WANG Li-li,HU Meng

(School of Mathematics and Statistics,Anyang Normal University,Anyang 455000,China)

ALMOST PERIODIC SOLUTION FOR A DYNAMICAL EQUATION WITH ALLEE EFFECTS ON TIME SCALES

WANG Li-li,HU Meng

(School of Mathematics and Statistics,Anyang Normal University,Anyang 455000,China)

This paper is concerned with an equation representing dynamics of a renewable resource subjected to Allee eff ects on time scales.By using exponential dichotomy of linear system and contraction mapping fixed point theorem,suffi cient conditions are established for the existence of unique positive almost periodic solution.Moreover,by constructing a suitable Lyapunov functional,we obtain suffi cient conditions for the global exponential stability of the almost periodic solution.

dynamical equation;Allee eff ect;almost periodic solution;global exponential stability;time scale

1 Introduction

Mathematicalecologicalsystem became one ofthe most important topics in the study of modern applied mathematics.During the last decade,Allee effects received much attention from researchers,largely because oftheir potentialrole in extinctions ofalready endangered, rare or dramatically declining species.The Allee effect refers to a decrease in population growth rate at low population densities.There were several mechanisms that create Allee effects in populations;see,for example[1–6].

Mathematicalcomponent of the available literature deals with differential equations or difference equations.Notice that,in the realworld,there are many species whose developing processes are both continuous and discrete.Hence,using the only differential equation or difference equation can’t accurately describe the law of their developments[7,8].Therefore there is a need to establish correspondent dynamic models on new time scales.

A time scale isa nonempty arbitrary closed subset ofreals.The theory oftime scales was first introduced by Hilger in[9],in order to unify continuous and discrete analysis.The studyof dynamic equations on time scales can combine the continuous and discrete situations,it unifies not only differential and difference equations,but also some other problems such as a mix of stop-start and continuous behaviors.

Although seasonality is known to have considerable impact on the species dynamics, to our knowledge there were few papers discussed the dynamics of a renewable resource subjected to Allee effects in a seasonally varying environment.Moreover,ecosystems are often disturbed by outside continuous forces in the real world,the assumption of almost periodicity ofthe parameters is a way ofincorporating the almost periodicity ofa temporally nonuniform environment with incommensurable periods(nonintegral multiples).In this paper,we introduce seasonality into the resource dynamic equation by assuming the involved coeffi cients to be almost periodic.

Motivated by the above statements,in the present paper,we shall study the following equation representing dynamics of a renewable resource x,that is subjected to Allee effects on time scales

where t ∈ T,T is an almost time scale;a(t)represents time dependent intrinsic growth rate ofthe resource;the nonnegative functions c(t)and b(t)stand for seasonaldependent carrying capacity and threshold function of the species respectively satisfying 0 ＜ b(t) ＜ c(t).All the coeffi cients a(t),b(t),c(t)are continuous,almost periodic functions.For the ecological justification of(1.1),one can refer to[5,6].

For convenience,we introduce the notation

where f is a positive and bounded function.Throughout this paper,we assume that the coeffi cients of equation(1.1)satisfy

This is the fi rst paper to study an almost equation representing dynamics of a renewable resource subjected to Allee effects on time scales.The aim of this paper is,by using exponential dichotomy of linear system and contraction mapping fi xed point theorem,to obtain suffi cient conditions for the existence of unique positive almost periodic solution of (1.1).We also investigate globalexponentialstability ofthe unique almost periodic solution by means of Lyapunov function.

2 Preliminaries

Let us first recall some basic definitions which can be found in[10].

Let T be a nonempty closed subset(time scale)of R.The forward and backward jump operators σ,ρ :T → T and the graininess μ :T → R+are defi ned,respectively,by

A point t ∈ T is called left-dense if t ＞ inf T and ρ(t)=t,left-scattered if ρ(t) ＜ t, right-dense if t ＜ sup T and σ(t)=t,and right-scattered ifσ(t) ＞ t.If T has a left-scattered maximum m,then Tk=T{m};otherwise Tk=T.If T has a right-scattered minimum m, then Tk=T{m};otherwise Tk=T.

A function f:T → R is right-dense continuous provided it is continuous at right-dense point in T and its left-side limits exist at left-dense points in T.

A function p:T → R is called regressive provided 1+ μ(t)p(t)/=0 for all t ∈ Tk.The set of allregressive and rd-continuous functions p:T → R will be denoted by R=R(T,R). Define the set R+=R+(T,R)={p ∈ R:1+ μ(t)p(t) ＞ 0,? t ∈ T}.

If r is a regressive function,then the generalized exponential function eris defi ned by

for all s,t ∈ T,with the cylinder transformation

Lemma 2.1(see[10])If p ∈ R and a,b,c ∈ T,then

Defi nition 2.1[11]A time scale T is called an almost periodic time scale if

Defi nition 2.2[12]Let x ∈ Rn,and A(t)be an n × n rd-continuous matrix on T,the linear system

is said to admitan exponentialdichotomy on T ifthere exist positive constant k,α,projection P and the fundamental solution matrix X(t)of(2.1),satisfying

where|·|0is a matrix norm on

Considering the following almost periodic system

where A(t)is an almost periodic matrix function,f(t)is an almost periodic vector function.

Lemma 2.2(see[12])If the linear system(2.1)admits exponential dichotomy,then system(2.2)has a unique almost periodic solutions x(t)as follows

where X(t)is the fundamentalsolution matrix of(2.1).

Defi nition 2.3The almost periodic solution x?ofequation(1.1)is said to be exponentially stable,if there exist positive constants α ＞ 0,α ∈ R+and N=N(t0) ≥ 1 such that for any solution x of equation(1.1)satisfying

3 Main Results

Clearly,the trivial solution x(t) ≡ 0 is an almost periodic solution of(1.1).Since the study deals with resource dynamics,we are interested in the existence of positive almost periodic solutions of the considered equation.

First,we make the following assumptions:

(H1) ?abc ∈ R+;

(H2)there exist two positive constants L1＞ L2＞ 0,such that

Theorem 3.1Assume that(H1)–(H3)hold,then equation(1.1)has a unique almost periodic solution.

ProofLet Z={z|z ∈ C(T,R),z is an almost periodic function}with the norm‖z‖ =sup|z(t)|,then Z is a Banach space.

t∈T

For z ∈ Z,we consider the almost periodic solution xz(t)of the nonlinear almost periodic differentialequation

Since i

tn

∈Tf [a(t)b(t)c(t)]≥ alblcl＞ 0,from Lemma 2.15[12]and(H1),the linear equation

admits exponentialdichotomy on T.

Hence by Lemma 2.2,equation(3.1)has exactly one almost periodic solution

Define an operator Φ :Z → Z,

Obviously,z is an almost periodic solution of equation(1.1)if and only if z is the fi xed point of operator Φ.

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Let ? ={z|z ∈ Z,L2≤ z(t) ≤ L1,t ∈ T}.

Now,we prove that Φ? ? ?.In fact,?z ∈ ?,we have

On the other hand,we have

Note that

Since the function g(u)=u2[bl+cl? u]is increasing on u ∈ [0,23(bl+cl)]and decreasing on u∈[23(bl+cl),+∞],then we have g(z(t)) ≥ g(L1)for t ∈ T,that is

Thus by(3.4),we obtain

It follows from(3.3)and(3.5)that

In addition,for z ∈ ?,equation(3.1)has exactly one almost periodic solution

Since xz(t)is almost periodic,then(Φz(mì))(t)is almost periodic.This,together with(3.6), implies Φz(mì) ∈ ?.So we have Φ? ? ?.

Next,we prove that Φ is a contraction mapping on ?.In fact,in view of(H1)–(H3), for any z1,z2∈ ?,

Next,we shallconstruct a suitable Lyapunov functionalto study the globalexponential stability of the almost periodic solution of(1.1).

Theorem 3.2Assume that(H1)–(H3)hold.Suppose further that

then equation(1.1)has a unique globally exponentially stable almost periodic solution.

ProofAccording to Theorem 3.1,we know that(1.1)has an almost periodic solution x?(t),and L2≤ x?(t) ≤ L1.Suppose that x(t)is arbitrary solution of(1.1)with initial condition x(t0) ＞ 0,t0∈ T.Now we prove x?(t)is globally exponentially stable.

Let V(t)=|x(t) ? x?(t)|.Calculating the upper right derivatives of V(t)along the solution of equation(1.1),from(H4)and(H5),then

Integrating(3.7)from t0to t,we get V(t) ≤ e?α(t,t0)V(t0),that is

From Definition 2.3,the almost periodic solution x?of(1.1)is globally exponentially stable.This completes the proof.

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時(shí)標(biāo)上具Allee效應(yīng)的動(dòng)力學(xué)方程的概周期解

王麗麗,胡 猛
(安陽師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,河南 安陽 455000)

本文研究了時(shí)標(biāo)上具Allee效應(yīng)的可再生資源動(dòng)力學(xué)方程的概周期解的存在性與穩(wěn)定性. 利用線性系統(tǒng)指數(shù)二分性與壓縮映射不動(dòng)點(diǎn)定理,得到了方程存在唯一概周期解的充分條件.此外,通過構(gòu)建適當(dāng)?shù)腖aypunov函數(shù),得到了概周期解是全局指數(shù)穩(wěn)定的充分條件.

動(dòng)力學(xué)方程;Allee效應(yīng); 概周期解; 指數(shù)穩(wěn)定; 時(shí)標(biāo)

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A < class="emphasis_bold">Article ID:0255-7797(2017)02-0283-08

0255-7797(2017)02-0283-08

?Received date:2015-12-13 Accepted date:2016-03-04

Foundation item:Supported by Key Project of Scientifi c Research in Colleges and Universities in Henan Province(15A110004;16A110008);Basic and Frontier Technology Research Project of Henan Province(142300410113).

Biography:Wang Lili(1981–),female,born at Xinxiang,Henan,lecturer,major in diff erential equations.