MULTIPLE PERIODIC SOLUTIONS FOR A DISCRETE TIME MODEL OF PLANKTON ALLELOPATHY JIANBAO ZHANG AND HUI FANG Received 19 May 2005; Revised 25 September 2005; Accepted 27 September 2005 We study a discrete time model of the growth of two species of plankton with compet- itive and allelopathic effects on each other N 1 (k +1)= N 1 (k)exp{r 1 (k) − a 11 (k)N 1 (k) − a 12 (k)N 2 (k) − b 1 (k)N 1 (k)N 2 (k)}, N 2 (k +1)= N 2 (k)exp{r 2 (k) − a 21 (k)N 1 (k) − a 22 (k) N 2 (k) − b 2 (k)N 1 (k)N 2 (k)}.Asetofsufficient conditions is obtained for the existence of multiple positive periodic solutions for this model. The approach is based on Mawhin’s continuation theorem of coincidence degree theory as well as some a prior i estimates. Some new results are obtained. Copyright © 2006 J. Zhang and H. Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, prov i ded the original work is properly cited. 1. Introduction Many researchers have noted that the increased population of one species of phytoplank- ton might affect the growth of one or several other species by the production of allelo- pathic toxins or stimulators, influencing bloom, pulses, and seasonal succession. The study of allelopathic interactions in the phytoplanktonic world has become an impor- tant subject in aquatic ecology. For detailed studies, we refer to [1, 2, 7, 9–11, 13]and references cited therein. Maynard-Smith [9] and Chattopadhyay [2] proposed the following two species Lotka- Volterra competition system, which descr ibes the changes of size and density of phyto- plankton: dN 1 dt = N 1  r 1 − a 11 N 1 (t) − a 12 N 2 (t) − b 1 N 1 (t)N 2 (t)  , dN 2 dt = N 1  r 2 − a 21 N 1 (t) − a 22 N 2 (t) − b 2 N 1 (t)N 2 (t)  , (1.1) where b 1 and b 2 are the rates of toxic inhibition of the first species by the second and vice versa, respectively . Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 90479, Pages 1–14 DOI 10.1155/ADE/2006/90479 2 Periodic solutions for a discrete plankton model Naturally, more realistic models require the inclusion of the periodic changing of envi- ronment (e.g., seasonal effects of weather, food supplies, etc). For such systems, as pointed out by Freedman and Wu [5] and Kuang [8], it would be of interest to study the existence of periodic solutions. This motivates us to modify system (1.1)totheform dN 1 dt = N 1 (t)  r 1 (t) − a 11 (t)N 1 (t) − a 12 (t)N 2 (t) − b 1 (t)N 1 (t)N 2 (t)  , dN 2 dt = N 1 (t)  r 2 (t) − a 21 (t)N 1(t) − a 22 (t)N 2 (t) − b 2 (t)N 1 (t)N 2 (t)  , (1.2) where r i (t), a ij (t) > 0, b i (t) > 0(i, j = 1,2) are continuous ω-periodic functions. The main purpose of this paper is to propose a discrete analogue of system (1.2)and to obtain sufficient conditions for the existence of multiple positive periodic solutions by employing the coincidence degree theory. To our knowledge, no work has been done for the existence of multiple positive periodic solutions for this model using this way. The paper is organized as follows. In Section 2, we propose a discrete analogue of sys- tem (1.2). In Section 3, motivated by the recent work of Fan and Wang [4]andChen[3], we study the existence of multiple positive periodic solutions of the difference equations derived in Section 2. 2. Discrete analogue of system (1.2) Assume that the average growth rates in (1.2) change at equally spaced time intervals and estimates of the population size are made at equally spaced time intervals, then we can incorporate this aspect in (1.2) and obtain the following system: dN 1 (t) dt 1 N 1 (t) = r 1  [t]  − a 11  [t]  N 1  [t]  − a 12  [t]  N 2  [t]  − b 1  [t]  N 1  [t]  N 2  [t]  , dN 2 (t) dt 1 N 2 (t) = r 2  [t]  − a 21  [t]  N 1  [t]  − a 22  [t]  N 2  [t]  − b 2  [t]  N 1  [t]  N 2  [t]  , (2.1) where t = 0,1,2, ,[t] denotes the integer part of t, t ∈ (0,+∞). By a solution of (2.1), we mean a function x = (x 1 ,x 2 ) T , which is defined for t ∈ [0,+∞), and p ossesses the following properties. (1) x is continuous on [0,+ ∞). (2) The derivatives dx 1 (t)/dt, dx 2 (t)/dt exist at each point t ∈ [0,+∞) w ith the pos- sible exception of the points t ∈{0,1, 2, }, where left-sided derivatives exist. The equations in (2.1) are satisfied on each inter v al [k, k +1)withk = 0,1,2, For k ≤ t<k+1,k = 0,1,2, , integrating (2.1)fromk to t,weobtain N 1 (t) = N 1 (k)exp  r 1 (k) − a 11 (k)N 1 (k) − a 12 (k)N 2 (k) − b 1 (k)N 1 (k)N 2 (k)  (t − k)  , N 2 (t) = N 2 (k)exp  r 2 (k) − a 21 (k)N 1 (k) − a 22 (k)N 2 (k) − b 2 (k)N 1 (k)N 2 (k)  (t − k)  . (2.2) J. Zhang and H. Fang 3 Letting t → k +1,wehave N 1 (k +1)= N 1 (k)exp  r 1 (k) − a 11 (k)N 1 (k) − a 12 (k)N 2 (k) − b 1 (k)N 1 (k)N 2 (k)  , N 2 (k +1)= N 2 (k)exp  r 2 (k) − a 21 (k)N 1 (k) − a 22 (k)N 2 (k) − b 2 (k)N 1 (k)N 2 (k)  , (2.3) for k = 0,1,2, Equation (2.3) is a discrete analogue of system (1.2). Notice that the periodicity of parameters of (2.1)issufficient, but not necessary for the periodicity of coefficients in (2.3). In system (2.3), we always assume that r i , a ij > 0, b i > 0(i, j = 1,2) are ω-periodic, that is, r i (k + ω) = r i (k), b i (k + ω) = b i (k), a ij (k + ω) = a ij (k), (2.4) for any k ∈ Z (the set of all integers), i, j = 1,2, where ω, a fixed positive integer, denotes the prescribed common period of the parameters in (2.3). 3. Existence of multiple positive periodic solutions In this section, in order to obtain the existence of multiple positive periodic solutions of (2.3), we first make the following preparations. Let X and Y be normed vector spaces. Let L :DomL ⊂ X → Y be a linear mapping and N : X → Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim kerL = codim Im L<∞ and ImL is closed in Z.IfL is a Fredholm mapping of index zero, then there exist continuous projectors P : X → X and Q : Y → Y such that ImP = ker L and ImL = ker Q = Im(I − Q). It follows that L | DomL ∩ kerP :(I − P)X → ImL is invertible and its inverse is denoted by K p .IfΩ is a bounded open subset of X, the mapping N is called L-compact on Ω if (QN)(Ω)is bounded and K p (I − Q)N : Ω → X is compact. Because ImQ is isomorphic to kerL,there exists an isomorphism J :ImQ → kerL. For convenience, we introduce Mawhin’s continuation theorem as follows. Lemma 3.1 [6, page 40] (Continuation theorem). Let L be a Fredholm mapping of index zero and let N : ¯ Ω → Z be L-compact on ¯ Ω.Suppose (a) Lx = λNx for every x ∈ domL ∩ ∂Ω and every λ ∈ (0,1); (b) QNx = 0 for every x ∈ ∂Ω ∩ Ker L,andBrouwerdegree deg B  JQN,Ω ∩ Ker L,0) = 0. (3.1) Then Lx = Nx has at least one s olution in domL ∩ ¯ Ω. Let Z, Z + , R, R + ,andR 2 denote the sets of all integers, nonnegative integers, real num- bers, nonnegative real numbers, and two-dimensional Euclidean vector space, respec- tively. 4 Periodic solutions for a discrete plankton model Suppose {g(k)} is an ω-periodic (ω ∈ Z + ) sequence of real numbers defined for k ∈ Z. Throughout this paper, we will use the following notation: I ω =  0,1, ,ω − 1  , g = 1 ω ω−1  k=0 g(k), ¯ R i = 1 ω ω−1  k=0   r i (k)   , α ij = ¯ a ji ¯ b i − ¯ a ii ¯ b j , α  ij = ¯ a ji ¯ b i − ¯ a ii ¯ b j e ¯ R j ω , α  ij =  ¯ a ji ¯ b i e ¯ R j ω − ¯ a ii ¯ b j  e ¯ R i ω , β ij = ¯ a ii ¯ a jj + ¯ b i ¯ r j − ¯ a ij ¯ a ji − ¯ b j ¯ r i , β  ij = ¯ a ii ¯ a jj e ¯ R j ω + ¯ b i ¯ r j − ¯ a ij ¯ a ji e ¯ R i ω − ¯ b j ¯ r i e ( ¯ R i + ¯ R j )ω , β  ij = ¯ a ii ¯ a jj e ¯ R i ω + ¯ b i ¯ r j e ( ¯ R i + ¯ R j )ω − ¯ a ij ¯ a ji e ¯ R j ω − ¯ b j ¯ r i , γ ij = ¯ r i ¯ a jj − ¯ r j ¯ a ij , γ  ij =  ¯ r i ¯ a jj e ¯ R j ω − ¯ r j ¯ a ij  e ¯ R i ω , γ  ij = ¯ r i ¯ a jj − ¯ r j ¯ a ij e ¯ R j ω , i, j = 1,2, i = j, N 1 (α,β,γ) = β −  β 2 − 4αγ 2α , N 2 (α,β,γ) = β +  β 2 − 4αγ 2α (α = 0, β 2 − 4αγ > 0  . (3.2) Define l 2 =  x =  x(k)  : x(k) ∈ R 2 , k ∈ Z  . (3.3) For a = (a 1 ,a 2 ) T ∈ R 2 ,define|a|=max{|a 1 |,|a 2 |}.Letl ω ⊂ l 2 denote the subspace of all ω-periodic sequences equipped with the usual supremum norm ·, that is, for x = { x(k):k ∈ Z}∈l ω , x=max k∈I ω |x(k)|. It is not difficult to show that l ω is a finite- dimensional Banach space. Let the linear operator S : l ω → R 2 be defined by S(x) = 1 ω ω−1  k=0 x(k), x =  x(k):k ∈ Z  ∈ l ω . (3.4) Then we obtain two subspaces l ω 0 and l ω c of l ω defined by l ω 0 =  x =  x(k)  ∈ l ω : S(x) = 0  , l ω c =  x =  x(k)  ∈ l ω : x(k) ≡ β,forsomeβ ∈ R 2 and ∀k ∈ Z  , (3.5) respectively. Denote by L : l ω → l ω the difference operator given by Lx ={(Lx)(k)} with (Lx)(k) = x(k +1)− x(k), for x ∈ l ω and k ∈ Z. (3.6) Let a linear operator K : l ω → l ω c be defined by Kx={(Kx)(k)} with (Kx)(k) ≡ S(x), for x ∈ l ω and k ∈ Z. (3.7) Then we have the following lemma. J. Zhang and H. Fang 5 Lemma 3.2 [12, Lemma 2.1]. (i) Both l ω 0 and l ω c are closed linear subspaces of l ω and l ω = l ω 0 ⊕ l ω c , diml ω c = 2. (ii) L is a bounded linear operator with ker L = l ω c and ImL = l ω 0 . (iii) K is a bounded linear operator with ker(L + K) ={0} and Im(L + K) = l ω . Lemma 3.3. Let g, r : Z → R be ω-periodic, that is, g(k +ω) = g(k), r(k + ω) = r(k). Assume that for any k ∈ Z, g(k +1) − g(k) ≤   r(k)   . (3.8) Then for any fixed k 1 ,k 2 ∈ I ω ,andanyk ∈ Z, one has g(k) ≤ g  k 1  + ω−1  s=0   r(s)   , g(k) ≥ g  k 2  − ω−1  s=0   r(s)   . (3.9) Proof. It is only necessary to prove that the inequalities hold for any k ∈ I ω .Forthefirst inequality, it is easy to see the first inequality holds if k = k 1 .Ifk>k 1 ,then g(k) − g  k 1  = k−1  s=k 1  g(s +1)− g(s)  ≤ k−1  s=k 1   r(s)   ≤ ω−1  s=0   r(s)   , (3.10) and hence, g(k) ≤ g(k 1 )+  ω−1 s =0 |r(s)|.Ifk<k 1 ,thenk + ω>k 1 . Therefore, g(k) − g  k 1  = g(k + ω) − g  k 1  = k+ω−1  s=k 1  g(s +1)− g(s)  ≤ k+ω−1  s=k 1   r(s)   ≤ k 1 +ω−1  s=k 1   r(s)   = ω−1  s=0   r(s)   , (3.11) equivalently, g(k) ≤ g(k 1 )+  ω−1 s =0 |r(s)|. Now we can claim that the first inequalit y is valid.  Similar to the above proof, we can prove that the second inequality is valid. In the following, we make the follow ing assumptions. (H 1 ) ¯ R i = (1/ω)  ω−1 k =0 |r i (k)|≥(1/ω)  ω−1 k =0 r i (k) > 0. (H 2 ) γ  ij = ¯ r i ¯ a jj − ¯ r j ¯ a ij e ¯ R j ω > 0, i = j, i, j = 1,2. (H 3 ) α  12 > 0. (H 4 ) β 12 /α 12 >β  12 /α  12 . Lemma 3.4 [13, Lemma 3.2]. Consider the following algebraic equations: ¯ a 11 N 1 + ¯ a 12 N 2 + ¯ b 1 N 1 N 2 = ¯ r 1 , ¯ a 21 N 1 + ¯ a 22 N 2 + ¯ b 2 N 1 N 2 = ¯ r 2 . (3.12) Assuming that (H 1 ), (H 2 ) hold, then the following conclusions hold. 6 Periodic solutions for a discrete plankton model (i) If α 12 > 0,then(3.12) have two positive solutions:  N i  α 12 ,β 12 ,γ 12  ,N 1  α 21 ,β 21 ,γ 21  , i = 1,2. (3.13) (ii) If α 21 > 0,then(3.12) have two positive solutions:  N 1  α 12 ,β 12 ,γ 12  ,N i  α 21 ,β 21 ,γ 21  , i = 1,2. (3.14) Lemma 3.5. Assume that (H 1 )–(H 3 ) hold, then the following conclusions hold. (i) β 12 > 0, β 2 12 − 4α 12 γ 12 > 0; (ii) β  12 > 0, β  2 12 − 4α  12 γ  12 > 0. Proof. (i) β 12 =  ¯ b 1 ¯ a 11 + ¯ a 12 ¯ r 1  γ 21 +  ¯ r 1 α 12 ¯ a 11 + ¯ a 11 γ 12 ¯ r 1  > 0, β 2 12 − 4α 12 γ 12 =  ¯ b 1 ¯ a 11 + ¯ a 12 ¯ r 1  2 γ 2 21 +  ¯ r 1 α 12 ¯ a 11 − ¯ a 11 γ 12 ¯ r 1  2 +2  ¯ b 1 ¯ a 11 + ¯ a 12 ¯ r 1  ¯ r 1 α 12 ¯ a 11 + ¯ a 11 γ 12 ¯ r 1  γ 21 > 0. (3.15) (ii) β  12 =  ¯ b 1 ¯ a 11 + ¯ a 12 ¯ r 1  γ  21 +  ¯ r 1 α  12 e ¯ R 1 ω ¯ a 11 + ¯ a 11 γ  12 ¯ r 1 e ¯ R 1 ω  > 0, β  2 12 − 4α  12 γ  12 =  ¯ b 1 ¯ a 11 + ¯ a 12 ¯ r 1  2 γ  2 21 +  ¯ r 1 α  12 e ¯ R 1 ω ¯ a 11 − ¯ a 11 γ  12 ¯ r 1 e ¯ R 1 ω  2 +2  ¯ b 1 ¯ a 11 + ¯ a 12 ¯ r 1  ¯ r 1 α  12 e ¯ R 1 ω ¯ a 11 + ¯ a 11 γ  12 ¯ r 1 e ¯ R 1 ω  γ  21 > 0. (3.16)  Lemma 3.6. Assume that (H 1 )–(H 4 ) hold, then the following conclusions hold, N 1  α 12 ,β 12 + m,γ 12 − n  <N 1  α 12 ,β 12 ,γ 12  <N 1  α  12 ,β  12 ,γ  12  <N 2  α  12 ,β  12 ,γ  12  <N 2  α 12 ,β 12 ,γ 12  <N 2  α 12 ,β 12 + m,γ 12 − n  , (3.17) where m = ¯ a 11 ¯ a 22  e ¯ R 1 ω − 1  + ¯ b 1 ¯ r 2  e ( ¯ R 1 + ¯ R 2 )ω − 1  > 0, n = ¯ a 12 ¯ r 2  e ¯ R 2 ω − 1  > 0. (3.18) Proof. Under the conditions that α>0, β>0, γ>0, β 2 − 4αγ > 0, we have N 1 (α,β,γ) = 2γ β +  β 2 − 4αγ = 2γ/α β/α +  β 2 /α 2 − 4(γ/α) = N 1  1, β α , γ α  , N 2 (α,β,γ) = β +  β 2 − 4αγ 2α = 1 2  β α +  β 2 α 2 − 4 γ α  = N 2  1, β α , γ α  . (3.19) J. Zhang and H. Fang 7 Thus N 1 (α,β,γ)(N 2 (α,β,γ)) is increasing (decreasing) in the first variable, decreasing (increasing) in the second variable, increasing (decreasing) in the third variable. Notice that α  12 >α 12 >α  12 > 0, γ  12 >γ 12 >γ  12 > 0, (3.20) we have γ  12 α  12 > γ 12 α 12 . (3.21) So from (3.19), (3.20), (3.21)and(H 1 )–(H 4 ), we obtain that N 1  α 12 ,β 12 + m,γ 12 − n  <N 1  α 12 ,β 12 ,γ 12  = N 1  1, β 12 α 12 , γ 12 α 12  <N 1  1, β  12 α  12 , γ  12 α  12  = N 1  α  12 ,β  12 ,γ  12  <N 2  α  12 ,β  12 ,γ  12  = N 2  1, β  12 α  12 , γ  12 α  12  <N 2  1, β 12 α 12 , γ 12 α 12  = N 2  α 12 ,β 12 ,γ 12  <N 2  α 12 ,β 12 + m,γ 12 − n  . (3.22)  Theorem 3.7. In addition to (H 1 )–(H 3 ), assume further that system (2.3)satisfies (H 5 ) N 1 (α 12 ,β 12 ,γ 12 ) <N 1 (α  12 ,β  12 ,γ  12 ) <N 2 (α 12 ,β 12 ,γ 12 ). Then system (2.3) has at least two positive ω-periodic solut ions. Proof. Since we are concerned with positive solutions of (2.3), we make the change of variables, N i (k) = exp  x i (k)  , i = 1,2. (3.23) Then (2.3)isrewrittenas x i (k +1)− x i (k) = r i (k) − a ii (k)exp  x i (k)  − a ij (k)exp  x j (k)  − b i (k)exp  x i (k)  exp  x j (k)  , (3.24) where i, j = 1,2, i = j.TakeX = Y = l ω ,(Lx)(k) = x(k +1)− x(k), and denote (ᏺx)(k) =  r 1 (k)−a 11 (k)exp  x 1 (k)  − a 12 (k)exp  x 2 (k)  − b 1 (k)exp  x 1 (k)  exp  x 2 (k)  r 2 (k)−a 22 (k)exp  x 2 (k)  − a 21 (k)exp  x 1 (k)  − b 2 (k)exp  x 2 (k)  exp  x 1 (k)   , (3.25) for any x ∈ X and k ∈ Z.ItfollowsfromLemma 3.2 that L is a bounded linear operator and ker L = l ω c ,ImL = l ω 0 ,dimkerL = 2 = codimImL, (3.26) then it follows that L is a Fredholm mapping of index zero. 8 Periodic solutions for a discrete plankton model Define Px = 1 ω ω−1  s=0 x(s), x ∈ X, Qy = 1 ω ω−1  s=0 y(s), y ∈ Y. (3.27) It is not difficult to show that P and Q are two continuous projectors such that ImP = kerL,ImL = kerQ = Im(I − Q). (3.28) Furthermore, the generalized inverse (of L) K p :ImL → kerP ∩ DomL exists and is given by K p (z) = k−1  s=0 z(s) − 1 ω ω−1  s=0 (ω − s)z(s). (3.29) Notice that Qᏺ, K p (I − Q)ᏺ are continuous and X is a finite-dimensional Banach space, it is not difficult to show that K p (I − Q)ᏺ(Ω)iscompactforanyopenboundedsetΩ ⊂ X. Moreover , Qᏺ( Ω)isbounded.Thus,ᏺ is L-compact on with any open bounded set Ω ⊂ X. Corresponding to the operator equation Lx = λᏺx, λ ∈ (0,1), we have x i (k +1)− x i (k) = λ  r i (k) − a ii (k)exp  x i (k)  − a ij (k)exp  x j (k)  − b i (k)exp  x i (k)  exp  x j (k)  , (3.30) where i, j = 1,2, i = j. Suppose that x = (x 1 (k),x 2 (k)) T ∈ X is a solution of (3.30)fora certain λ ∈ (0,1). Summing on both sides of (3.30)from0toω − 1aboutk,weget 0 = ω−1  k=0  x i (k +1)− x i (k)  = λ ω−1  k=0  r i (k) − a ii (k)exp  x i (k)  − a ij (k)exp  x j (k)  − b i (k)exp  x i (k)  exp  x j (k)  , (3.31) that is ¯ r i ω = ω−1  k=0  a ii (k)exp  x i (k)  + a ij (k)exp  x j (k)  + b i (k)exp  x i (k)  exp  x j (k)  , (3.32) where i, j = 1,2, i = j. It follows from (3.30)that x i (k +1)− x i (k) <   r i (k)   , k ∈ Z, i = 1,2. (3.33) Since x(t) ∈ X, there exist ξ i , η i ∈ I ω such that x i  ξ i  = min k∈I ω  x i (k)  , x i  η i  = max k∈I ω  x i (k)  , i = 1,2. (3.34) J. Zhang and H. Fang 9 From (3.32), (3.34), one obtains ¯ a 11 exp  x 1  η 1  + ¯ a 12 exp  x 2  η 2  + ¯ b 1 exp  x 1  η 1  exp  x 2  η 2  ≥ ¯ r 1 , (3.35) ¯ a 21 exp  x 1  ξ 1  + ¯ a 22 exp  x 2  ξ 2  + ¯ b 2 exp  x 1  ξ 1  exp  x 2  ξ 2  ≤ ¯ r 2 . (3.36) WecanderivefromLemma 3.3,(3.33)and(3.36)that x 2  η 2  ≤ x 2  ξ 2  + ¯ R 2 ω ≤ ln ¯ r 2 − ¯ a 21 exp  x 1  ξ 1  ¯ a 22 + ¯ b 2 exp  x 1  ξ 1  + ¯ R 2 ω, (3.37) which, together with (3.35), leads to exp  x 1  η 1  ≥ ¯ r 1 − ¯ a 12 exp  x 2  η 2  ¯ a 11 + ¯ b 1 exp  x 2  η 2  ≥ ¯ r 1  ¯ a 22 + ¯ b 2 exp  x 1  ξ 1  − ¯ a 12 exp  ¯ R 2 ω  ¯ r 2 − ¯ a 21 exp  x 1  ξ 1  ¯ a 11  ¯ a 22 + ¯ b 2 exp  x 1  ξ 1  + ¯ b 1 exp  ¯ R 2 ω  ¯ r 2 − ¯ a 21 exp  x 1  ξ 1  . (3.38) From Lemma 3.3 and (3.33), we have x 1  ξ 1  >x 1  η 1  − ¯ R 1 ω. (3.39) This is exp  x 1  ξ 1  > exp  x 1  η 1  exp  − ¯ R 1 ω  , (3.40) which, together with (3.38), leads to exp  ¯ R 1 ω  exp  x 1  ξ 1  > ¯ r 1  ¯ a 22 + ¯ b 2 exp  x 1  ξ 1  − ¯ a 12 exp  ¯ R 2 ω  ¯ r 2 − ¯ a 21 exp  x 1  ξ 1  ¯ a 11  ¯ a 22 + ¯ b 2 exp  x 1  ξ 1  + ¯ b 1 exp  ¯ R 2 ω  ¯ r 2 − ¯ a 21 exp  x 1  ξ 1  , (3.41) which implies α  12 exp  2x 1  ξ 1  − β  12 exp  x 1  ξ 1  + γ  12 < 0. (3.42) So from (3.20), one obtains α 12 exp  2x 1  ξ 1  −  β 12 + m  exp  x 1  ξ 1  + γ 12 − n<0, (3.43) where m = ¯ a 11 ¯ a 22  e ¯ R 1 ω − 1  + ¯ b 1 ¯ r 2  e ( ¯ R 1 + ¯ R 2  ω − 1  > 0, n = ¯ a 11 ¯ r 2  e ¯ R 2 ω − 1  > 0. (3.44) According to (i) of Lemma 3.5,weobtain  β 12 + m  2 − 4α 12  γ 12 − n  >β 2 12 − 4α 12 γ 12 > 0. (3.45) 10 Periodic solutions for a discrete plankton model Therefore, the equation α 12 x 2 −  β 12 + m  x + γ 12 − n = 0 (3.46) has two positive solutions N i  α 12 ,β 12 + m,γ 12 − n  , i = 1,2. (3.47) Thus, we have N 1  α 12 ,β 12 + m,γ 12 − n  < exp  x 1  ξ 1  <N 2  α 12 ,β 12 + m,γ 12 − n  . (3.48) In a similar way as the above proof, we can conclude from ¯ a 21 exp  x 1  η 1  + ¯ a 22 exp  x 2  η 2  + ¯ b 2 exp  x 1  η 1  exp  x 2  η 2  ≥ ¯ r 2 , ¯ a 11 exp  x 1  ξ 1  + ¯ a 12 exp  x 2  ξ 2  + ¯ b 1 exp  x 1  ξ 1  exp  x 2  ξ 2  ≤ ¯ r 1 , (3.49) that α  12 exp  2x 1  η 1  − β  12 exp  x 1  η 1  + γ  12 > 0. (3.50) According to (ii) of Lemma 3.5,onehas β  2 12 − 4α  12 γ  12 > 0. (3.51) Therefore, the equation α  12 x 2 − β  12 x + γ  12 = 0 (3.52) has two positive solutions N i  α  12 ,β  12 ,γ  12  , i = 1,2. (3.53) Thus, we have exp  x 1  η 1  >N 2  α  12 ,β  12 ,γ  12  ,orexp  x 1  η 1  <N 1  α  12 ,β  12 ,γ  12  . (3.54) It follows from Lemma 3.3,(3.33)and(3.48)that x 1  η 1  ≤ x 1  ξ 1  + ¯ R 1 ω < lnN 2  α 12 ,β 12 + m,γ 12 − n  + ¯ R 1 ω := H. (3.55) On the other hand, it follows from (3.32)and(3.34)that ¯ a ii ωexp  x i  ξ i  ≤ ω−1  k=0 a ii (k)exp  x i (k)  < ¯ r i ω, i = 1,2, (3.56) [...]... University of California Press, California, 1974 [8] Y Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, vol 191, Academic Press, Massachusetts, 1993 [9] J Maynard-Smith, Models in Ecology, Cambridge University Press, Cambridge, UK, 1974 [10] A Mukhopadhyay, J Chattopadhyay, and P K Tapaswi, A delay differential equations model of plankton allelopathy,... has at least two positive ω -periodic solutions Acknowledgment This research is supported by the National Natural Science Foundation of China (No 10161007, 10561004) References [1] R Arditi, L R Ginzburg, and H R Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent predation models, The American Naturalist 138 (1991), 1287–1296 [2] J Chattopadhyay, Effect of toxic substances... allelopathy, Mathematical Biosciences 149 (1998), no 2, 167–189 [11] E L Rice, Allelopathy, Academic Press, New York, 1984 14 Periodic solutions for a discrete plankton model [12] R Y Zhang, Z C Wang, Y Chen, and J Wu, Periodic solutions of a single species discrete population model with periodic harvest/stock, Computers & Mathematics with Applications 39 (2000), no 1-2, 77–90 [13] J Zhen and Z Ma, Periodic. .. Periodic solutions for delay differential equations model of plankton allelopathy, Computers & Mathematics with Applications 44 (2002), no 3-4, 491–500 Jianbao Zhang: Center for Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan 650093, China E-mail address: jianbaozhang@163.com Hui Fang: Center for Nonlinear Science Studies, Kunming University of Science and Technology,... on a two-species competitive system, Ecological Modelling 84 (1996), no 1–3, 287–289 [3] Y Chen, Multiple periodic solutions of delayed predator-prey systems with type IV functional responses, Nonlinear Analysis: Real World Applications 5 (2004), no 1, 45–53 [4] M Fan and K Wang, Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system, Mathematical and Computer Modelling... Freedman and J Wu, Periodic solutions of single-species models with periodic delay, SIAM Journal on Mathematical Analysis 23 (1992), no 3, 689–701 [6] R E Gaines and J L Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol 568, Springer, Berlin, 1977 [7] J A Hellebust, Extracellular Products in Algal Physiology and Biochemistry, edited by W D P Stewart, University... Lemma 3.4 and (3.66) that xi ∈ Ωi , i = 1,2 With the help of (3.48), (3.54), (3.55), (3.63) and (H5 ), it is ¯ ¯ easy to see that Ω1 Ω2 = φ and Ωi satisfies the requirement (a) in Lemma 3.1 for i = 1,2 Moreover, Qᏺx = 0 for x ∈ ∂Ωi KerL, i = 1,2 A direct computation gives degB JQᏺ,Ωi ∩ KerL,0 = 0 (3.69) Here J is taken as the identity mapping since ImQ = Ker L So far we have proved that Ωi satisfies all... J Zhang and H Fang 13 A direct computation gives that ¯ ¯ r1 = 0.0002 = R1 , ¯ a2 1 = 0.0002, α12 > 1.91629, ¯ ¯ r2 = 0.00041 = R2 , ¯ a2 2 = 10000, ¯ a1 1 = 0.0001, ¯ ¯ 1 = b2 = 20000, b γ12 > 1.57284, ¯ a1 2 = 1000, ω = 100, γ21 > 1.9194 × 10−10 , (3.71) α12 β12 − α12 β12 > 0.00904 So according to Corollary 3.8, the above system has at least two positive 100 -periodic solutions Similar to the proof of. .. ∈ R2 Note that Qᏺ x1 ,x2 = ¯ ¯ ¯ ¯ r1 − a1 1 exp x1 − a1 2 exp x2 − b1 exp x1 exp x2 ¯ ¯ ¯ ¯ r2 − a2 1 exp x1 − a2 2 exp x2 − b2 exp x1 exp x2 (3.64) According to Lemma 3.4, we can show that Qᏺx = 0 has two distinct solutions i x = ln Ni α12 ,β12 ,γ12 , lnN1 α21 ,β21 ,γ21 , i = 1,2 (3.65) Choose C > 0 such that C > lnN1 α21 ,β21 ,γ21 (3.66) 12 Periodic solutions for a discrete plankton model Let ⎧ ⎨... all the assumptions in Lemma 3.1 Hence (3.24) has at least two ω -periodic solu¯ ˘ ˘ ˘ ˘ tions xi with xi ∈ DomL Ωi (i = 1,2) Obviously xi (i = 1,2) are different Let N ji (k) = i i ˘i ˘ ˘ ˘ exp(x j (k)), i, j = 1,2 Then N i = (N1 , N2 ) (i = 1,2) are two different positive ω -periodic solutions of (2.3) The proof is complete With the help of Lemma 3.6 and Theorem 3.7, we have the following Corollary 3.8 . Ginzburg, and H. R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent predation models, The American Naturalist 138 (1991), 1287–1296. [2] J. Chattopadhyay, Effect of. Mukhopadhyay, J. Chattopadhyay, and P. K. Tapasw i, Adelaydifferential equations model of plankton allelopathy, Mathematical Biosciences 149 (1998), no. 2, 167–189. [11] E. L. Rice, Allelopathy, Academic. growth rates in (1.2) change at equally spaced time intervals and estimates of the population size are made at equally spaced time intervals, then we can incorporate this aspect in (1.2) and obtain