ONe of the most interesting questions considering ecological

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1 On an Almost Periodic Gilpin-Ayala Competition Model of Phytoplankton Allelopathy Lijun Xu and Yongzhi Liao Abstract By using some analytical techniques modified inequalities and Mawhin s continuous theorem of coincidence degree theory some sufficient conditions for the existence of positive almost periodic solutions to a class of Gilpin-Ayala competition model of phytoplankton allelopathy are established Further by constructing a suitable Lyapunov functional the global asymptotical stability of almost periodic solution for the model is also studied The work of this paper extends and improves some results in recent years Finally some examples and simulations are given to illustrate the main results in this paper Index Terms Almost periodic oscillation; Coincidence degree; Gilpin-Ayala; Phytoplankton allelopathy I INTRODUCTION ONe of the most interesting questions considering ecological populations is how species which grow in the same environment influence each other since they usually compete for food territory and other requirements A specific type of ecological interaction corresponding to this issue is called facultative mutualism and it means each species enhances the average growth rate of the other although each one of them can survive in the absence of other species The first model regards ecological population systems was suggested independently by Lotka and Volterra in the 9s and was described by the following differential equation ẋ i = r i x i x i x j a ij i j = i j K i K j where x i and r i are the population size and exponential rate of the growth of the ith species respectively K i is the carrying capacity of the ith species in the absence of its competitor-the jth species and a ij is the linear reduction (in terms of K i ) of the ith species rate of growth by its competitor-the jth species i j = i j During the last decades ecological population systems have been intensively studied and there exist a lot of excellent papers in this field Most of them are mainly grounded on the classical Lotka-Volterra competition system but they have many different forms (see for example 34) However regardless of this fact the Lotka-Volterra competition models have often been severely criticized One of the remarks refers to the fact this model is linear ie the rate of change in the size of each species is a linear function of sizes of the interacting species Particularly in 973 Gilpin and Ayala 5 pointed out the Lotka-Volterra Manuscript received May 9 6; revised January 7 Lijun Xu is with School of Mathematics and Computer Science Panzhihua University Panzhihua Sichuan 67 China (pzhxlj@6com) Yongzhi Liao is with School of Mathematics and Computer Science Panzhihua University Panzhihua Sichuan 67 China (mathyzliao@6com) systems are the linearization of the per capita growth rates ẋ i /x i about the point of equilibrium They claimed a little more complicated model was needed in order to yield more realistic solutions Hence they proposed the following competition model: ẋ = r x ( ) x θ K x a K ẋ = r x ( x ) θ () K x a K where x i is the population density of the ith species r i is the intrinsic exponential growth rate of the ith species K i is the environmental carrying capacity of species i in the absence of competition θ i provides a nonlinear measure of interspecific interference i = a and a provides a measure of interspecific interference In recent years many generalizations and modifications of system () have been proposed and studied (see 6- for more detail) The aim of this paper is to consider the following almost periodic Gilpin-Ayala competition model of phytoplankton allelopathy with time-varying delays: ẋ(t) = x(t) r (t) a (t)x α (t) b (t)y β (µ (t)) c (t)x γ (ν (t))y δ (σ (t)) () ẏ(t) = y(t) r (t) a (t)y α (t) b (t)x β (µ (t)) c (t)x γ (ν (t))y δ (σ (t)) where α i β i γ i δ i are nonnegative constants r i a i b i c i t µ i (t) t ν i (t) and t σ i (t) are nonnegative almost periodic functions i = Let R Z and N denote the sets of real numbers integers and positive integers respectively In real world phenomenon the environment varies due to the factors such as seasonal effects of weather food supplies mating habits harvesting So it is usual to assume the periodicity of parameters in the systems However if the various constituent components of the temporally nonuniform environment is with incommensurable (nonintegral multiples) periods then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions Hence if we consider the effects of the environmental factors almost periodicity is sometimes more realistic and more general than periodicity It is well known Mawhin s continuation theorem of coincidence degree theory is an important method to investigate the existence of positive periodic solutions of some kinds of non-linear ecosystems (see -) However it is difficult to be used to investigate the existence of positive almost periodic solutions of non-linear ecosystems Therefore to the best of the author s knowledge so far there are scarcely any papers concerning with the existence of

2 positive almost periodic solutions of system () by using Mawhin s continuation theorem 3-4 Motivated by the above reason the main purpose of this paper is to establish some new sufficient conditions on the existence of positive almost periodic solutions of system () by using Mawhin s continuous theorem of coincidence degree theory Related to a continuous function f we use the following notations: f l = inf f(s) f = sup f(s) f M = sup f(s) f = lim T T T f(s) ds Throughout this paper we always make the following assumption for system (): (H ) r i > and ā i > i = The paper is organized as follows In Section some basic definitions and necessary lemmas are given In Section 3 by using Mawhin s continuous theorem of coincidence degree theory some new sufficient conditions for the existence of at least one positive almost periodic solution of system () are established In Section 4 some simple applications are stated Finally some illustrative examples are given in Section 5 II PRELIMINARIES Definition (5 6) x C(R R n ) is called almost periodic if for any ϵ > it is possible to find a real number l = l(ϵ) > for any interval with length l(ϵ) there exists a number τ = τ(ϵ) in this interval such x(t τ) x(t) < ϵ t R where is arbitrary norm of R n τ is called to the ϵ-almost period of x T (x ϵ) denotes the set of ϵ-almost periods for x and l(ϵ) is called to the length of the inclusion interval for T (x ϵ) The collection of those functions is denoted by AP (R R n ) Let AP (R) := AP (R R) Next we present and prove several useful lemmas which will be used in later section Lemma (4) Assume x AP (R) C (R) with ẋ C(R) For arbitrary interval a b with b a = ω > let a b and I = s b : ẋ(s) then ones have x(t) x() ẋ(s) ds I t b Lemma (4) If x AP (R) then for arbitrary interval a b with I = b a = ω > there exist a b ( a and b ) such x() = x( ) and x() x(s) s Lemma 3 (4) If x AP (R) then for n N there exists α n R such x(α n ) x n x where x = sup x(s) For x AP (R) we denote by x = m(x) = lim T T T x(s) ds a(x ϖ) = lim T T Λ(x) = ϖ R : lim T T T T x(s)e iϖs ds x(s)e iϖs ds the mean value and the set of Fourier exponents of x respectively Lemma 4 (5) Assume x AP (R) and x > then for t R there exists a positive constant T independent of t such T tt t x(s) ds x 3 x T T III MAIN RESULTS The method to be used in this paper involves the applications of the continuation theorem of coincidence degree This requires us to introduce a few concepts and results from Gaines and Mawhin 7 Let X and Y be real Banach spaces L : DomL X Y be a linear mapping and N : X Y be a continuous mapping The mapping L is called a Fredholm mapping of index zero if ImL is closed in Y and dimkerl = codimiml < If L is a Fredholm mapping of index zero and there exist continuous projectors P : X X and Q : Y Y such ImP = KerL KerQ = ImL = Im(I Q) It follows L DomL KerP : (I P )X ImL is invertible and its inverse is denoted 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 exists an isomorphism J : ImQ KerL Lemma 5 (7) Let Ω X be an open bounded set L be a Fredholm mapping of index zero and N be L-compact on Ω If all the following conditions hold: (a) Lx λnx x Ω DomL λ ( ); (b) QNx x Ω KerL; (c) degjqn Ω KerL where J : ImQ KerL is an isomorphism Then Lx = Nx has a solution on Ω DomL Now we are in the position to present and prove our result on the existence of at least one positive almost periodic solution for system () From (H ) for a R there exists a constant ω ( ) independent of a such T at a r i (s) ds ri 3 r i at āi a i (s) ds T a 3ā i T ω i = ( ) In addition we introduce two numbers two functions and a assumption as follows: k i := α i ln 3 r i ā i 3 r iω i = (s) = r (s) e β k b (s) (s) = r (s) e β k b (s) s R

3 where ω is defined as in ( ) (H ) i > i = Theorem Assume (H )-(H ) hold then system () admits at least one positive almost periodic solution Proof: Under the invariant transformation (x y) T = (e u e v ) T system () reduces to u(t) = a (t) b (t)e α u(t) c (t)e β v(µ (t)) d (t)e γ u(ν (t)) e δ v(σ (t)) := F (t) v(t) = a (t) b (t)e αv(t) c (t)e βu(µ(t)) d (t)e γu(ν(t)) e δv(σ(t)) := F (t) (3) It is easy to see if system (3) has one almost periodic solution (u v) T then (x y) T = (e u e v ) T is a positive almost periodic solution of system () Therefore to complete the proof it suffices to show system (3) has one almost periodic solution Set X = Y = V V where V = (u v) T AP (R) : ϖ Λ(u) Λ(v) ϖ γ V = z = (u v) T (k k ) T k k R where γ is a given positive constant Define the norm z X = max sup u(s) sup v(s) z X = Y then X and Y are Banach spaces with the norm X Set L : DomL X Y Lz = L(u v) T = ( u v) T where DomL = z = (u v) T X : u v C (R) u v C(R) and u(t) F (t) N : X Y Nz = N = v(t) F (t) With these notations system (3) can be written in the form Lz = Nz z X It is not difficult to verify KerL = V ImL = V is closed in Y and dim KerL = = codim ImL Therefore L is a Fredholm mapping of index zero (see Lemma in 4) Now define two projectors P : X X and Q : Y Y as u m(u) u P z = P = = Qz z = X = Y v m(v) v Then P and Q are continuous projectors such ImP = KerL and ImL = KerQ = Im(I Q) Furthermore through an easy computation we find the inverse K P : ImL KerP DomL of L P has the form t u K P z = K P = u(s) ds m t u(s) ds v t v(s) ds m t v(s) ds Then QN : X Y and K P (I Q)N : X X read u m(f ) QNz = QN = v m(f ) fu(t) Qfu(t) K P (I Q)Nz = z ImL fv(t) Qfv(t) where f(x) is defined by fx(t) = t Nx(s) QNx(s) ds Then N is L-compact on Ω (see Lemma 3 in 4) In order to apply Lemma 5 we need to search for an appropriate open-bounded subset Ω Corresponding to the operator equation Lz = λz λ ( ) we have u (t) = λ r (t) a (t)e αu(t) b (t)e βv(µ(t)) c (t)e γu(ν(t)) e δv(σ(t)) (3) v (t) = λ r (t) a (t)e αv(t) b (t)e βu(µ(t)) c (t)e γu(ν(t)) e δv(σ(t)) Suppose (u v) T DomL X is a solution of system (3) for some λ ( ) By Lemma 3 there exist two sequences T n : n N and P n : n N such u(t n ) x n u u = sup u(s) n N (3) v(p n ) v n v v = sup v(s) n N (33) For n N we consider T n ω T n and P n ω P n where ω is defined as in ( ) By Lemma there exist T n ω T n ( T n ω and T n ) such u() = u( ) and u() u(s) s (34) Integrating the first equation of system (3) from to leads to r (s) a (s)e αu(s) b (s)e βv(µ(s)) c (s)e γ u(ν (s)) e δ v(σ (s)) ds = (35) which yields from (34) a (s)e αu(n u ) ds which yields from ( ) u() ln α a (s)e αu(s) ds r (s) ds r (s) ds ln 3 r (36) α ā a (s) ds Similar to the argument as in (36) there exists ζ P n ω P n so v(ζ) α ln 3 r ā (37) Let I = s T n : u(s) It follows from system (3) u(s) ds = λ r (s) a (s)e αu(s) b (s)e βv(µ(s)) I I c (s)e γ u(ν (s)) e δ v(σ (s)) ds

4 Tn λr (s) ds I r (s) ds T n ω 3 r ω (38) Let I = s ζ P n : v(s) Similar to the argument as in (38) we can easily obtain I v(s) ds Pn P n ω r (s) ds 3 r ω (39) By Lemma it follows from (36)-(39) u(t) u() u(s) ds ln 3 r I α ā := k t T n 3 r ω v(t) v(ζ) v(s) ds ln 3 r 3 r ω I α ā := k t ζ P n which imply u(t n ) k and v(p n ) k In view of (3)-(33) letting n inequalities leads to u = v = in the above lim u(t n n ) k (3) lim v(p n n ) k (3) In view of (35) by the integral mean value theorem there exists s such r (s ) a (s )e α u(s ) b (s )e β v(µ (s )) c (s )e γ u(ν (s )) e δ v(σ (s )) = which implies p := inf r (s) a (s)e αk b (s)e βk c (s)e γk e δk inf r (s) a (s)e αu(s) b (s)e βv(µ(s)) c (s)e γu(ν(s)) e δv(σ(s)) r (s ) a (s )e α u(s ) b (s )e β v(µ (s )) = c (s )e γu(ν(s)) e δv(σ(s)) Similarly we can easily obtain q := inf r (s) a (s)e α k b (s)e β k c (s)e γ k e δ k Substituting (3)-(3) into system (3) we obtain u(t) λ r (t) a (t)e αk b (t)e βk c (t)e γk e δk λp p v(t) λ r (t) a (t)e αk b (t)e βk c (t)e γk e δk λq q Under the invariant transformation (u v) T = (ln x ln y) T for x > and y > the above system changes to e ẋ(t) p p x(t) t x(t) ẏ(t) q y(t) e q t y(t) (3) For any s t integrating (3) from s to t leads to e u(s) e p(ts) e u(t) e v(s) e q(ts) e v(t) (33) Substituting (33) into system (3) leads to u(t) λ r (t) a (t)e αu(t) b (t)e βv(µ(t)) c (t)e p(tν(t))γ e δv(σ(t)) e γu(t) v(t) λ r (t) a (t)e αv(t) b (t)e β u(µ (t)) c (t)e γ u(ν (t)) e q (tσ (t))δ e δ v(t) From (H ) and Lemma 4 for a R there exists a constant ˆω ω ) independent of a such at i i (s) ds T a 3 i (34) where T ˆω i = For n Z by Lemma we can conclude there exist n ˆω n ˆω ˆω ( n ˆω and n ˆω ˆω ) such u() = u( ) and u() u(s) s (35) Integrating the first inequality of system (34) from to leads to r (s) b (s)e β v(µ (s)) ds c (s)e p (sν (s))γ e δ v(σ (s)) e γ u(s) which implies from (36) = (s) ds (r (s) b (s)e βk ) ds (r (s) b (s)e βv(µ(s)) ) ds ( a (s)e α u(s) c (s)e γ p (sν (s)) e δ v(σ (s)) e γ u(s) a (s)e α u(s) ds (36) ) ds

5 ( a (s)e αu() c (s)e γp(sν(s)) e δv(σ(s)) e γu() ) ds (37) If e u() < then maxe αu() e γu() e κl u() where κ l = minα γ By (37) we obtain from ( ) and (35) u() κ l ln (s) ds a (s) c (s)e γp(sν(s)) e δv(σ(s)) ds κ l ln a M cm ep (sν (s)) M γ e k δ = κ l Γ (38) where Γ = ln a M cm ep(sν(s))m γ e k δ If e u() then maxe αu() e γu() e κm u() where κ M = maxα γ Similar to (38) we obtain from ( ) and (35) u() κ M Γ (39) From (38)-(39) we obtain u() min κ l Γ κ M Γ := Π (3) Similar to the argument as in (3) it is not difficult to obtain there exists ς n ˆω n ˆω ˆω so v(ς) min υ l Γ υ M Γ := Π (3) where υ l = minα γ υ M = maxα γ and Γ = ln a M cm eq(sσ(s))m δ e k γ Further it follows from system (3) n ˆωˆω n ˆω n ˆωˆω u(s) ds = λ r (s) a (s)e α u(s) n ˆω b (s)e β v(µ (s)) c (s)e γ u(ν (s)) e δ v(σ (s)) ds r M a M e k α b M e k β c M e k γ e k δ n ˆωˆω ˆω := Θ (3) v(s) ds n ˆω r M a M e k α b M e k β c M e k γ e k δ In view of (3)-(33) it follows u(t) u() n ˆωˆω n ˆω u(s) ds ˆω := Θ (33) Π Θ := k 3 t n ˆω n ˆω ˆω (34) v(t) v(ς) n ˆωˆω n ˆω v(s) ds Π Θ := k 4 t n ˆω n ˆω ˆω (35) Obviously k 3 and k 4 are constants independent of n So it follows from (34)-(35) u = inf u(s) = inf min u(s) k 3 (36) n Z s n ˆωn ˆωˆω v = inf v(s) = inf min v(s) k 4 (37) n Z s n ˆωn ˆωˆω Set K = k k k 3 k 4 then w = (u v) T < K Clearly K is independent of λ ( ) Consider the algebraic equations QNz = for z = (u v ) T R as follows: m(r ) m(a )e α u m(b )e β v m(c )e γu e δv = m(r ) m(a )e α v m(b )e β u (38) m(c )e γu e δv = From the first equation of system (38) we have m(r ) m(a )e α u which implies u ln r < k (39) α ā Similarly we also obtain v ln r < k (33) α ā Furthermore by the first equation of system (38) and (33) we have m( ) = m(r ) e β k m(b ) m(r ) m(b )e β v = m(a )e αu m(c )e γu e δv ( a M c M e k δ) maxe κl u e κm u where κ l = minα γ and κ M = maxα γ Therefore we obtain Similarly we have u k 3 (33) v k 4 (33) where υ l = minα γ υ M = maxα γ In view of (39)-(33) we can easily obtain z X = u v < K Let Ω = z X : z X < K then Ω satisfies conditions (a) and (b) of Lemma 5 Finally we will show condition (c) of Lemma 5 is

6 satisfied Let us consider the homotopy H(ι z) = ιqnz ( ι)f z (ι z) R where ( ) ( u m(r ) m ( a F z = F = )e α u v m(r ) m(a )e αv ) From the above discussion it is easy to verify H(ι z) on Ω KerL ι By the invariance property of homotopy direct calculation produces deg ( JQN Ω KerL ) = deg ( F Ω KerL ) where deg( ) is the Brouwer degree and J is the identity mapping since ImQ = KerL Obviously all the conditions of Lemma 5 are satisfied Therefore system (3) has one almost periodic solution is system () has at least one positive almost periodic solution This completes the proof IV GLOBAL ASYMPTOTICAL STABILITY Lemma 6 (33) Assume a > b > and ẋ x(b ax α ) where α is positive constant then b α lim sup x(t) t a Lemma 7 Assume (H ) holds then any positive solution (x y) T of system () satisfies where lim sup x(t) M t r M = a lim sup y(t) N t α r α N = a Proof: In view of the first equation of system () we have ẋ(t) x(t) r a xα (t) which implies from Lemma 6 r α lim sup x(t) := M t Further from the second equation of system () it follows ẏ(t) y(t) r a yα (t) which implies from Lemma 6 r α lim sup y(t) := N t This completes the proof Theorem Assume (H )-(H ) hold Suppose further a c γ K δγα α ν b β K β α α µ > a c γ K δγα α σ a a c γ K δγα α ν c γ K δγα α σ b β K β α α > where K = minm N Then system () has a unique positive almost periodic solution which is globally asymptotically stable Proof: By Theorem we know system () has at least one positive almost periodic solution (x y ) T Suppose (x y) T is another positive solution of system () From (H 3 ) there must exist ϵ > and θ > such a c γ (K ϵ) δγα α ν c γ (K ϵ) δ γ α α ν b β (K ϵ) β α α µ > θ a c γ (K ϵ) δ γ α α σ c γ (K ϵ) δ γ α α σ b β (K ϵ) βα α > θ From Lemma 7 there must exist T > such Define where x(t) < K ϵ y(t) < K ϵ t T V (t) = 3 V i (t) i= V (t) = ln x(t) ln x ln y(t) ln y V (t) t c = γ (K ϵ) δγα ν (t) α ν x α (s) x α (s) ds t c γ (K ϵ) δ γ α ν (t) α ν x α (s) x α (s) ds t b β (K ϵ) β α µ (t) α µ x α (s) x α (s) ds = V 3 (t) t σ (t) t σ (t) t µ (t) c γ (K ϵ) δγα α σ y α (s) y α (s) ds c γ (K ϵ) δ γ α α σ y α (s) y α (s) ds b β (K ϵ) β α α µ y α (s) y α (s) ds For t T calculating the upper right derivative of V i (i = 3) along the solution of system () it follows D V (t)

7 a xα (t) x α b yβ (µ (t)) y β (µ (t)) c yδ (σ (t)) x γ (ν (t)) x γ (ν (t)) c x γ (ν (t)) y δ (σ (t)) y δ (σ (t)) a yα (t) y α b xβ (µ (t)) x β (µ (t)) c yδ (σ (t)) x γ (ν (t)) x γ (ν (t)) c x γ (ν (t)) y δ (σ (t)) y δ (σ (t)) a xα (t) x α b β (K ϵ) β α y α (µ (t)) y α (µ (t)) α c γ (K ϵ) δ γ α x α (ν (t)) x α (ν (t)) α c γ (K ϵ) δγα y α (σ (t)) y α (σ (t)) α a yα (t) y α b β (K ϵ) βα x α (µ (t)) x α (µ (t)) α c γ (K ϵ) δ γ α x α (ν (t)) x α (ν (t)) α c γ (K ϵ) δγα y α (σ (t)) y α (σ (t)) α D V (t) c γ (K ϵ) δγα α ν x α (t) x α c γ (K ϵ) δ γ α x α (ν (t)) x α (ν (t)) α c γ (K ϵ) δγα α ν x α (t) x α c γ (K ϵ) δγα x α (ν (t)) x α (ν (t)) α b β (K ϵ) β α α µ x α (t) x α b β (K ϵ) βα x α (µ (t)) x α (µ (t)) α D V 3 (t) c γ (K ϵ) δγα α σ y α (t) y α c γ (K ϵ) δ γ α y α (σ (t)) y α (σ (t)) α c γ (K ϵ) δ γ α α σ y α (t) y α c γ (K ϵ) δγα y α (σ (t)) y α (σ (t)) α b β (K ϵ) β α y α (t) y α α b β (K ϵ) βα α µ y α (µ (t)) y α (µ (t)) Together with the above inequalities it follows D V (t) a c γ (K ϵ) δγα α ν c γ (K ϵ) δ γ α α ν b β (K ϵ) β α α µ x α (t) x α a c γ (K ϵ) δγα α σ c γ (K ϵ) δ γ α α σ b β (K ϵ) β α y α (t) y α α θ x α (t) x α y α (t) y α Therefore V is non-increasing Integrating the last inequality from T to t leads to is T t V (t) θ x α (s) x α (s) T y α (s) y α (s) ds V (T ) < t T x α (s) x α (s) y α (s) y α (s) ds < which implies lim s xα (s) x α (s) y α (s) y α (s) = Thus the almost periodic solution of system () is asymptotical stability The global asymptotical stability implies the almost periodic solution is unique This completes the proof V SOME APPLICATIONS Application V Consider the following logistic equation Ṅ(t) = N(t) r(t) a(t)n(t) b(t)n(t µ) (5) where µ is a constant r a and b are nonnegative almost periodic functions with m(r) > and m(a b) > In 8 Xie and Li obtained Corollary Eq (5) admits at least one positive almost periodic solution Corollary Assume all the coefficients in Eq (5) are π-periodic then Eq (5) admits at least one positive π-periodic solution As regards the existence of a periodic solution in Eq (5) Freedman and Wu 9 proved an existence result for E- q (5) by introducing a different assumption: Theorem 3 Assume r(t) > a(t) > b(t) for all t π and suppose the delayed functional equation r(t) a(t)ψ(t) b(t)ψ(t µ) =

8 has a positive and continuously differentiable π-periodic solution then Eq (5) has a positive π-periodic solution Further Lisena 3 obtained Theorem 4 Assume m(r) > a(t) > b(t) for all t π and suppose there exists a positive and continuously differentiable π-periodic function ϕ such a(t)ϕ(t) b(t)ϕ(t µ) = t π Then Eq (4) has a positive π-periodic solution Remark Obviously Corollary improves some conditions in Theorems 3-4 Application V Consider non-autonomous model of phytoplankton allelopathy as follows: ẋ(t) = x(t) r (t) a (t)x(t) b (t)y(t) c (t)x(t)y(t) (5) ẏ(t) = y(t) r (t) a (t)y(t) b (t)x(t) c (t)x(t µ)y(t) where all the coefficients of (5) are nonnegative almost periodic functions Let k i := ln 3 r i ā i 3 r iω i = (s) = r (s) e k b (s) (s) = r (s) e k b (s) s R where ω is defined as in ( ) Similar to the proof of Theorem we can easily show Corollary 3 Assume m( i ) > (i = ) then system (5) admits at least one almost periodic solution Remark In 3 the authors obtained a existence result for the almost periodic solutions of system (5) on condition r r a and a are strictly positive However Corollary 3 broaden such condition Therefore Corollary 3 improves the work in 3 Application V3 Consider the following a delayed Logistic equation Ṅ(t) = N(t) r(t) a(t)n p (t) b(t)n q (t µ(t)) (53) where p q are positive constants µ r a b are nonnegative π-periodic functions m(r) > and m(a b) > Similar to the proof of Theorem we can easily obtain Corollary 4 Eq (53) admits at least one positive πperiodic solution In 3 Chen obtained Theorem 5 Assume p q then Eq (53) admits at least one positive π-periodic solution Remark 3 It is clear Corollary 4 removes the condition p q in Theorem 5 Therefore our result improves the work in 3 VI EXAMPLES AND SIMULATIONS Example Consider the following Gilpin-Ayala competition model of phytoplankton allelopathy with time-varying delays with different periods: ẋ(t) = x(t) sin 3t x(t) sin ( t) e y (µ(t)) x (µ(t))y 3 (µ(t)) ẏ(t) = y(t) 3 cos ( (6) t)y(t) cos 3t e x 7 (ν(t)) x 4 (ν(t))y 3 (ν(t)) where µ(t) = t sin t and ν(t) = t sin t t R Corresponding to system () we have r = r = 3 ā = π bm = b M = e ā = α = α = β = β = 7 Further for a R we can choose ω = 3π 3 so ( ) holds is at a (s) ds T π 3 π a at a (s) ds T a By a easy calculation we obtain Hence T ω = 3π 3 k := ln 3π 3π k := ln 9π 3 3π 8973 e > 3 e > Therefore all the conditions of Theorem are satisfied By Theorem system (6) admits at least one positive almost periodic solution (see Figures -) x(t) time t Fig State variable x of system (5)

9 y(t) time t Fig State variable y of system (5) 3π Remark 4 In system (6) sin 3t is 3 -periodic function and cos ( t) is π -periodic function So system (6) is with incommensurable periods Through all the coefficients of system (6) are periodic functions the positive periodic solutions of system (6) could not possibly exist However by Theorem the positive almost periodic solutions of system (6) exactly exist VII CONCLUSION In this paper some sufficient conditions are established for the existence uniqueness and global asymptotical stability of almost periodic solution for a Gilpin-Ayala competition model of phytoplankton allelopathy The main result obtained in this paper are completely new Besides the method used in this paper may be used to study the almost periodic dynamics of many other biological models such as predatorprey models facultative mutualism models and so on ACKNOWLEDGMENTS This work are supported by the Funding for Applied Technology Research and Development of Panzhihua City under Grant 5CY-S-4 and Natural Science Foundation of Education Department of Sichuan Province under Grant 5ZB49 REFERENCES A Lotka Elements of Physical Biology Williams and Wilkins Baltimore Md 94 V Volterra Lecons Sur la Theorie Mathematique de la Lutte pour la Vie Gauthier-Villars Paris 93 3 X He K Gopalsamy Persistence attractivity and delay in facultative mutualism J Math Anal Appl 5: H Bereketoglu I Gyori Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay J Math Anal Appl : FJ Ayala ME Gilpin JG Eherenfeld Competition between species: Theoretical models and experimental tests Theor Popul Biol 4: M Fan K Wang Global periodic solutions of a generalized n-species Gilpin-Ayala competition model Comput Math Appl 4: SW Zhang DJ Tan LS Chen The periodic n-species Gilpin-Ayala competition system with impulsive effect Chaos Solitons & Fractals 6: FD Chen LP Wu Z Li Permanence and global attractivity of the discrete Gilpin-Ayala type population model Comput Math Appl 53: MX He Z Li FD Chen Permanence extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses Nonlinear Anal: RWA : Q Wang MM Ding ZJ Wang HY Zhang Existence and attractivity of a periodic solution for an n-species Gilpin-Ayala impulsive competition system Nonlinear Anal: RWA : TW Zhang YK Li Positive periodic solutions for a generalized impulsive n-species Gilpin-Ayala competition system with continuously distributed delays on time scales Int J Biomath 4: 3-34 CJ Xu MX Liao Stability and bifurcation analysis in a seir epidemic model with nonlinear incidence rates IAENG International Journal of Applied Mathematics 4: CJ Xu QM Zhang L Lu Chaos Control in A 3D Ratio-dependent Food Chain System IAENG International Journal of Applied Mathematics 44: M Fan K Wang Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system Math Comput Model 35: M Fan K Wang Global existence of positive periodic solutions of periodic predator-prey system with infinite delays J Math Anal Appl 6: - 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