Existence of positive periodic solution of n-dimensional Lotka–Volterra system with delays

Abstract

With the help of a continuation theorem based on coincidence degree, easily verifiable criteria are established for the existence of positive periodic solutions of n-D Lotka–Volterra system with delays. And sufficient conditions are obtained for the persistence of the delay system. Finally, computer simulations are presented to illustrate the conclusions.

Introduction

Recently, more and more researchers have been studying the periodic solution of Lotka–Volterra system with delay owing to the application of the functional differential equation theory in math-biology and they obtained some results [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Wright [1] and Kuang [2], respectively, researched into the single population differential models with delays and acquired the sufficient conditions of global asymptotic stability of a positive equilibrium point. Goh [3] considered the sufficient condition of the global stability of a positive equilibrium point about the 2-D non-autonomous facultative mutualism system with delay. And the others, like Gopalsamy and He [4], he considered the sufficient condition of the global attractability of a positive equilibrium point about the 2-D autonomous facultative mutualism system with delay. Mukherjee [5] studied the persistence and global attractability of the 2-D autonomous facultative mutualism system with multiple delays. Lately, the persistence, boundedness and global attractability of a positive equilibrium point of the non-autonomous facultative mutualism system with multiple delays has been studied by Yang and Jiang [6]. Also Yang et al. [7] considered the existence of periodic positive solution of the non-autonomous facultative mutualism system with multiple delays, and Wang and Wang [8] discussed the existence of periodic positive solution of 2-D Lotka–Volterra system with multiple delays.

However, the variation of the environment plays an important role in many biological and ecological systems. In particular, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, the assumption of periodicity of the parameters in a way incorporates the periodicity of the environment (e.g. seasonal effects of weather condition, food supplies, temperature, mating habits, harvesting etc.). In view of this, it is realistic to assume that the parameters in the models are periodic functions of period ω. Therefore, we discuss the following ns-D non-autonomous Lotka–Volterra system with multiple delays,

$$x_i^{\prime }(t)=x_i(t)\left(r_i(t)-a_{\mathit{ii}}(t)x_i(t-{\tau }_{\mathit{ii}}(t))+\sum _{{\displaystyle \genfrac{}{}{0ex}{}{j=1}{j\ne i}}}^n{a}_{\mathit{ij}}(t)x_j(t-{\tau }_{\mathit{ij}}(t))\right)\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

where x_i(t) = φ_i(t) ⩾ 0, t ∈ [−τ, 0], τ = max_{i,j=1,2,…,n}{sup_t∈[0,ω]τ_ij(t)}, φ_i ∈ C([−τ, 0], R⁺), φ_i(0) > 0 and r_i(t), a_ij(t), τ_ij(t) are rigorous positive ω-periodic continuous bounded functions in R⁺.

The main purpose of this paper is to derive a set of easily verifiable sufficient conditions for the existence of positive periodic solution of (1), and the conditions of persistence. The method used here will be the coincidence degree theory developed by Gaines and Mawhin [12].

Definition 1

x^∗(t) is said to be a positive solution of system (1) if $x^{\ast }(t)=(x_1^{\ast }(t)\text{,}{x}_2^{\ast }(t)\text{,}\dots \text{,}{x}_n^{\ast }(t){)}^{\mathrm{T}}>0$ in R⁺.

Lemma 1

The domain $R^{{+}^n}=\{(x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_n{)}^T|x_i>0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\}$ is invariant with respect to (1).

In order to obtain the existence of a positive periodic solution of the system, we first make the following preparations.

Let X, Z be two real Banach space, Consider an operator equation

$$\mathit{Lx}=\lambda \mathit{Nx}\text{,}\lambda \in (0\text{,}1)\text{,}$$

where L : DomL ∩ X → Z is a linear operator and λ is a parameter. Let P and Q denote two projectors such that

$$P:X\cap \mathit{DomL}\to \mathit{KerL}\text{,}Q:Z\to Z/\mathrm{Im}L\text{.}$$

Denote by J : ImQ → KerL is an isomorphism of ImQ onto KerL. Recall that a linear operator L:DomL ∩ X → Z with KerL = L⁻¹(0) and ImL = L(DomL), will be called a Fredholm operator if the following two conditions hold: (I) KerL has a finite dimension; (II) ImL is closed and has a finite codimension.

We shall say that a mapping N is L compact on Ω if the mapping $\mathit{QN}:\overline{\Omega }\to Z$ is continuous, $\mathit{QN}(\overline{\Omega })$ is bounded, and $K_p(I-Q)N:\overline{\Omega }\to X$ is compact, i.e., it is continuous and $K_p(I-Q)N(\overline{\Omega })$ is relatively compact, where K_p : ImL → DomL ∩ KerP is an inverse of the restriction L_p of L to DomL ∩ KerP, so that LK_p = I and K_pL = I − P. In the sequel, we will use the following result of continuation theorem [12].

Lemma 2 Continuation Theorem

Let X, Z be two real Banach spaces and L a Fredholm mapping of index zero. Assume that $N:\overline{\Omega }\to Z$ is L compact on $\overline{\Omega }$ with Ω open bounded in X. Furthermore suppose,

• for each λ ∈ (0, 1), every solution x of Lx ≠ λNx is such that x ∈ DomL ∩ ∂Ω;

• QNx ≠ 0 for each x ∈ KerL ∩ ∂Ω and Brouwer–Degree,

  $$\mathrm{deg}\{\mathit{JQNx}\text{,}\Omega \cap \mathit{KerL}\text{,}0\}\ne 0\text{.}$$

Then operator equation Lx = Nx has at least one solution in $\mathit{DomL}\cap \overline{\Omega }$.