MUtualism, an interaction of two-species of organisms

Transcription

1 Engineering Letters, 6:, EL_6 Dynamics of Two-species Harvesting Model of Almost Periodic Facultative Mutualism with Discrete and Distributed Delays Yongzhi Liao Abstract By using Mawhins continuous theorem of coincidence degree theory, some new sufficient conditions for the existence of at least four positive almost periodic solutions to a class of two-species harvesting model of facultative mutualism with both discrete and distributed delays are established Further, by constructing a suitable Lyapunov functional, the local asymptotical stability for the model is also studied An example is also given to illustrate the main result in this paper Index Terms Multiplicity; Almost periodicity; Coincidence degree; Facultative mutualism; Local asymptotical stability I INTRODUCTION MUtualism, an interaction of two-species of organisms benefits both see ], is found in many types of communities Some notable examples can be found in papers -6] Mutualism may be obligate or facultative An obligate mutualist is a species which requires the presence of another species for its survival, eg, some species of Acacia require the ant P seudomyrmex in order to survive see 7] A facultative mutualist is one which benefits in some way from the association with another species but will survive in its absence, eg, blue-green algae can grow and reproduce in the absence of zooplankton grazers, but growth and reproduction are enhanced by the presence of the zooplankton see 8] Despite the fact mutualisms are common in nature, attempts to model such interactions mathematically are somewhat scant in the literature, in other words, in theoretical population biology mutualism has received very little attention compared to given to predator-prey interactions or competition among species see 9-3] Inspired by a delayed single-species population growth model with so-called hereditary effect see 4-6] as follows: y t = yt rt atyt btyt µt ], the net birth rate rt, the self-inhibition rate at, the reproduction rate bt, and the delay µt are nonnegative continuous functions Clearly, such a system involves a positive feedback term btyt µt, which is due to gestation see 7-8] In ], Liu et al proposed a delayed two-species system Manuscript received November 7, 6; revised April 5, 7 Yongzhi Liao is with School of Mathematics and Computer Science, Panzhihua University, Panzhihua, Sichuan 67, China mathzhli@63com modelling facultative mutualism as follows: y t = y t r t a ty t b ty t µ t + c ty t ν t ], y t = y t r t a ty t b ty t µ t + c ty t ν t ], r i, a i, b i, c i, µ i and ν i are continuous periodic functions, i =, It is obvious the type of ecological interaction corresponding to is a two-species system of facultative mutualism, is, each species can persist in the absence of the other, however, each species enhances the average growth rate of the other From, we can see the mutualism increases the average growth rates of two species with two different time delays ν t and ν t This assumption corresponds to the possible fact the mutualism has not an effect when they species are infants, and hence they have to mature for some time before they are capable of increasing the average growth rate of two-species In ], some sufficient conditions are derived for the existence and globally asymptotic stability of positive periodic solutions of system, by using Mawhin s continuous theorem and Lyapunov functional In fact, time delays often occur and vary in an irregular fashion, and sometimes they may be not continuously differentiable In addition, if at any time there are always individuals entering the mature population, the delay will be continuously distributed Moreover, in many earlier studies, it has been shown harvesting has a strong impact on dynamic evolution of a population, eg, see 9-] So the study of the population dynamics with harvesting is becoming a very important subject in mathematical bioeconomics By considering the continuous distributed delay and harvesting term, the aim of this paper is to consider the following two-species harvesting model of facultative mutualism with both discrete and distributed delays: y t = y t r t a ty t b t k sy t + s ds +c ty t ν t ] h t, y t = y t r t a ty t b t k sy t + s ds +c ty t ν t ] h t, h i i =, denote harvesting rate, r i, a i, b i, c i, ν i and h i are continuous nonnegative almost periodic functions For the last few years, by utilizing Mawhin s continuation theorem of coincidence degree theory, many scholars are concerning with the existence of multiple positive periodic solutions for some non-linear ecosystems with harvesting

2 Engineering Letters, 6:, EL_6 terms, eg, see 3-7] However, in real world phenomenon, 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 8-3] 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 in science and engineering 33-36] Unlike the periodic oscillation, owing to the complexity of the almost periodic oscillation, it is hard to study the existence of positive almost periodic solutions of non-linear ecosystems by using Mawhin s continuation theorem Therefore, to the best of the author s knowledge, so far, there are scarcely any papers concerning with the multiplicity of positive almost periodic solutions of system by using Mawhin s continuation theorem Stimulated by the above reason, the main purpose of this paper is to establish sufficient conditions for the existence of multiple positive almost periodic solutions to system by applying Mawhin s continuation theorem of coincidence degree theory If b i and ν i, i =,, then system reduces to the following form y t = y t r t a ty t +c ty t ] h t, y t = y t r t a ty t +c ty t ] h t 3 By using the continuation theorem of coincidence degree, Hu and Zhang ] established the existence of four positive periodic solutions for system 4 Related to a continuous function f, we use the following notations: f l = inf s R fs, f = sup fs, s R f u = sup fs, s R f = lim fs ds The paper is organized as follows In Section, we give some basic definitions and necessary lemmas which will be used in later sections In Section 3, some sufficient conditions for the existence of at least four positive almost periodic solutions to system are obtained by means of Mawhin s continuous theorem of coincidence degree theory An example is also given to illustrate the main result of this paper II PRELIMINARIES Mawhin s Continuous Theorem 4] 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,, J : ImQ KerL is an isomorphism Then Lx = Nx has a solution on Ω DomL Definition 39] Let x CR = CR, R x is said to be almost periodic on R, if for >, the set T x, = τ : xt + τ xt <, t R is relatively dense, ie, for >, it is possible to find a real number l = l >, for any interval length l, there exists a number τ = τ T x, in this interval such xt + τ xt <, t R τ is called to the -almost period of x, T x, denotes the set of -almost periods for x and l is called to the length of the inclusion interval for T x, Let AP R denote the set of all real valued almost periodic functions on R and AP R, R n = x, x,, x n T : x i AP R, i =,,, n, n N + Lemma 3] Assume x AP R C R with x CR, for >, there is a point ξ, + such xξ x, x ] and x ξ = Lemma 3] Assume x AP R C R with x CR, for >, there is a point η, + such xη x, x + ] and x η = III MULTIPLICITY Now we are in the position to present and prove our result on the existence of at least four positive almost periodic solutions for system Under the invariant transformation y, y T = e x, e x T, system reduces to x t = r t a te xt b te +c te xt νt ht, e x t x t = r t a te xt For x AP R, we denote by k sx t+s ds k sx t+s ds b te +c te xt νt ht e x t x = mx = lim ax, ϖ = lim Λx = ϖ R : lim xs ds, xse iϖs ds, xse iϖs ds 3 the mean value, the Bohr transform and the set of Fourier exponents of x, respectively Set X = Y = V V, V = w = x, x T AP R, R : ϖ Λx Λx, ϖ θ, V = w = x, x T k, k T, k, k R,

3 Engineering Letters, 6:, EL_6 θ is a given positive constant Define the norm w = max sup x s, sup x s, w X = Y s R s R Lemma 3 3] X and Y are Banach spaces endowed with Lemma 4 3] Let L : X Y, Lw = Lx, x T = x, x T, then L is a Fredholm mapping of index zero Lemma 5 3] Define N : X Y, P : X X and Q : Y Y by x Nw = N x r t a te xt k µ b te sx t+s ds +c te xt νt ht = e x t r t a te xt k µ b te sx t+s ds, +c te xt νt ht e x t x mx P w = P = x mx T = lim x s ds T x = Qw, w X = Y s ds Then N is L-compact on ΩΩ is an open and bounded subset of X Theorem Assume H h l i >, i =, H a l a l > c u c u, H r l b u e ρ + c l e ρ4 > a u hu, rl b u e ρ + c l e ρ3 > a u hu, r u ρ := ln a l + c u r u ] r u a l al, ρ := ln + c u ] e ρ cu cu a l, ρ 3 := ln r u +, ρ 4 := ln cu r u eρ +, cu eρ then system admits at least four positive almost periodic solutions Proof: It is easy to see if system 3 has one almost periodic solution x, x T, then ȳ, ȳ T =e x, e x T is a positive almost periodic solution of system Therefore, to complete the proof it suffices to show system 3 has four almost periodic solutions In order to use the Mawhin s continuous theorem, we set the Banach spaces X and Y as those in Lemma 3 and L, N, P, Q the same as those defined in Lemmas 4 and 5, respectively It remains to search for an appropriate open and bounded subset Ω X Corresponding to the operator equation Lw = λw, λ,, we have x t = λ r t a te xt b te +c te xt νt ht ] e x, t x t = λ r t a te xt k sx t+s ds k sx t+s ds b te +c te xt νt ht ] e x t 3 Suppose x, x T DomL X is a solution of system 3 for some λ,, DomL = x X : x C R, x CR By Lemma 3, for,, there are two points ξ, ξ, + such x ξ =, x ξ x, x ]; x ξ =, x ξ x, x ], 33 x = sup s R x s and x = sup s R x s Further, in view of H, we may assume the above is small enough so a l a l > e c u c u From system 3, it follows from 33 a ξ +b ξ e xξ = r ξ + c ξ a ξ +b ξ k se xξ e xξ = r ξ + c ξ s ds + h ξ e xξ ν ξ k se xξ e xξ, 34 s ds + h ξ e xξ ν ξ In view of 34, we have from 33 That is, a l e x a ξ e xξ r ξ + c ξ r u + c u e x e xξ 35 e xξ ν ξ a l e x e r u + e c u e x 36 Similarly, we obtain from 35 a l e x e r u + e c u e x 37 Substituting 37 into 36 leads to a l a l e x e r u a l + e c u e r u + e c u e x ], which implies a l a l e c u c u ] e x e r u a l + e c u r u is equivalent to e x r u a l + e c u ln r u ] a l al e c u cu Letting in the above inequality leads to r x u ln a l + c u r u ] a l al := ρ 38 cu cu Substituting 38 into 37 leads to a l e x e r u + e c u e ρ Letting in the above inequality leads to r x u ln + c u ] e ρ := ρ 39 a l

4 Engineering Letters, 6:, EL_6 Also, by Lemma 4, for,, there are two points η, η, + such x η =, x η x, x + ]; x η =, x η x, x + ], 3 x = inf s R x s and x = inf s R x s From system 3, it follows from 3 a η +b η e xη = r η + c η a η +b η k se xη e xη = r η + c η s ds + h η e xη ν η k se xη e xη, 3 s ds + h η e xη ν η e xη 3 In view of 3, we have from 38-3 That is, h l e x + h η e xη r η + c η r u + c u e ρ e x is equivalent to x ln h l e r u + cu eρ h l e r u + cu eρ e xη ν η Letting in the above inequality leads to x ln r u + := ρ 3 33 cu eρ By a parallel argument as in 33, we can obtain x ln r u + := ρ 4 34 cu eρ ] In view of 34 and 3, we have a ξ e xξ + b ξ k se xξ + h ξ e xξ rl + c l e ρ4, a η e xη + b η which yield k se xη + h η e xη rl + c l e ρ4, s ds s ds a u e xξ r l b u e ρ + c l e ρ4 e xξ + h u, a u e xη r l b u e ρ + c l e ρ4 e xη + h u, which imply from H x ξ ln κ +, x ξ ln κ, x η ln κ +, x η ln κ, κ ± := rl b u e ρ + c l e ρ4 ± a u hu a u Letting in the above inequalities lead to x ln κ + or x ln κ, x ln κ + or x ln κ 35 By a similar argument as in 35, we obtain x ln κ + or x ln κ, x ln κ + or x ln κ, 36 κ ± := rl b u e ρ + c l e ρ3 ± a u hu a u From and 33-36, it follows ρ 3 x t ln κ or ln κ + x t ρ, 37 ρ 4 x t ln κ or ln κ + x t ρ 38 Obviously, ln κ ±, ln κ ±, ρ, ρ, ρ 3 and ρ 4 are independent of λ Let ε i = ln κi + ln κi 4 i =, and Ω = w = x, x T X : ρ 3 < x t < ln κ + ε, Ω = Ω 3 = Ω 4 = ρ 4 < x t < ln κ + ε, t R, w = x, x T X : ρ 3 < x t < ln κ + ε, ln κ + ε < x t < ρ +, t R, w = x, x T X : ln κ + ε < x t < ρ +, ρ 4 < x t < ln κ + ε, t R, w = x, x T X : ln κ + ε < x t < ρ +, ln κ + ε < x t < ρ +, t R Then Ω, Ω, Ω 3 and Ω 4 are bounded open subsets of X, Ω i Ω j =, i j, i, j =,, 3, 4 Therefore, Ω, Ω, Ω 3 and Ω 4 satisfy condition a of Mawhin s continuous theorem Now we show condition b of Mawhin s continuous

5 Engineering Letters, 6:, EL_6 theorem holds, ie, we prove QNw for all w = x, x T Ω i KerL = Ω i R, i =,, 3, 4 If it is not true, then there exists at least one constant vector w = x, x T Ω i i =,, 3, 4 such r ā e x b e x + c e x h =, r ā e x b e x + c e x h = e x Similar to the argument as in 37-38, it follows ρ 3 x ln κ or ln κ + x ρ, e x ρ 4 x ln κ or ln κ + x ρ Then w Ω R or w Ω R or w Ω 3 R or w Ω 4 R This contradicts the fact w Ω i i =,, 3, 4 This proves condition b of Mawhin s continuous theorem holds Finally, we will show condition c of Mawhin s continuous theorem is satisfied Let us consider the homotopy Hι, w = ιqnw + ιφw, ι, w, ] R, Φw = Φ x = x r ā e x + c e ρ h e x r ā e x + c e ρ h e x From the above discussion it is easy to verify Hι, w on Ω i KerL, ι, ], i =,, 3, 4 Further, Φw = has four distinct solutions: x, x T = ln x, ln x T, y, y T = ln x +, ln x+ T, u, u T = ln x +, ln x T, v, v T = ln x, ln x+ T, x ± x ± = r + c e ρ ± r + c e ρ 4ā h ā, = r + c e ρ ± r + c e ρ 4ā h ā It is easy to verify Therefore ρ 3 < ln x < ln κ < ln κ+ < ln x+ < ρ, ρ 4 < ln x < ln κ < ln κ+ < ln x+ < ρ x, x T Ω, v, v T Ω, u, u T Ω 3, y, y T Ω 4 A direct computation yields deg Φ, Ω i KerL, ā e x + h e x ā e x + h e x h ā e x + e x h ā e x + e x Φx,x T = r ā e x + c e ρ Φx,x T = r ā e x + c e ρ, Φx,x T = i =,, 3, 4 Thus deg Φ, Ω KerL, r ā x + c e ρ r ā x + c e ρ =, deg Φ, Ω KerL, r ā x + c e ρ r ā x + + c e ρ =, deg Φ, Ω 3 KerL, r ā x + + c e ρ r ā x + c e ρ =, deg Φ, Ω 4 KerL, r ā x + + c e ρ r ā x + + c e ρ = By the invariance property of homotopy, we have deg JQN, Ω i KerL, = deg QN, Ω i KerL, = deg Φ, Ω i KerL,, i =,, 3, 4, deg,, is the Brouwer degree and J is the identity mapping since ImQ = KerL Obviously, all the conditions of Mawhin s continuous theorem are satisfied Therefore, system 3 has four almost periodic solutions, is, system has at least four positive almost periodic solutions This completes the proof From the proof of Theorem 3, we can show Corollary Assume H -H hold Suppose further r i, a i, b i, c i, ν i and h i of system are continuous nonnegative periodic functions with different periods, i =,, then system admits at least four positive almost periodic solutions Corollary Assume H -H hold Suppose further r i, a i, b i, c i, ν i and h i of system are continuous nonnegative ω-periodic functions, i =,, then system admits at least four positive ω-periodic solutions Remark In system, when b i and ν i, i =,, Hu and Zhang ] obtained Corollary, but they couldn t obtain Corollary Therefore, the main result in this paper extends their work in article ] IV LOCAL ASYMPTOTICAL STABILITY In this section, we will construct some suitable Lyapunov functions to study the local asymptotical stability of system Theorem Assume H 3 there exist two positive constants y and y such Θ = min, r i ā iy i b i y i c iy i > Then system is locally asymptotically stable

6 Engineering Letters, 6:, EL_6 Proof: Let B := y = y, y T R : y i yi, i =, Assume yt = y t, y t T B and y t = yt, yt T B are any two solutions of system In view of system, for t R +, i =,,, n, we have y t y t = r t y t y t ] a t y t + y t ] y t y t ] b ty t k s y t + s yt + s ] ds b t k syt + s ds y t yt ] +c tyt ν y t yt ] +c tyt y t ν yt ν ] r t y t yt ] ā y y t yt similarly, Let V t = b y k s y t + s yt + s ds b y y t yt c ty y t yt c y y t ν yt ν, y t yt r t y t yt ] ā y y t yt b y k s y t + s yt + s ds b y y t yt c ty y t yt c y y t ν yt ν bi yi V 3 t = V t = V t V t V 3 t, V t = µ i c i yi y i t yi t, t t+s t k i l s yi l y i l dl ds, t ν i y 3 i s y 3 is ds Hence we can obtain from H 7 -H 9 D + V t r i ā i yi b i yi c i yi y i t yi t = Θ y i t yi t Integrating the last inequality from T to t leads to t V T + Θ y i s yi s s V t < +, T is, + T y i s y i s s < +, which implies lim y is yi s = s + Thus, system is locally asymptotically stable This completes the proof Theorem 3 Assume H -H 3 hold Then the unique almost periodic solution of system is locally asymptotically stable V AN EXAMPLE Example Consider the following almost periodic model: y t = y t a ty t b t ln es y t + s ds +y t ] 5, y t = y t 5 a ty t b t 3 ln 3 es y t + s ds +y t cos t ] 69, a + 5 cos 3t =, a 3 b b = + cos t Obviously, r l = r u = r l = r u =, c l = c u = c l = c u =, a l =, a u = 5, a l = a u = 3, b l = b u =, b l =, b u = 3, h l = h u = 5, h l = h u = 69 It is easy to see a l a l = 3 > = c u c u So H holds By a easy calculation, we can obtain Then e ρ 33, e ρ 87, e ρ3, e ρ4 r l b u e ρ + c l e ρ4 = 9538 > 75 = a u hu, r l b u e ρ + c l e ρ3 = 9639 > 587 = a u hu, which imply H holds Therefore, all the conditions in Theorem are satisfied By Theorem, system 4 admits at least four positive almost periodic solutions Remark Clearly, system 4 is with incommensurable periods Through all the coefficients of system 4 are periodic functions, the positive periodic solutions could not possibly exist However, by the work in this paper, the positive almost periodic solutions of system 4 exactly exist

7 Engineering Letters, 6:, EL_6 VI CONCLUSIONS By using a fixed point theorem of coincidence degree theory, some criterions for the multiplicity of positive almost periodic solution to a kind of two-species harvesting model of facultative mutualism with both discrete and distributed delays are obtained Theorems gives the sufficient conditions for the multiplicity of positive almost periodic solution of system Theorems gives the sufficient conditions for the local asymptotical stability of system The results obtained in this paper are new and different from the results in, ] Therefore, The method used in this paper provides a possible method to study the multiplicity of positive almost periodic solution and the local asymptotical stability of the models in biological populations ACKNOWLEDGEMENTS 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 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