Existence of periodic solutions in predator prey and competition dynamic systems

Transcription

1 Nonlinear Analsis: Real World Applications 7 (2006) Eistence of periodic solutions in predator pre and competition dnamic sstems Martin Bohner a,, Meng Fan b, Jimin Zhang b a Department of Mathematics and Statistics, Universit of Missouri Rolla, Rolla, MO 65401, USA b School of Mathematics and Statistics, and Ke Laborator for Vegetation Ecolog, Northeast Normal Universit, Changchun, Jilin , PR China Received 30 October 2005; accepted 1 November 2005 Abstract In this paper, we sstematicall eplore the periodicit of some dnamic equations on time scales, which incorporate as special cases man population models (e.g., predator pre sstems and competition sstems) in mathematical biolog governed b differential equations and difference equations. Easil verifiable sufficient criteria are established for the eistence of periodic solutions of such dnamic equations, which generalize man known results for continuous and discrete population models when the time scale T is chosen as R or Z, respectivel. The main approach is based on a continuation theorem in coincidence degree theor, which has been etensivel applied in studing eistence problems in differential equations and difference equations but rarel applied in dnamic equations on time scales. This stud shows that it is unnecessar to eplore the eistence of periodic solutions of continuous and discrete population models in separate was. One can unif such studies in the sense of dnamic equations on general time scales Elsevier Ltd. All rights reserved. MSC: 92D25; 39A12 Kewords: Time scales; Periodic solution; Coincidence degree; Predator pre sstem; Beddington DeAngelis response; Holling-tpe response; Competition sstem; Gilpin Aala sstem 1. Introduction In the past decades, mathematical ecolog has seen much progress, especiall in population dnamics. Most natural environments are phsicall highl variable, and in response, birth rates, death rates, and other vital rates of populations, var greatl in time. Theoretical evidence to date suggests that man population and communit patterns represent intricate interactions between biolog and variation in the phsical environment (see 4 and other papers in the same issue). Therefore, the focus in theoretical models of population and communit dnamics must be not onl on how populations depend on their own population densities or the population densities of other organisms, but also on how Supported b the National Natural Science Foundation of PR China (No ), the Ke Project on Science and Technolog of the Education Ministr of PR China, and the Universit of Missouri Research Board. Corresponding author. Tel.: ; fa: addresses: bohner@umr.edu (M. Bohner), mfan@nenu.edu.cn (M. Fan) /$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi: /j.nonrwa

2 1194 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) populations change in response to the phsical environment. When the environmental fluctuation is taken into account, a model must be nonautonomous, and hence, of course, more difficult to analze in general. But, in doing so, one can and should also take advantage of the properties of those varing parameters. For eample, one ma assume the parameters are periodic or almost periodic for seasonal reasons. Due to the recognition that temporal fluctuations in the phsical environment are a major driver of population fluctuations, there has been more and more theoretical attention to predict the characteristic of the resultant population fluctuations. A ver basic and important problem in the stud of a population growth model with a periodic environment is the global eistence and stabilit of a positive periodic solution, which plas a similar rôle as a globall stable equilibrium does in an autonomous model. Thus, it is reasonable to seek conditions under which the resulting periodic nonautonomous sstem would have a positive periodic solution that is globall asmptoticall stable. Much progress has been seen in this direction. Careful investigation reveals that it is similar to eplore the eistence of periodic solutions for nonautonomous population models governed b ordinar differential equations and their discrete analogue in the approaches, the methods and the main results. For eample, etensive research reveals that man results concerning the eistence of periodic solutions of predator pre sstems modelled b differential equations can be carried over to their discrete analogues based on the coincidence theor, for eample, 5,7,9 11,14,15,18,19. It is natural to ask whether we can eplore such an eistence problem in a unified wa. The theor of calculus on time scales (see 2,3 and references cited therein) was initiated b Stefan Hilger in his Ph.D. Thesis in in order to unif continuous and discrete analsis, and it has a tremendous potential for applications and has recentl received much attention since his foundational work. It has been created in order to unif the stud of differential and difference equations. Man results concerning differential equations carr over quite easil to corresponding results for difference equations, while other results seem to be completel different from their continuous counterparts. The stud of dnamic equations on general time scales can reveal such discrepancies and help avoid proving results twice once for differential equations and once again for difference equations. The two main features of the calculus on time scales are unification and etension. To prove a result for a dnamic equation on a time scale is not onl related to the set of real numbers or set of integers but those pertaining to more general time scales. The principle aim of this paper is to sstematicall unif the eistence of periodic solutions of population models modelled b ordinar differential equations and their discrete analogues in form of difference equations and to etend these results to more general time scales. The approach is based on a continuation theorem in coincidence degree, which has been widel applied to deal with the eistence of periodic solutions of differential equations and difference equations. This paper is the first one to appl coincidence degree theor to eplore the eistence of periodic solutions of dnamic equations on time scales. The setup of this paper is as follows. In the coming section, we present some preliminar results such as the calculus on time scales and the continuation theorem in coincidence degree theor. Then we sstematicall eplore the eistence of periodic solutions of dnamic equations on time scales of predator pre tpe and competition tpe. This stud reveals that, when we deal with the eistence of positive periodic solutions of population models, it is unnecessar to prove results for differential equations and separatel again for difference equations. One can unif such problems in the frame of dnamic equations on time scales. 2. Preliminaries In this section, we briefl give some elements of the time scales calculus, recall the continuation theorem from coincidence degree theor, and prove an auiliar result that is needed in the paper. First, let us present some foundational definitions and results from the calculus on time scales so that the paper is self-contained. For more details, one can see 2,3,13. Notation 2.1. Throughout this paper, the smbol T denotes a time scale, i.e., an arbitrar nonempt closed subset of the real numbers R. Let ω > 0. Throughout, the time scale T is assumed to be ω-periodic, i.e., t T implies t +ω T. In particular, the time scale T under consideration is unbounded above and below. Some eamples of such time scales are { R, Z, 2k, 2k + 1, k + 1 }. n k Z k Z n N

3 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) Definition 2.1. We define the forward jump operator σ : T T, the backward jump operator ρ : T T, and the graininess μ : T R + =0, ) b σ(t) := inf{s T : s>t}, ρ(t) := sup{s T : s<t} and μ(t) = σ(t) t for t T, respectivel. If σ(t) = t, then t is called right-dense (otherwise: right-scattered), and if ρ(t) = t, then t is called left-dense (otherwise: left-scattered). Definition 2.2. Assume f : T R is a function and let t T. Then we define f Δ (t) to be the number (provided it eists) with the propert that given an ε > 0, there is a neighborhood U of t (i.e., U = (t δ,t + δ) T for some δ > 0) such that f(σ(t)) f(s) f Δ (t)σ(t) s ε σ(t) s for all s U. In this case, f Δ (t) is called the delta (or Hilger) derivative of f at t. Moreover, f is said to be delta or Hilger differentiable on T if f Δ (t) eists for all t T.A function F : T R is called an antiderivative of f : T R provided F Δ (t)=f(t) for all t T. Then we define s r f(t)δt = F(s) F(r) for r, s T. Definition 2.3. A function f : T R is said to be rd-continuous if it is continuous at right-dense points in T and its left-sided limits eist (finite) at left-dense points in T. The set of rd-continuous functions f : T R will be denoted b C rd (T). Lemma 2.1. Ever rd-continuous function has an antiderivative. Lemma 2.2. If a,b T, α, β R and f, g C rd (T), then (a) b a αf(t)+ βg(t)δt = α b a f(t)δt + β b a g(t)δt; (b) if f(t) 0 for all a t <b, then b a f(t)δt 0; (c) if f(t) g(t) on a,b) := {t T : a t <b}, then b a f(t)δt b a g(t)δt. Notation 2.2. To facilitate the discussion below, we now introduce some notation to be used throughout this paper. Let = min{0, ) T}, I ω =, + ω T, g u = sup g(t), g = 1 g(s)δs = 1 g(s)δs, ω I ω ω t T g l = inf t T g(t), where g C rd (T) is an ω-periodic real function, i.e., g(t + ω) = g(t) for all t T. Net, let us recall the continuation theorem in coincidence degree theor borrowing notations and terminolog from 12, which will come into pla later on. Notation 2.3. Let X, Z be normed vector spaces, L : Dom L X Z be a linear mapping, N : X Z be a continuous mapping. The mapping L will be called a Fredholm mapping of inde zero if dim Ker L=codim Im L<+ and Im L is closed in Z. IfL is a Fredholm mapping of inde zero and there eist continuous projections P : X X and Q : Z Z such that Im P = Ker L, ImL = Ker Q = Im (I Q), then it follows that L Dom L Ker P : (I P)X Im L is invertible. We denote the inverse of that map b K P.IfΩ 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 Im Q is isomorphic to Ker L, there eists an isomorphism J : Im Q Ker L.

4 1196 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) Lemma 2.3 (Continuation Theorem). Let L be a Fredholm mapping of inde zero and N be L-compact on Ω. Suppose (a) For each λ (0, 1), ever solution z of Lz = λnz is such that z/ Ω; (b) QNz = 0 for each z Ω Ker L and the Brouwer degree deg{jqn,ω Ker L, 0} = 0. Then the operator equation Lz = Nz has at least one solution ling in Dom L Ω. In order to achieve the priori estimation in the case of dnamic equations on a time scale T, we first prove the following inequalities, which will be ver essential in this paper. Lemma 2.4. Let t 1,t 2 I ω and t T. If g : T R is ω-periodic, then g(t) g(t 1 ) + g Δ (s) Δs and g(t) g(t 2 ) g Δ (s) Δs. Proof. We onl show the first inequalit as the proof of the second inequalit is similar to the proof of the other one. Since g is ω-periodic, without an loss of generalit, it suffices to show that the inequalit is valid for t I ω.ift = t 1, then the first inequalit is obviousl true. If t>t 1, then one has t g(t) g(t 1 ) g(t) g(t 1 ) = g Δ t (s)δs g Δ (s) Δs g Δ (s) Δs, t 1 t 1 and hence g(t) g(t 1 ) + g Δ (s) Δs. If t<t 1, then t1 g(t 1 ) g(t) g(t 1 ) g(t) = g Δ t1 (s)δs g Δ (s) Δs g Δ (s) Δs, t t that is g(t) g(t 1 ) + g Δ (s) Δs. The proof is complete. Remark 2.1. If T = R, then the inequalities are standard for the Riemann integral, while if T = Z, then Lemma 2.4 reduces to the inequalities established b Fan and Wang 10, Lemma Predator pre dnamic sstems The dnamic relationship between predators and their pres has long been and will continue to be one of the dominating themes in both ecolog and mathematical biolog due to its universal eistence and importance 1. Understanding the dnamical relationship between predator and pre is a central goal in ecolog, and one significant component of the predator pre relationship is the predator s rate of feeding upon pre, i.e., the so-called predator s functional response. In general, the functional response can be classified into two tpes: pre-dependent and predatordependent. Pre dependent means that the functional response is onl a function of the pre s densit, while predatordependent means that the functional response is a function of both the pre s and the predator s densities. Although the predator-dependent models that are considered fit those data reasonabl well, no single functional response best describes all of the data sets. Theoretical studies have shown that the dnamics of models with predator-dependent functional responses can differ considerabl from those with pre-dependent functional responses. Due to the fact that man results concerning the eistence of periodic solutions of predator pre sstems modelled b differential

5 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) equations can be carried over to their discrete analogues, in this section, we unif the eistence of periodic solutions of predator pre sstems with different functional response in the framework of dnamic equations on time scales Predator pre dnamic sstems with Beddington DeAngelis functional response First, we focus on predator pre sstems with Beddington DeAngelis functional response on time scales T of the form Δ c(t) ep{(t)} (t) = a(t) b(t) ep{(t)} α(t) + β(t) ep{(t)}+γ(t) ep{(t)}, (3.1) Δ f(t)ep{(t)} (t) = d(t) + α(t) + β(t) ep{(t)}+γ(t) ep{(t)}, where a, b, c, d, f, α, β, γ C rd (T) are ω-periodic such that a,d,γ l > 0 and b(t), c(t), f (t), α(t), β(t) 0 for all t T. (3.2) Remark 3.1. Let (t) = ep{(t)} and ỹ(t) = ep{(t)}. IfT = R, then (3.1) reduces to the standard predator pre sstem with Beddington DeAngelis functional response governed b the ordinar differential equations c(t)ỹ(t) (t) = (t) a(t) b(t) (t), ỹ (t) =ỹ(t) d(t) + α(t) + β(t) (t) + γ(t)ỹ(t), f(t) (t) α(t) + β(t) (t) + γ(t)ỹ(t) where (t) and ỹ(t) denote the densit of the pres and the predators. The predator pre sstems of form (3.3) have been etensivel studied 6.IfT = Z, then (3.1) is reformulated as c(t)ỹ(t) (t + 1) = (t)ep a(t) b(t) (t), ỹ(t + 1) =ỹ(t)ep d(t) + α(t) + β(t) (t) + γ(t)ỹ(t), f(t) (t) α(t) + β(t) (t) + γ(t)ỹ(t) which is the discrete time predator pre sstem with Beddington DeAngelis functional response and is also a discrete analogue of (3.3). Since (3.1) incorporates (3.3) and (3.4) as special cases, we call (3.1) the predator pre dnamic sstem with Beddington DeAngelis functional response on time scales. In order to eplore the eistence of periodic solutions of (3.1), first we should embed our problem in the frame of coincidence degree theor. Define L ω ={(u, v) C(T, R 2 ) : u(t + ω) = u(t), v(t + ω) = v(t) for all t T}, (u, v) =ma u(t) +ma v(t) for (u, v) L ω. t I ω t I ω It is not difficult to show that L ω is a Banach space when it is endowed with the above norm. Let L ω 0 ={(u, v) Lω : u = 0, v = 0}, L ω c ={(u, v) Lω : (u(t), v(t)) (h 1,h 2 ) R 2 for t T}. Then it is eas to show that L ω 0 and Lω c are both closed linear subspaces of Lω, L ω = L ω 0 Lω c, and dim Lω c = 2. Theorem 3.1. Assume (3.2). If a c/γ > 0 and (f dβ u )(a c/γ) ep{ (a + a )ω} bdα u > 0, (3.5) then (3.1) has at least one ω-periodic solution. (3.3) (3.4)

6 1198 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) Proof. Let X = Z = L ω and define N L = = N1 N 2 Δ Δ c(t) ep{(t)} a(t) b(t) ep{(t)} α(t) + β(t) ep{(t)}+ γ(t) ep{(t)} =, f(t)ep{(t)} d(t) + α(t) + β(t) ep{(t)}+ γ(t) ep{(t)}, P = Q =. Then Ker L = L ω c,iml = Lω 0, and dim Ker L = 2 = codim Im L. Since Lω 0 is closed in Lω, it follows that L is a Fredholm mapping of inde zero. It is not difficult to show that P and Q are continuous projections such that Im P =Ker L and Im L = Ker Q = Im (I Q). Furthermore, the generalized inverse (to L) K P : Im L Ker P Dom L eists and is given b K p = X X Y Y t t where X(t) = (s)δs and Y(t)= (s)δs. Thus and QN 1 = ω K p (I Q)N a(t) b(t) ep{(t)} 1 ω d(t) + t N 1(s)Δs 1 = ω t N 2(s)Δs 1 ω c(t) ep{(t)} Δt α(t) + β(t) ep{(t)}+ γ(t) ep{(t)} f(t)ep{(t)}, Δt α(t) + β(t) ep{(t)}+ γ(t) ep{(t)} ( t N 1(s)ΔsΔt t 1 ω t N 2(s)ΔsΔt ( t 1 ω ) (t )Δt ) (t )Δt N 1 N 2. Obviousl, QN and K p (I Q)N are continuous. Since X is a Banach space, using the Arzelà Ascoli theorem, it is eas to show that K p (I Q)N(Ω) is compact for an open bounded set Ω X. Moreover, QN(Ω) is bounded. Thus, N is L-compact on Ω with an open bounded set Ω X. Now we are in the position to search for an appropriate open, bounded subset Ω for the application of the continuation theorem, Lemma 2.3. For the operator equation L = λn, L = λn, λ (0, 1), wehave Δ c(t) ep{(t)} (t) = λ a(t) b(t) ep{(t)}, α(t) + β(t) ep{(t)}+ γ(t) ep{(t)} (3.6) Δ f(t)ep{(t)} (t) = λ d(t) +. α(t) + β(t) ep{(t)}+ γ(t) ep{(t)} Assume that (, ) X is an arbitrar solution of sstem (3.6) for a certain λ (0, 1). Integrating both sides of (3.6) over the interval, + ω, we obtain aω = +ω b(t) ep{(t)}+ dω = c(t) ep{(t)} α(t) + β(t) ep{(t)}+ γ(t) ep{(t)} Δt. f(t)ep{(t)} α(t) + β(t) ep{(t)}+ γ(t) ep{(t)} Δt, (3.7)

7 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) From (3.6) and (3.7), we obtain Δ (t) Δt λ a(t) Δt + b(t) ep{(t)}δt c(t) ep{(t)} + α(t) + β(t) ep{(t)}+γ(t) ep{(t)} Δt = λ(a + a )ω <(a + a )ω, Δ f(t)ep{(t)} (t) Δt λ d(t) Δt + α(t) + β(t) ep{(t)}+γ(t) ep{(t)} Δt = λ(d + d )ω <(d + d )ω. Note that since (, ) X, there eist ξ i, η i, + ω, i {1, 2}, such that (ξ 1 ) = min t,+ω (η 1) = ma t,+ω (ξ 2 ) = min t,+ω (η 2) = ma (t). t,+ω (3.8) From (3.8) and the first equation of (3.7), we have aω b(t) ep{(η 1 )}+ c(t) Δt = bω ep{(η γ(t) 1 )}+(c/γ)ω. B the first part of the assumption in (3.5), we can conclude that b>0 necessaril must hold. Then (η 1 ) ln{(a c/γ)/b} =:l 1, and therefore, using the second inequalit in Lemma 2.4, (t) (η 1 ) Δ (t) Δt >l 1 (a + a )ω =: H 2. (3.9) On the other hand, from (3.8) and the first equation of (3.7), we also obtain aω b(t) ep{(ξ 1 )}Δt = bω ep{(ξ 1 )}, which reduces to (ξ 1 ) ln{a/b} =:L 1, and hence, using the first inequalit in Lemma 2.4, (t) (ξ 1 ) + Δ (t) Δt <L 1 + (a + a )ω =: H 1, which, together with (3.9), leads to ma t,+ω (t) ma{ H 1, H 2 } =: B 1. From (3.8) and the second equation of (3.7), we can derive that f(t)ep{(t)} +ω f(t)e H 1 dω β l Δt ep{(t)}+γ l ep{(t)} β l e H 1 + γ l ep{(ξ 2 )} Δt = ωfe H1 β l e H 1 + γ l ep{(ξ 2 )}, so ep{(ξ 2 )} ((f dβ l )e H 1)/dγ l. Thus f dβ l > 0 necessaril must hold. Then (ξ 2 ) ln{(f dβ l )e H 1/dγ l }=:L 2. Hence, b using the first inequalit in Lemma 2.4, (t) (ξ 2 ) + Δ (t) Δt <L 2 + (d + d )ω =: H 3. (3.10)

8 1200 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) We can also derive from the second equation of (3.7) that f(t)ep{(t)} +ω f(t)e H 2 dω α u + β u ep{(t)}+γ u Δt ep{(η 2 )} α u + β u e H 2 + γ u ep{(η 2 )} Δt. Then it follows that ep{(η 2 )} (f dβu )((a c/γ)/b) ep{ (a + a )ω} dα u =: l dγ u 2. B the second part of the assumption in (3.5), we can conclude that l2 > 0 so that (η 2) ln(l2 ) =: l 2 and hence, b using the second inequalit in Lemma 2.4, (t) (η 2 ) Δ (t) Δt >l 2 ω(d + d ) =: H 4, which, together with (3.10), leads to ma t,+ω (t) ma{ H 3, H 4 } =: B 2. Obviousl, B 1 and B 2 are both independent of λ. Let B =B 1 +B 2 +B 3, where B 3 > 0 is taken sufficientl large such that B 3 l 1 + L 1 + l 2 + L 2. Net let us consider the algebraic equations a b ep{} 1 ω d + 1 ω νc(t) ep{} Δt = 0, α(t) + β(t) ep{}+γ(t) ep{} f(t)ep{} α(t) + β(t) ep{}+γ(t) ep{} Δt = 0 (3.11) for (, ) R 2, where ν 0, 1 is a parameter. B carring out similar arguments as above, it is not difficult to show that an solution (, ) of (3.11) with ν 0, 1 satisfies l 1 L 1 and l 2 L 2. (3.12) Now we define Ω ={(, ) X : (, ) <B}. Then it is clear that Ω satisfies the requirement (a) of Lemma 2.3. If (, ) Ω Ker L = Ω R 2, then (, ) is a constant vector in R 2 with (, ) = + =B. Then from (3.12) and the definition of B,wehave QN a b ep{} 1 ω = d + 1 ω c(t) ep{} α(t) + β(t) ep{}+γ(t) ep{} Δt = f(t)ep{} α(t) + β(t) ep{}+γ(t) ep{} Δt 0. 0 Moreover, note that J = I since Im Q = Ker L. In order to compute the Brouwer degree, let us consider the homotop H ν (, ) = νqn(, ) + (1 ν)g(, ) for ν 0, 1, where a b ep{} G(, ) = d 1 f(t)ep{} ω α(t) + β(t) ep{}+γ(t) ep{} Δt. From (3.12), it is eas to show that 0 / H ν ( Ω Ker L) for ν 0, 1. Moreover, one can easil show that the algebraic equation G(, ) = 0 has a unique solution in R 2. B the invariance propert of homotop, direct calculation produces deg(j QN, Ω Ker L, 0) = deg(qn, Ω Ker L, 0) = deg(g, Ω Ker L, 0) = 0, where deg(,, ) is the Brouwer degree. B now we have proved that Ω satisfies all requirements of Lemma 2.3. Thus Lz = Nz has at least one solution in Dom L Ω, i.e., (3.1) has at least one ω-periodic solution in Dom L Ω. The proof is complete. Remark 3.2. If T = R, then (3.1) is the continuous predator pre sstem with Beddington DeAngelis functional response and Theorem 3.1 is 6, Theorem 3.2.

9 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) Remark 3.3. If α(t) 0, then (3.1) reduces to the ratio-dependent predator pre sstem. If in addition T = R or T = Z, then (3.1) reduces to the continuous or the discrete ratio-dependent predator pre sstem, and Theorem 3.1 unifies and generalizes the main results in 9,10. Eample 3.1. Consider two insect populations (one of them the predator, the other one the pre) that are both continuous while in season (sa during the si warm months of the ear), die out in (sa) winter, while their eggs are incubating or dormant, and then both hatch in a new season, both of them giving rise to nonoverlapping populations. This situation can be modelled using the time scale T = k Z 2k, 2k + 1 with ω = 1. If the model assumes a Beddington DeAngelis functional response as in (3.1), and if the assumptions (3.2) and (3.5) hold for the underling parameters, then, b Theorem 3.1, there eists a 1-periodic solution of (3.1) Predator pre dnamic sstems with Holling-tpe functional response Consider the predator pre sstem on time scales with Holling-tpe functional response Δ c(t) ep{(t)} (t) = a(t) b(t) ep{(t)} 1 + m(t) ep{(t)}, Δ (t) = d(t) + f(t)ep{(t)} 1 + m(t) ep{(t)} (3.13) and Δ (t) = a(t) b(t) ep{(t)} Δ (t) = d(t) + f(t)ep{2(t)} 1 + m(t) ep{2(t)}, c(t) ep{(t) + (t)} 1 + m(t) ep{2(t)}, (3.14) where a, b, c, d, m, f C rd (T, R) are nonnegative ω-periodic. Remark 3.4. In sstems (3.13) and (3.14), if T = R or T = Z, then (3.13) and (3.14) reduce to a continuous or discrete predator pre sstems with Holling-tpe II or III functional responses, which have been etensivel studied in the literature 15,19. Following a similar strateg as in 8,20, one can reach the following two conclusions. Since we supplied the details for sstems with Beddington DeAngelis functional response in the time scales case and since the proofs of the following two theorems are similar to those in 8,20, the details are omitted here. Theorem 3.2. If f>m ν d and a>bd/(f m ν d) ep{2aω}, then sstem (3.13) has at least one ω-periodic solution. Theorem 3.3. If f>m ν d and a>bd/(f m ν d) 1/2 ep{2aω}, then sstem (3.14) has at least one ω-periodic solution Semi-ratio-dependent predator pre dnamic sstems Consider the nonautonomous semi-ratio-dependent predator pre sstem on time scales { Δ (t) = a(t) b(t) ep{(t)} c(t, ep{(t)}) ep{(t) (t)}, Δ (t) = d(t) e(t) ep{(t) (t)}, (3.15)

10 1202 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) where c(t, ) is a pre-dependent functional response, which can be either of the following forms (in order, we call them the functional response of tpe 1 5) m(t); m(t) A + ; m(t) n A + n, n 2; m(t) 2 ; m(t)(1 ep{ A}). (A + )(B + ) In (3.15), we consider the following assumptions. (H 1 ) a,b,d,e C rd (T, R + ) are nonnegative ω-periodic; (H 2 )c: T R + R + is rd-continuous and ω-periodic with respect to the first variable, and is differentiable with respect to the second variable, and ( c/ )(t,)>0 for an t T, and ( c/ )(t, ) is bounded with respect to t; (H 3 ) there eists an ω-periodic function C 0 C rd (T, R + ) with c(t, ) C 0 (t) for an t T; (H 4 ) there eists an ω-periodic function C 1 C rd (T, R + ) with c(t, ) C 1 (t) for an t T. B carring out similar arguments as in 11,18, we present, without including the proofs, the following two results. Theorem 3.4. Assume that (H 1 ), (H 2 ), and (H 3 ) hold. If be>c 0 d ep{(a + a +d + d )ω}, then (3.15) has at least one ω-periodic solution. Theorem 3.5. Assume that (H 1 ), (H 2 ), and (H 4 ) hold. If e l a>c1 u d, then (3.15) has at least one ω-periodic solution. Remark 3.5. If T = R or T = Z, then (3.15) reduces to the continuous or discrete semi-ratio predator pre sstem investigated in 14,18 or 11, respectivel. Theorems 3.4 and 3.5 unif and generalize the main results in 18 and Competition dnamic sstems Competition poses an important rôle in mathematical ecolog, and it has been studied etensivel. In this section, we outline how to conclude the eistence of periodic solutions of competition dnamic sstems on time scales. Consider the generalized n-species Gilpin Aala competition sstem with several deviating arguments on time scales Δ i (t) = r i(t) n a ij (t) ep{θ ij j (t)} for i {1, 2,...,n}, (4.1) j=1 where r i,a ij C rd (T, R + ), i, j {1, 2,...,n}, are ω-periodic and bounded above and below b positive constants. Remark 4.1. If T=R or T=Z and ỹ i (t)=ep{ i (t)}, then (4.1) reduces to the continuous or discrete time Gilpin Aala competition sstem with several deviating arguments n ỹ i (t) =ỹ i(t) r i (t) a ij (t)(ỹ j (t)) θ ij for i {1, 2,...,n}, t R j=1 or n ỹ i (t + 1) =ỹ i (t) ep r i(t) a ij (t)(ỹ j (t)) θ ij j=1 for i {1, 2,...,n}, t Z. Therefore, we might as well recognize sstem (4.1) as a generalized n-species Gilpin Aala competition dnamic sstem with several deviating arguments on a time scale. The model (4.1) is ver general and includes man ecological

11 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) models as special cases on time scales, e.g., if n = 1 and θ ij 1, then (4.1) is a logistic equation on time scales; if θ ij 1, then (4.1) is the classical n-species Lotka Volterra competition sstem with periodic coefficients on time scales; if θ ij 1 for i = j, then (4.1) is the classical Gilpin Aala competition model with periodic environment on time scales. To prove Theorem 4.1, we proceed similarl as in the proof of Theorem 3.1, where the spaces L ω etc. now take the following forms: L ω ={ = ( 1, 2,..., n ) C(T, R n ) : (t + ω) = (t) for all t T}, { n ( ) } 2 1/2 = ma i (t) for L ω. t I ω i=1 It is not difficult to show that L ω is a Banach space when it is endowed with the above norm. Let L ω 0 ={ Lω : = 0}, L ω c ={ Lω : (t) h R n for t T}. Then it is eas to show that L ω 0 and Lω c are both closed linear subspaces of Lω, L ω = L ω 0 Lω c, and dim Lω c = n. With these alterations compared to the proof of Theorem 3.1, the proof of our last result is ver similar to the proof of 7, Theorem 2.1. Theorem 4.1. If the sstem of algebraic equations n g(u) = r i a ij u θ ij j = 0 j=1 n 1 has finitel man solutions u = (u 1,...,u n ) Rn + with u i > 0 and u sgn J g(u ) = 0, and if n ( ) θij /θ rj jj r i > a ij ep{θ ij 2r j ω}, a jj j=1,j =i then sstem (4.1) has at least one ω-periodic solution. References 1 A.A. Berrman, The origins and evolution of predator pre theor, Ecolog 73 (1999) M. Bohner, A. Peterson, Dnamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, M. Bohner, A. Peterson, Advances in Dnamic Equations on Time Scales, Birkhäuser, Boston, P. Chesson, Understanding the role of environmental variation in population and communit dnamics, Theor. Popul. Biol. 64 (2003) M. Fan, S. Agarwal, Periodic solutions for a class of discrete time competition sstems, Nonlinear Stud. 9 (3) (2002) M. Fan, Y. Kuang, Dnamics of a nonautonomous predator pre sstem with the Beddington DeAngelis functional response, J. Math. Anal. Appl. 295 (1) (2004) M. Fan, K. Wang, Global periodic solutions of a generalized n-species Gilpin Aala competition model, Comput. Math. Appl. 40 (10 11) (2000) M. Fan, K. Wang, Global eistence of a positive periodic solution to a predator pre sstem with Holling tpe II functional response, Acta Math. Sci. Ser. A, Chin. Ed. 21 (4) (2001) M. Fan, K. Wang, Periodicit in a delaed ratio-dependent predator pre sstem, J. Math. Anal. Appl. 262 (1) (2001) M. Fan, K. Wang, Periodic solutions of a discrete time nonautonomous ratio-dependent predator pre sstem, Math. Comput. Modelling 35 (9 10) (2002) M. Fan, Q. Wang, Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator pre sstems, Discrete Contin. Dnam. Sst. Ser. B 4 (3) (2004) R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568, Springer, Berlin, Heidelberg, New York, S. Hilger, Analsis on measure chains a unified approach to continuous and discrete calculus, Results Math. 18 (1990) H.F. Huo, Periodic solutions for a semi-ratio-dependent predator pre sstem with functional responses, Appl. Math. Lett. 18 (2005) Y.K. Li, Periodic solutions of a periodic dela predator pre sstem, Proc. Amer. Math. Soc. 127 (5) (1999)

12 1204 M. Bohner et al. / Nonlinear Analsis: Real World Applications 7 (2006) Q. Wang, M. Fan, K. Wang, Dnamics of a class of nonautonomous semi-ratio-dependent predator pre sstems with functional responses, J. Math. Anal. Appl. 278 (2) (2003) R. Xu, M.A.J. Chaplain, F.A. Davidson, Periodic solutions for a predator pre model with Holling-tpe functional response and time delas, Appl. Math. Comput. 161 (2) (2005) S.L. Yuan, Z. Jin, Z. Ma, Global eistence of a positive periodic solution to a predator pre sstem, J. Xi an Jiaotong Univ. 34 (10) (2000)