑄𝑁₁,𝑢₂,𝑢₃,0^𝑇𝑢;Ω∩Ker𝐿;0=signdetexp^∗₁𝑢000exp^∗₂0𝑓₁𝑢^∗0𝑓₂𝑢^∗, (3.62) where 𝑓₁𝑢^∗=1𝜔^𝜔−1_𝑘=0𝑑(𝑘)𝜂(𝑘)exp𝑢^∗₁1+𝜈₁(𝑘)exp𝑢^∗₃1+𝜔₁(𝑘)exp𝑢^∗₁+𝜈₁(𝑘)exp𝑢^∗₃²𝑓>0,₂𝑢^∗1=−𝜔^𝜔−1_𝑘=0𝑑(𝑘)𝜂(𝑘)𝜈₁(𝑘)exp𝑢^∗₁+𝑢^∗₃1+𝜔₁(𝑘)exp𝑢^∗₁+𝜈₁(𝑘)exp𝑢^∗₃²<0. (3.63) Then, it follows from (3.62) that deg𝐽𝑄𝑁𝑢₁,𝑢₂,𝑢₃^𝑇;Ω∩ker𝐿;0=sign−exp𝑢₁+𝑢^∗₂𝑓₂𝑢^∗=−1<0. (3.64) By now, we have proved that Ω verifies all requirements of Lemma 3.1. Then it follows that 𝐿𝑢=𝑁𝑢 has at least one solution in Dom𝐿∩Ω, that is to say, (3.6) has at least one 𝜔-periodic solution in Dom𝐿∩Ω, say 𝑢^∗={𝑢^∗(𝑘)}={(𝑢^∗₁(𝑘),𝑢^∗₂(𝑘),𝑢^∗₃(𝑘))^𝑇}. Let 𝑥^∗₁(𝑘)=exp{𝑢^∗₁(𝑘)},𝑥^∗₂(𝑘)=exp{𝑢^∗₂(𝑘)}, and 𝑥^∗₃(𝑘)=exp{𝑢^∗₃(𝑘)}. Then we know that (𝑥^∗₁(𝑘),𝑥^∗₂(𝑘),𝑥^∗₃(𝑘))^𝑇 is an 𝜔-periodic solution of system (2.3) with strictly positive components. We complete the proof.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 10961008), the Scientific Research Fund of Hunan Provincial Education Department (no. 10C0560), and Doctoral Foundation of Guizhou College of Finance and Economics (2010).