# Stability Analysis of Three-Species Almost Periodic Competition Models with Grazing Rates and Diffusions

Authors: Chang-you Wang; Rui-fang Wang; Ming Yi; Rui Li
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

Almost periodic solution of a three-species competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions to ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, which extend and include corresponding results obtained by Q. C. Lin (1999), F. D. Chen and X. X. Chen (2003), and Y. Q. Liu, S. L, Xie, and Z. D. Xie (1996).


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## Body

## 1. Introduction

In this paper, we study the following three-species competition system with grazing rates and diffusions:(1.1)∂v1(x,t)∂t=k1(t)Δv1(x,t)+v1(x,t)[a1(t)-b1(t)v1(x,t)-c1(t)v2(x,t)-d1(t)v3(x,t)]+f1(t),∂v2(x,t)∂t=k2(t)Δv2(x,t)+v2(x,t)[a2(t)-b2(t)v1(x,t)-c2(t)v2(x,t)-d2(t)v3(x,t)]+f2(t),∂v3(x,t)∂t=k3(t)Δv3(x,t)+v3(x,t)[a3(t)-b3(t)v1(x,t)-c3(t)v2(x,t)-d3(t)v3(x,t)]+f3(t),
where (x,t)∈Ω×R+, Ω⊆Rn is the bounded open subset of Rn with smooth boundary ∂Ω, which represent the habitat domain for three species. System (1.1) is supplement with boundary conditions and initial conditions:(1.2)∂vi(x,t)∂n=0,i=1,2,3,(x,t)∈∂Ω×R+,(1.3)vi(x,0)=vi0(x)≥0,vi0(x)≢0,i=1,2,3,x∈Ω¯,
where ∂/∂n denotes the outward normal derivation on ∂Ω, and vi(x,t) represent the density of ith species at point x=(x1,…,xn) and the time of t. Here, ki(t), ai(t), bi(t), ci(t), di(t), and fi(t) (i=1,2,3) denote the diffusivity rates, competition rates, and grazing rates, respectively. They are almost periodic functions in real number field R. Δ is a Laplace operator on Ω.System (1.1)–(1.3) describes the interaction among three species and is an important model in biomathcmatics, which has been intensively investigated, and much attention is carried to the problem [1–8]. When there is no diffusion, Jiang [1] and Lin [2] studied the existence, uniqueness, and stability on periodic solution and almost periodic solution for two-species competition system under the condition that the coefficients are the periodic function and almost periodic function, respectively; F. D. Chen and X. X. Chen [3] extended the results in [2] to n-species case. When there are no diffusion and grazing rates, Zhang and Wang [4, 5] investigated the existence of a positive periodic solution for a two-species nonautonomous competition Lotka-Volterra patch system with time delay and the existence of multiple positive periodic solutions for a generalized delayed population model with exploited term by using the continuation theorem of coincidence degree theory; Hu and Zhang [6] established criteria for the existence of at least four positive periodic solutions for a discrete time-delayed predator-prey system with nonmonotonic functional response and harvesting by employing the continuation theorem of coincidence degree theory. When there are no grazing rates, Pao and Wang [7] proved the stability for invariable coefficient case by utilizing the method of upper and lower solutions. Liu et al. [8] showed the stability on the periodic solution for n-species competition system with grazing rates and diffusions. Nevertheless, generally speaking, the system does not always change strictly according to periodic laws, sometimes it changes according to almost periodic laws, and it is important to survey almost periodic solution for the multispecies competition system with grazing rates and diffusions. To sum up, we pay more attention to almost periodic solution of a three-species competition system (1.1)–(1.3) with grazing rates and diffusions; in this paper, by using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we obtain sufficient conditions which ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, which extend and include corresponding results obtained in [2, 3, 8]. Many other results on the periodic solution and almost periodic solution can be found in [9–16].


## 2. Preliminary

Firstly, we give out some definitions and lemmas.Definition 2.1.
Suppose thatf(t) is a continuous function in R. Then f(t) is said to be almost periodic in t∈R if for every ε>0 corresponds T(ε)>0 such that for any interval I whose length is equal to T(ε) there is at least one τ∈I such that
(2.1)|f(t+τ)-f(t)|≤ε,∀t∈R.Definition 2.2.
If a smooth functionV(t)=(v1(t),v2(t),v3(t)) satisfies (1.1) in R+, and every component of V(t) is the almost periodic function, we called that V(t) is a spatial homogeneity almost periodic solution for (1.1), which is denoted by V(t,T(ε)).Definition 2.3.
For any nonnegative smooth initial data(2.2)V(x,0)=(v1(x,0),v2(x,0),v3(x,0))=(v10(x),v20(x),v30(x))≥0,V(x,0)≢0,x∈Ω,
if there exists a unique positive solution V(x,t)=(v1(x,t),v2(x,t),v3(x,t)) for the system (1.1) with boundary conditions (1.2), and limt→∞(Vi(x,t)-Vi(t,T(ε)))=0, i=1,2,3, uniformly for x∈Ω¯, we called that spatial homogeneity almost periodic solution V(t,T(ε)) is globally asymptotically stable.Definition 2.4.
Suppose thatV¯(x,t)≡(v¯1(x,t),v¯2(x,t),v¯3(x,t)), V̲(x,t)≡(v̲1(x,t),v̲2(x,t),v̲3(x,t)); if V¯(x,t)≥V̲(x,t) and
(2.3)∂v¯1(x,t)∂t≥k1(t)Δv¯1(x,t)+v¯1(x,t)[a1(t)-b1(t)v¯1(x,t)-c1(t)v̲2(x,t)-d1(t)v̲3(x,t)]+f1(t),(x,t)∈Ω×R+,∂v¯2(x,t)∂t≥k2(t)Δv¯2(x,t)+v¯2(x,t)[a2(t)-b2(t)v̲1(x,t)-c2(t)v¯2(x,t)-d2(t)v̲3(x,t)]+f2(t),(x,t)∈Ω×R+,∂v¯3(x,t)∂t≥k3(t)Δv¯3(x,t)+v¯3(x,t)[a3(t)-b3(t)v̲1(x,t)-c3(t)v̲2(x,t)-d3(t)v¯3(x,t)]+f3(t),(x,t)∈Ω×R+,∂v¯i(x,t)∂n≥0,i=1,2,3,(x,t)∈∂Ω×R+,v¯i(x,0)≥vi0(x),i=1,2,3,x∈Ω¯,∂v̲1(x,t)∂t≤k1(t)Δv̲1(x,t)+v̲1(x,t)[a1(t)-b1(t)v̲1(x,t)-c1(t)v¯2(x,t)-d1(t)v¯3(x,t)]+f1(t),(x,t)∈Ω×R+,∂v̲2(x,t)∂t≤k2(t)Δv̲2(x,t)+v̲2(x,t)[a2(t)-b2(t)v¯1(x,t)-c2(t)v̲2(x,t)-d2(t)v¯3(x,t)]+f2(t),(x,t)∈Ω×R+,∂v̲3(x,t)∂t≤k3(t)Δv̲3(x,t)+v̲3(x,t)[a3(t)-b3(t)v¯1(x,t)-c3(t)v¯2(x,t)-d3(t)v̲3(x,t)]+f3(t),(x,t)∈Ω×R+,∂v̲i(x,t)∂n≤0,i=1,2,3,(x,t)∈∂Ω×R+,v̲i(x,0)≤vi0(x),i=1,2,3,x∈Ω¯,
we called V¯(x,t) andV̲(x,t) a pair of ordered upper and lower solutions for systems (1.1)–(1.3).Lemma 2.5 (see [12, 17]).
Suppose thatV¯(x,t)andV̲(x,t) are a pair of ordered upper and lower solution for systems (1.1)–(1.3), then there exists a unique solution V(x,t) for systems (1.1)–(1.3). Moreover, one has
(2.4)V¯(x,t)≥V(x,t)≥V̲(x,t).For the almost periodic functionF(t) in R, one denotes F̃=sup{F(t),t∈R}, F~=inf{F(t),t∈R}, and M[F]=lim(t-s)→∞{∫stF(τ)dτ/(t-s)}. When F(t) is T-periodic function, one denotes M[F]=∫0TF(s)ds/T.


## 3. Main Results and Proofs

Now we are in a position to state our main results and give our proofs.Theorem 3.1.
Ifa~i,b~i,c~i,d~i,f~i are positive numbers, and
(3.1)(b̃i+c̃i+d̃i)a~i≤L=min⁡{b~1f̃1,c~2f̃2,d~3f̃3,(d~1+c~1)ã1,(b~2+d~2)ã2,(b~3+c~3)ã3}
are satisfied for i=1,2,3, then there exists a strictly positive spatial homogeneity almost periodic solution V(t)=(v̂1(t),v̂2(t),v̂3(t)) for (1.1).Proof.
By the conditions in Theorem3.1, we have
(3.2)0<c̃1+d̃1La~1-b̃1≤1,0<b̃2+d̃2La~2-c̃2≤1,0<b̃3+c̃3La~3-d̃3≤1.
Let(3.3)m=L⋅max⁡{c̃1+d̃1La~1-b̃1,b̃2+d̃2La~2-c̃2,b̃3+c̃3La~3-d̃3}.
Then we have 0<m≤L, and
(3.4)(c̃1+d̃1)Lm≤La~1-b̃1,(b̃2+d̃2)Lm≤La~2-c̃2,(b̃3+c̃3)Lm≤La~3-d̃3.
Therefore
(3.5)b̃1+(c̃1+d̃1)Lm-f~1m2≤La~1,c̃2+(b̃2+d̃2)Lm-f~2m2≤La~2,d̃3+(b̃3+c̃3)Lm-f~3m2≤La~3.
Furthermore, by the given conditions in Theorem 3.1, one has
(3.6)b~1-f̃1L2≥0,c~2-f̃2L2≥0,d~3-f̃3L2≥0,(c~1+d~1)mL≥mã1,(b~2+d~2)mL≥mã2,(b~3+c~3)mL≥mã3.
Thus
(3.7)b~1+(c~1+d~1)mL-f̃1L2≥mã1,c~2+(b~2+d~2)mL-f̃2L2≥mã2,d~3+(b~3+c~3)mL-f̃3L2≥mã3.
Combining (3.5) and (3.7), we have
(3.8)b̃1+(c̃1+d̃1)Lm-f~1m2≤La~1,b~1+(c~1+d~1)mL-f̃1L2≥mã1,c̃2+(b̃2+d̃2)Lm-f~2m2≤La~2,c~2+(b~2+d~2)mL-f̃2L2≥mã2,d̃3+(b̃3+c̃3)Lm-f~3m2≤La~3,d~3+(b~3+c~3)mL-f̃3L2≥mã3.
Let(3.9)HLm={(ϕ(t),φ(t),γ(t)):ϕ,φ,γarealmostperiodicint∈R,0<m≤ϕ,φ,γ≤L}.
We consider the following system corresponding to (1.1):
(3.10)v̇1=v1(a1(t)-b1(t)v1-c1(t)v2-d1(t)v3)+f1(t),t∈R+,v̇2=v2(a2(t)-b2(t)v1-c2(t)v2-d2(t)v3)+f2(t),t∈R+,v̇3=v3(a3(t)-b3(t)v1-c3(t)v2-d3(t)v3)+f3(t),t∈R+.
Let zi=1/vi, i=1,2,3; then (3.10) becomes
(3.11)ż1=b1(t)-a1(t)z1+c1(t)z1z2+d1(t)z1z3-f2(t)z12,ż2=c2(t)-a2(t)z2+b2(t)z2z1+d2(t)z2z3-f2(t)z22,ż3=d3(t)-a3(t)z3+b3(t)z3z1+c3(t)z3z2-f3(t)z32.
For any (ϕ(t),φ(t),γ(t))∈HLm, by M[b1]>0, M[c2]>0, M[d3]>0, we observe [18] that
(3.12)ż1=b1(t)-a1(t)z1+c1(t)ϕ(t)φ(t)+d1(t)ϕ(t)γ(t)-f1(t)ϕ2(t),ż2=c2(t)-a2(t)z2+b2(t)φ(t)ϕ(t)+d2(t)φ(t)γ(t)-f2(t)φ2(t),ż3=d3(t)-a3(t)z3+b3(t)γ(t)ϕ(t)+c3(t)γ(t)φ(t)-f3(t)γ2(t),
have almost periodic solution:
(3.13)ẑ1(t)=∫-∞te-∫sta1(r)dr[b1(s)+c1(s)ϕ(s)φ(s)+d1(s)ϕ(s)γ(s)-f1(s)ϕ2(s)]ds,ẑ2(t)=∫-∞te-∫sta2(r)dr[c2(s)+b2(s)φ(s)ϕ(s)+d2(s)φ(s)γ(s)-f2(s)φ2(s)]ds,ẑ3(t)=∫-∞te-∫sta2(r)dr[d3(s)+b3(s)γ(s)ϕ(s)+c3(s)γ(s)φ(s)-f3(s)γ2(s)]ds.
By the system (3.13), we define a mapping A:
(3.14)A(ϕ(t),φ(t),γ(t))=(ẑ1(t),ẑ2(t),ẑ3(t)),∀(ϕ(t),φ(t),γ(t))∈HLm.
Combining (3.8) and (3.13), we have
(3.15)ẑ1(t)≥∫-∞te-ã1(t-s)[b~1+(c~1+d~1)mL-f̃1L2]ds=1ã1[b~1+(c~1+d~1)mL-f̃1L2]≥m,ẑ1(t)≤∫-∞te-a~1(t-s)[b̃1+(c̃1+d̃1)Lm-f~1m2]ds=1a~1[b̃1+(c̃1+d̃1)Lm-f~1m2]≤L,ẑ2(t)≥∫-∞te-ã2(t-s)[c~2+(b~2+d~2)mL-f̃2L2]ds=1ã2[c~2+(b~2+d~2)mL-f̃2L2]≥m,ẑ2(t)≤∫-∞te-a~2(t-s)[c̃2+(b̃2+d̃2)Lm-f~2m2]ds=1a~2[c̃2+(b̃2+d̃2)Lm-f~2m2]≤L,ẑ3(t)≥∫-∞te-ã3(t-s)[d~3+(b~3+c~3)mL-f̃3L2]ds=1ã3[d~3+(b~3+c~3)mL-f̃3L2]≥m,ẑ3(t)≤∫-∞te-a~3(t-s)[d̃3+(b̃3+c̃3)Lm-f~3m2]ds=1a~3[d̃3+(b̃3+c̃3)Lm-f~3m2]≤L.
Therefore, (ẑ1(t),ẑ2(t),ẑ3(t))∈HLm, that is, AHLm⊂HLm. If A is uniformly boundness and equicontinuous, by Ascoli-Arzela theorem, A is compact mapping.
It is obvious to obtain uniformly boundedness. In fact, for any(ϕ(t),φ(t),γ(t))∈HLm, by (3.15) we have (ẑ1(t),ẑ2(t),ẑ3(t))=A(ϕ(t),φ(t),γ(t))∈HLm; that is, it satisfies
(3.16)0<(m,m,m)≤A(ϕ(t),φ(t),γ(t))=(ẑ1(t),ẑ2(t),ẑ3(t))≤(L,L,L).
Next we prove equicontinuous. For any(ϕ(t),φ(t),γ(t))∈HLm, we denote (ẑ1(t),ẑ2(t),ẑ3(t))=A(ϕ(t),φ(t),γ(t)),t1<t2, and then
(3.17)|ẑ1(t1)-ẑ1(t2)|=|∫-∞t1e-∫st1a1(r)dr[b1(s)+c1(s)ϕ(s)φ(s)+d1(s)ϕ(s)γ(s)-f1(s)ϕ2(s)]-∫-∞t2e-∫st2a1(r)dr[b1(s)+c1(s)ϕ(s)φ(s)+d1(s)ϕ(s)γ(s)-f1(s)ϕ2(s)]|.
Leth1(t)=b1(t)+c1(t)(ϕ(t)/φ(t))+d1(t)(ϕ(t)/γ(t))-f1(t)ϕ2(t); we obtain
(3.18)|ẑ1(t2)-ẑ1(t1)|=|∫-∞t2e-∫st2a1(r)drh1(s)ds-∫-∞t1e-∫st1a1(r)drh1(s)ds|≤|∫t1t2e-∫st2a1(r)drh1(s)ds|+|∫-∞t1e-∫st1a1(r)dr(e-∫t1t2a1(r)dr-1)h1(s)ds|.
Recalling (ϕ(t),φ(t),γ(t))∈HLm, we deduce that there exists a positive number M such that |h1(s)|≤M; then (3.18) becomes
(3.19)|ẑ1(t2)-ẑ1(t1)|≤Me-∫ξ1t2a1(r)dr|t2-t1|+1a~1M|1-e-∫t1t2a1(r)dr|,
where ξ1∈(t1,t2).
Similarly, we have(3.20)|ẑ2(t2)-ẑ2(t1)|≤Ne-∫ξ2t2a2(r)dr|t1-t2|+1a~2N|1-e-∫t1t2a2(r)dr|,
where ξ2∈(t1,t2), and N is a positive number.
By a completely analogous argument, we obtain(3.21)|ẑ3(t2)-ẑ3(t1)|≤Pe-∫ξ3t2a3(r)dr|t1-t2|+1a~3P|1-e-∫t1t2a3(r)dr|,
where ξ3∈(t1,t2), and P is a positive number.
By (3.19)–(3.21), for any (ϕ(t),φ(t),γ(t))∈HLm, we derive
(3.22)lim⁡ξ→0sup⁡|t1-t2|≤ξ|A(ϕ,φ,γ)(t1)-A(ϕ,φ,γ)(t2)|=0,uniformlyforx∈Ω¯.
Thus,A is a compact mapping which maps HLm into itself; by Schauder fixed point theorem, there exists a fixed point (ϕ(t),φ(t),γ(t))∈HLm for A; namely, (3.11) has a solution; therefore there exists a strictly positive almost periodic solution (v1*(t),v2*(t),v3*(t))=(1/ϕ(t),1/φ(t),1/γ(t)),t∈R+ for system (3.10). It is obvious that (v1*(t),v2*(t),v3*(t)),t∈R+ is also the spatial homogeneity almost periodic solution for (1.1).Theorem 3.2.
Under the conditions of Theorem3.1, suppose that system (1.1) satisfies the following conditions:
(3.23)sup⁡t≥0(b3(t)+b2(t)-b1(t))=-ε1<0,sup⁡t≥0(c3(t)+c1(t)-c2(t))=-ε2<0,sup⁡t≥0(d1(t)+d2(t)-d3(t))=-ε3<0.
Then there exists a strictly positive spatial homogeneity almost periodic solution (v1*(t),v2*(t),v3*(t)) for (1.1), and the corresponding solution for systems (1.1)–(1.3) is globally asymptotically stable; that is, the solution (v1(x,t),v2(x,t),v3(x,t)),(x,t)∈Ω¯×R+ satisfies
(3.24)lim⁡t→∞(vi(x,t)-vi*(t))=0,i=1,2,3,uniformlyforx∈Ω¯.Proof.
We have obtained the existence by Theorem3.1; next we pay more attention to the stability. Concerning (3.24), we have two cases on initial data vi0(x),i=1,2,3. (1)
vi0(x)>0,x∈Ω¯.(2)
There exists a pointx0∈Ω¯, such that v10(x0)=0,v20(x0)=0 or v30(x0)=0.
For the case (1), letli=minx∈Ω¯vi0(x), ri=maxx∈Ω¯vi0(x), i=1,2,3; then 0<li≤vi0(x)≤ri. Suppose that (v¯1(t),v¯2(t),v¯3(t)) and (v̲1(t),v̲2(t),v̲3(t)) are the solution for (3.10) corresponding to initial datum (v¯1(0),v¯2(0),v¯3(0))=(r1,r2,r3) and (v̲1(0),v̲2(0),v̲3(0))=(l1,l2,l3), respectively; then there are a pair of ordered upper and lower solutions (v¯1(t),v¯2(t),v¯3(t)) and (v̲1(t),v̲2(t),v̲3(t)) for (1.1)–(1.3); by Lemma 2.5, there exists a unique solution (v1(x,t),v2(x,t),v3(x,t)),(x,t)∈Ω¯×R+ for system (1.1)–(1.3), which satisfies
(3.25)(v̲1(t),v̲2(t),v̲3(t))≤(v1(x,t),v2(x,t),v3(x,t))≤(v¯1(t),v¯2(t),v¯3(t)).
If we have(3.26)lim⁡t→∞[v¯i(t)-vi*(t)]=lim⁡t→∞[v̲i(t)-vi*(t)]=0,i=1,2,3,
then (3.24) holds. Therefore, if we want to obtain (3.26), we only need to prove that the solution (v1(t),v2(t),v3(t)) for (3.10) with arbitrary positive initial data (v1(0),v2(0),v3(0))=(v10,v20,v30) satisfies
(3.27)lim⁡t→∞(vi(t)-vi*(t))=0,i=1,2,3.
Because of the initial datum(v10,v20,v30)>0 and grazing rates (f~1,f~2,f~3)>0, by the practical meaning in biology, we know that (v1(t),v2(t),v3(t))>0. Now let
(3.28)Pi(t)=ln⁡vi(t),Qi(t)=ln⁡vi*(t),i=1,2,3.
Then one has
(3.29)ddt(P1(t)-Q1(t))=-b1(t)(eP1(t)-eQ1(t))-c1(t)(eP2(t)-eQ2(t))-d1(t)(eP3(t)-eQ3(t))+(1v1(t)-1v1*(t))f1(t),ddt(P2(t)-Q2(t))=-b2(t)(eP1(t)-eQ1(t))-c2(t)(eP2(t)-eQ2(t))-d2(t)(eP3(t)-eQ3(t))+(1v2(t)-1v2*(t))f2(t),ddt(P3(t)-Q3(t))=-b3(t)(eP1(t)-eQ1(t))-c3(t)(eP2(t)-eQ2(t))-d3(t)(eP3(t)-eQ3(t))+(1v3(t)-1v3*(t))f3(t).
Namely,
(3.30)ddt(P1(t)-Q1(t))=-(b1(t)+f1(t)v1(t)v1*(t))(eP1(t)-eQ1(t))-c1(t)(eP2(t)-eQ2(t))-d1(t)(eP3(t)-eQ3(t)),ddt(P2(t)-Q2(t))=-b2(t)(eP1(t)-eQ1(t))-(c2(t)+f2(t)v2(t)v2*(t))(eP2(t)-eQ2(t))-d2(t)(eP3(t)-eQ3(t)),ddt(P3(t)-Q3(t))=-b3(t)(eP1(t)-eQ1(t))-c3(t)(eP2(t)-eQ2(t))-(d3(t)+f3(t)v3(t)v3*(t))(eP3(t)-eQ3(t)).
Consider the following Lyapunov function:
(3.31)U(t)=∑i=13|Pi(t)-Qi(t)|,t≥0.
LetD+U represent the right derivation on function U(t); we have
(3.32)D+U(t)=∑i=13D+|Pi(t)-Qi(t)|=∑i=13sgn⁡(Pi(t)-Qi(t))ddt(Pi(t)-Qi(t))=sgn⁡(P1(t)-Q1(t))×[-(b1(t)+f1(t)v1(t)v1*(t))(eP1(t)-eQ1(t))-c1(t)(eP2(t)-eQ2(t))-d1(t)(eP3(t)-eQ3(t))]+sgn⁡(P2(t)-Q2(t))×[-b2(t)(eP1(t)-eQ1(t))-(c2(t)+f2(t)v2(t)v2*(t))(eP2(t)-eQ2(t))-d2(t)(eP3(t)-eQ3(t))]+sgn⁡(P3(t)-Q3(t))×[-b3(t)(eP1(t)-eQ1(t))-c3(t)(eP2(t)-eQ2(t))-(d3(t)+f3(t)v3(t)v3*(t))(eP3(t)-eQ3(t))]≤(b3(t)+b2(t)-b1(t))|eP1(t)-eQ1(t)|+(c1(t)+c3(t)-c2(t))|eP2(t)-eQ2(t)|+(d1(t)+d2(t)-d3(t))|eP3(t)-eQ3(t)|≤-ε1|v1(t)-v1*(t)|-ε2|v2(t)-v2*(t)|-ε3|v3(t)-v3*(t)|.
Integrated by the time, we have
(3.33)U(t)+∑i=13εi∫0t|vi(s)-vi*(s)|ds≤U(0).
By the nonnegative of U(t) and the boundedness of U(0), we obtain that the U(t) is bounded, and
(3.34)∫0t|vi(t)-vi*(t)|ds,i=1,2,3,
convergences, by (3.32) we get D+U(t)<0, then the limit
(3.35)lim⁡t→∞U(t)=l
exists, and U(t)≥l. If l>0, then at least one of the following three inequalities
(3.36)|P1(t)-Q1(t)|>l4,|P2(t)-Q2(t)|>l4,|P3(t)-Q3(t)|>l4
holds. Without loss of generality, we assume that |P1(t)-Q1(t)|>l/4. Thus there is no point of intersection between P1(t) and Q1(t). Suppose that P1(t)>Q1(t); then we have P1(t)-Q1(t)>l/4. Thus
(3.37)∫0t|v1(t)-v1*(t)|ds=∫0t|eP1(s)-eQ1(s)|ds=∫0teQ1(s)|eP1(s)-Q1(s)-1|ds≥m∫0t(eP1(s)-Q1(s)-1)ds>m∫0t(el/4-1)ds=m(el/4-1)t⟶+∞,
which contradicts with the convergence of ∫0t|vi(s)-vi*(s)|ds. Therefore l=0; consequently
(3.38)lim⁡t→∞|vi(t)-vi*(t)|=0,i=1,2,3.
Then we obtain (3.27).
For the case (2), firstly, choose three sufficient large positive numbersM1,M2,andM3, such that
(3.39)f1(t)≤-M1(a1(t)-b1(t)M1),t>0,f2(t)≤-M2(a2(t)-c2(t)M2),t>0,f3(t)≤-M3(a3(t)-d3(t)M3),t>0,
and Mi≥maxx∈Ω¯vi0(x), i=1,2,3. Let v~i=0, ṽi=Mi, i=1,2,3; then we have
(3.40)∂ṽ1∂t-k1(t)Δṽ1-ṽ1[a1(t)-b1(t)ṽ1-c1(t)v~2-d1(t)v~3]-f1(t)≥0,∂v~1∂t-k1(t)Δv~1-v~1[a1(t)-b1(t)v~1-c1(t)ṽ2-d1(t)ṽ3]-f1(t)≤0,∂ṽ2∂t-k2(t)Δṽ2-ṽ2[a2(t)-b2(t)v~1-c2(t)ṽ2-d2(t)v~3]-f2(t)≥0,∂v~2∂t-k2(t)Δv~2-v~2[a2(t)-b2(t)ṽ1-c2(t)v~2-d2(t)ṽ3]-f2(t)≤0,∂ṽ3∂t-k3(t)Δṽ3-ṽ3[a3(t)-b3(t)v~1-c3(t)v~2-d3(t)ṽ3]-f3(t)≥0,∂v~3∂t-k3(t)Δv~3-v~3[a3(t)-b3(t)ṽ1-c3(t)ṽ2-d3(t)v~3]-f3(t)≤0.
Namely, v~i=0, and ṽi=Mi, i=1,2,3 are a pair of ordered upper and lower solutions for systems (1.1)–(1.3). By Lemma 2.5, there exists a unique solution (v1(x,t),v2(x,t),v3(x,t)) for systems (1.1)–(1.3), which satisfy
(3.41)0≤vi(x,t)≤Mi,i=1,2,3;(x,t)∈Ω¯×[0,∞).
Secondly, we choose positive numbersδ1,δ2,andδ3 such that
(3.42)δ1+a1(t)-b1(t)v1(x,t)-c1(t)v2(x,t)-d1(t)v3(x,t)>0,(x,t)∈Ω¯×[0,∞),δ2+a2(t)-b2(t)v1(x,t)-c2(t)v2(x,t)-d2(t)v3(x,t)>0,(x,t)∈Ω¯×[0,∞),δ3+a3(t)-b3(t)v1(x,t)-c3(t)v2(x,t)-d3(t)v3(x,t)>0,(x,t)∈Ω¯×[0,∞).
Accordingly, we have
(3.43)∂v1∂t-k1(t)Δv1+δ1v1=v1[δ1+a1(t)-b1(t)v1-c1(t)v2-d1(t)v3]+f1(t)≥0,∂v2∂t-k2(t)Δv2+δ2v2=v2[δ2+a2-b2(t)v1-c2(t)v2-d2(t)v3]+f2(t)≥0,∂v3∂t-k3(t)Δv3+δ3v3=v3[δ3+a3-b3(t)v1-c3(t)v2-d3(t)v3]+f3(t)≥0.
Next, we prove vi(x,t)>0 in Ω¯×(0,∞) for i=1,2,3. Firstly, we show vi(x,t)>0 in Ω×(0,∞). If there exists one point (x0,t0)∈Ω×(0,∞) such that vi(x0,t0)=0, by extremum principle, we have vi(x,t)≡0 in Ω¯×[0,t0). However vi(x,0)=vi0(x)≥0, and not being constant zero, we obtain a contradiction. Therefore we have vi(x,t)>0 in Ω×(0,∞). Then we show vi(x,t)>0 in ∂Ω×(0,∞). If there exists a point (x0,t0)∈∂Ω×(0,∞) such that vi(x0,t0)=0, by the extremum principle, we have ∂vi(x,t)/∂n<0, where (x,t)∈∂Ω×(0,∞), which is contrary with boundary conditions (1.2). Thus we have vi(x,t)>0 in Ω¯×(0,∞).
For a fixed numberλ>0, by (3.41), we have
(3.44)0<vi(x,λ)≤Mi,i=1,2,3,x∈Ω¯.
Because vi(x,t+λ) satisfy system (1.1) in Ω¯×(0,∞) and the conditions (1.2) in ∂Ω×(0,∞), thereby (v1(x,t+λ),v2(x,t+λ),v3(x,t+λ)) is regarded as a solution for system (1.1) under initial data (v̂10(x),v̂20(x),v̂30(x))=(v1(x,λ),v2(x,λ),v3(x,λ)), nevertheless, we have v̂i0>0 in Ω̅,i=1,2,3; combining the conclusions in case (1), we have
(3.45)lim⁡t⟶∞⁡(vi(x,t+λ)-vi*(t))=0,i=1,2,3,uniformlyforx∈Ω.̅
By the arbitrariness of λ, we obtain
(3.46)lim⁡t⟶∞⁡(vi(x,t)-vi*(t))=0,i=1,2,3,uniformlyforx∈Ω.̅
Ifki(t),ai(t),bi(t),ci(t),di(t),andfi(t)(i=1,2,3) of (1.1) are periodic functions in real number field R, respectively, then we have the following results.Corollary 3.3.
Ifa~i,b~i,c~i,d~i,f~i are positive numbers, and
(3.47)(b̃i+c̃i+d̃i)a~i≤L=min⁡{b~1f̃1,c~2f̃2,d~3f̃3,(d~1+c~1)ã1,(b~2+d~2)ã2,(b~3+c~3)ã3}
are satisfied for i=1,2,3, then there exists a strictly positive spatial homogeneity periodic solution V(t)=(v̂1(t),v̂2(t),v̂3(t)) for (1.1).Corollary 3.4.
Under the conditions of Corollary3.3, suppose that system (1.1) satisfies the following conditions:
(3.48)sup⁡t≥0(b3(t)+b2(t)-b1(t))=-ε1<0,sup⁡t≥0(c3(t)+c1(t)-c2(t))=-ε2<0,sup⁡t≥0(d1(t)+d2(t)-d3(t))=-ε3<0.
Then there exists a strictly positive spatial homogeneity periodic solution (v1*(t),v2*(t),v3*(t)) for (1.1), and the corresponding solution for systems (1.1)–(1.3) is globally asymptotically stable; that is, the solution (v1(x,t),v2(x,t),v3(x,t)),(x,t)∈Ω¯×R+ satisfies
(3.49)lim⁡t→∞(vi(x,t)-vi*(t))=0,i=1,2,3,uniformlyforx∈Ω¯.


## 4. Conclusion

This paper presents the use of upper and lower solutions method for systems of nonlinear reaction-diffusion equations. This method is a powerful tool for solving nonlinear differential equations in mathematical physics, chemistry, and engineering, and so forth. The technique constructing a pair of upper and lower solutions and Lyapunov function provides a new efficient method to handle the nonlinear structure.We have dealt with the problem of almost periodic solution for a three-species competition system with grazing rates and diffusions. The general sufficient conditions have been obtained to ensure the existence and stability of the strictly positive space homogenous almost periodic solution for the nonlinear reaction-diffusion equations. These criteria generalize and improve some known results. In particular, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of nonlinear three-species competition system.



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*Source: 783136-2011-08-04.xml*