Permanence and global stability of a May cooperative system with strong and weak cooperative partners

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1 Zhao et al. Advances in Difference Equations 08 08:7 R E S E A R C H Open Access ermanence and global stability of a May cooperative system with strong and weak cooperative partners Liang Zhao * BinQin and Fengde Chen * Correspondence: zhaoliang00809@yeah.net College of Information and Statistics Guangxi University of Finance and Economics Nanning.R. China Full list of author information is available at the end of the article Abstract In this paper a May cooperative system with strong and weak cooperative partners is proposed. First by using differential inequality theory we obtain the permanence and non-permanence of the system. Second we discuss the existence of the positive equilibrium point and boundary equilibrium point after that by constructing suitable Lyapunov functions it is shown that the equilibrium points are globally asymptotically stable in the positive octant. Finally examples together with their numerical simulations show the feasibility of the main results. MSC: 34C05; 34C5 Keywords: Cooperative system; Strong; Weak; Global stability Introduction Cooperative system is an important system in the field of biology and the importance of the system is the same as for prey predator and competitive systems. Many scholars have done research on the cooperative ecosystem see []. May []describedacooperative system with the following equations: dx = r x x c x a + b x. dx = r x x a + b x c x where x x are the densities of the species x x at time t respectivelyr i refers to the intrinsic rate of population x i i =andb i i = refers to the coefficients of cooperation r i a i b i c i i = are positive constants. His research shows that the cooperative system has a unique positive equilibrium point and it is globally asymptotically stable. Cui and Chen [] think that a non-autonomous form is more reasonable. They put forward the following cooperation system: dx = r x tx c tx a t+b tx. dx = r x tx c tx a t+b tx The Authors 08. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made.

2 Zhao et al. Advances in Difference Equations 08 08:7 age of 3 where the function r i t a i t b i t c i t i = are continuous functions and bounded above and below by positive constants. Under the premise of r i t a i t b i t c i t i = are periodic function they get the sufficient conditions which guarantee the global asymptotic stability of positive periodic solutions of this system. In view of the influence of time delay species interactions and feedback Chen Liao and Huang [3] proposed the following n-species cooperationsystem: dx i t du i t [ = r i tx i t a i t+ n j=j i b ijt 0 ] x i t T ij K ij sx j t + s ds c itx i t 0 d i tu i tx i t e i tx i t H i su i t + s ds τ i 0 = α i tu i t+β i tx i t+r i t G i sx i t + s ds η i.3 where x i t i =...nis the density of cooperation species X i u i t i =...n is the feedback control variable. The authors obtained the sufficient conditions which guarantee the permanence by using differential inequality theory. For more work as regards the system we can refer to [4 6]. In the real world individual organisms are associated with a strong and weak differential. Mohammadi [3] proposed a Leslie Gower predator prey model: d = b αh dh d = =α c c H r a H.4 where b α c c r a are positive constants the predators can distinguish between strong and weak prey and predator eats only weak prey when a prey becomes weak it does not become strong again; by constructing a suitable Lyapunov function it is shown that the unique equilibrium point is stable in the positive octant. Conversely in many cooperative ecosystems partners like strong partners because the strong partners are more conducive to their survival. This shows that the cooperative object should only be part instead of the whole. There are two populations: The partner H whosetotaldensityish is divided into two categories H. denotes the strong partner density and H denotes the weak. Of the other partner the total density is. The May cooperative model. is our basic model and we consider the following assumptions to improve the model: A The partner can distinguish between strong partner and weak partner H and the partner cooperates only with strong partner. A When provided with food resources the weak partner H has no negative influence on the stronger partner that is to say the weak partners can only eat the food after thestrongoneshaveusedenough.

3 Zhao et al. Advances in Difference Equations 08 08:7 age 3 of 3 A 3 Due to the lack of sufficient food resources once it becomes weak the weak partner H and their descendants will no longer be strong. A 4 The rate of becoming weak is described by the simple mass action α H. By the above assumptions we propose a model as follows: d = c αh dh d = r = H α + d eh c.5 where r i a i b i c i d i = are positive constants. The structureof this articleasfollows. In Sect. we will introduce several useful lemmas and prove permanence and non-permanence. In Sect. 3 we will discuss the existence of the equilibrium point. In Sect. 4 global stability of equilibrium points is studied. In Sect. 5 two examples are given to show the feasibility of our results. We end this paper by a brief discussion. ermanence and non-permanence In view of the actual ecological implications of system.5 we assume that the initial value H i 0 > 0 i =0 > 0 in system.5. Obviously any solution of system.5 remains positive for all t 0. Lemma. see [4] Let a >0b >0. I If dx xb ax then lim inf t + xt b for t 0 and x0 > 0. a II If dx xb ax then lim sup t + xt b for t 0 and x0 > 0. a Lemma. see [5] Let a >0b >0. If dx x b ax then lim t + xt=0for t 0 and x0 > 0. Theorem. If the assumptions B and B hold B M = αd e >0 α a c +b ea c c +b c +c B > then system.5 is permanent. roof Let t H t t T be any positive solution of system.5 from the second equation of system.5 it followsthat dh H d eh. According to Lemma.we have lim inf H t d def = H t + i > 0.. e

4 Zhao et al. Advances in Difference Equations 08 08:7 age 4 of 3 For any positive constant ε small enough it follows from. that there exists a large enough T >0suchthat H t>h i ε t T.. From the third equation we have d r c. According to Lemma.we have lim sup t def = s > 0..3 t + c For any positive constant ε small enough it follows from.3 that there exists a large enough T > T such that t s + ε t T..4 By applying.and.4 from the first equation of system.5 we have d According to Lemma.we have s + ε c αhi ε t T. lim sup t αhi ε s + ε t + +a c + b c s + ε. Letting ε 0 and by applying.and.3 a c + b lim sup t M def = H s > 0..5 t + a c c + b c + c For any positive constant ε small enough it follows from.5 that there exists a large enough T 3 > T such that t H s + ε t T 3..6 Then the second equation of.5leads to dh H α H s + ε + d eh t T3. According to Lemma.we have lim sup t + H t αhs + ε+d. e

5 Zhao et al. Advances in Difference Equations 08 08:7 age 5 of 3 Letting ε 0 in the above inequality leads to lim sup H t αhs + d def = H s t + e..7 For any positive constant ε small enough it follows from.7 that there exists a large enough T 4 > T 3 such that H t H s + ε t T 4..8 Then substituting.8 into the first equation of.5 we have d c αhs + ε t T 4. a According to Lemma.we have lim inf t αhs + ε a t + a c +. Letting ε 0 and by applying.5and.7 lim inf t M α H s def = H t + i > 0..9 e From the third equation of system.5 it follows that d r c. a According to Lemma.we have lim inf t a def = i..0 t + a c and.0 show that if the assumptions B B holdthen system.5ispermanent. Theorem. If the assumption B 3 holds B 3 M = αd e <0 then the weak partners H and partners are permanent the strong partners are nonpermanent. roof Let t H t T be any positive solutions of system.5fort 0. From the proof Theorem.weknow d s + ε c αhi ε t T. Noting that condition B 3 impliesthat αhi ε <0.

6 Zhao et al. Advances in Difference Equations 08 08:7 age 6 of 3 According to Lemma.we have lim t + t = 0.. By applying. from the second equation of system.5 it is easy to prove that lim t + H t= d e and. show that if the assumptions B 3 hold then the weak partners H and partners are permanent the strong partners are non-permanent. 3 Existence of equilibrium point Theorem 3. If the assumption B holds then system.5 have a unique positive equilibrium point. roof We determine the positive equilibrium of the system.5 through solving the following equations: a +b c αh =0 α + d eh =0 a +b c =0. 3. Here we transform Eqs. 3. into the following form: H a +b c =0 a +b c =0 α + d eh =0 3. where a = Ma b = Mb c =c + αd /M from the first and second equations of 3. e we have DH + E + F = where D = b a c c + b c + c F = a a c + b + E = [ a c ++c a + a a c + a b b a c + b ]. From the terms D and F of 3.3 we know that there is a unique positive solutions H. Substitute H into the second and third equations of 3.. Then system.5hasaunique positive equilibrium point E H H. Theorem 3. Clearly the system.5 has an equilibrium point E 0 H.

7 Zhao et al. Advances in Difference Equations 08 08:7 age 7 of 3 4 Global stability Theorem 4. If the assumptions B and B 4 hold B 4 α < a c e a +b H s then the positive equilibrium of system.5 is globally asymptotically stable. roof Inspired by the idea of Li Han and Chen [7]andLeon[8] the following Lyapunov function is presented: We define L : { H R 3 + : >0H >0 >0} R by L H =η H H θ H a + b θθ dθ + η θ a + b θθ dθ + η 3 H H H ln H H where η = η = b H a + b H /r b η 3 =/a. The function L H is defined continuous and positive definite for all H > 0. and the minimum L H = 0 occurs at the equilibrium point H H T. Calculating the derivative of L along the solution t H t t T of the system.5 we have where dl = η H c αh + η 3 H H αh + d eh + η r = η H αh H c H H + η 3 H H α H H e H H c + + η r = η H αh H c + H a + b H H c H H b H + + η r c + At+Bt At= η b H + + η r b a + b H α Bt= H H + η 3 H H α H H e H H b H a + b H η H H η r H H H H c H H + η 3 α H H H η3 e H H. Now we prove At Bt are negative definite.

8 Zhao et al. Advances in Difference Equations 08 08:7 age 8 of 3 Let At def = a +b a +b Y T AY where Y = H T and η η b H A = a +b + η r b η b H a +b + η r b a +b H η r a +b H. 4. Note first that both of the off-diagonal elements of matrix A are negative and η b H η η r + η r b a + b H = η η r b H = η η r b H thus At 0. Noting that ab θa + b θ b a + b H b a + b H θ > 0 it follows + >0 η b H η r b a + b H α Bt H H θ + H H r c H H θ θ + η 3α H H θ r c a + b H s η 3 αθ + H H η3 e H H α η 3α a θ θ η 3 e η 3αθ αθ a H H H H. Denote δ = c a +b H s α a θ η 3α θ and δ = η 3 c η 3αθ αθ a.thentaking η 3 = αea +αeb H s θ = θ = a a c c + a α + b H sα gives δ = ea c e a α b H s α a a c e + a α + b H s α δ = a c e a α b H s α a ea + b H s. From B 4 we know that δ i >0i =.ItiseasytoseethatBt 0. Obvious dl <0forallH >0H >0 > 0 except the equilibrium point H H where dl = 0. According to the Lyapunov asymptotic stability theorem [9] the equilibrium point H H is globally asymptotically stable in the interior of R 3 +.Thiscompletesthe proof. Theorem 4. If the assumptions B 3 and B 5 hold B 5 α < c e then the equilibrium point E 0 H system is globally asymptotically stable.

9 Zhao et al. Advances in Difference Equations 08 08:7 age 9 of 3 roof We define L : { H R 3 + : >0H >0 >0} R by L H =η H 0 a + b θ dθ + η θ a + b θθ dθ + η 3 H H H ln H H where η =η = a /r b η 3 =/a. The function L H is defined continuous and positive definite for all H > 0. and the minimum L H =0occurs at the equilibrium point H H T. Calculating the derivative of L along the solution t H t t T of the system.5 we have where dl = η + η r = η + η r c αh c αh H c + η 3 H H α eh H c + a = η αh H c + η 3 H H α eh H + η r Ct+Dt Ct= α Dt= η 3 α + η 3 H H α + d eh c + b a η H η r + η r b a H H Now we prove Ct Dt isnegativedefinite. Let Ct def = a +b a +b Y T CY where Y = T c H η 3eH H. η η r r b a C = η r b. 4. a η r Note first that both of the off-diagonal elements of matrix A are negative and η r b η η r = η r η η r b a 4a = 3η r >0 4 thus Ct 0.

10 Zhao et al. Advances in Difference Equations 08 08:7 age 0 of 3 Noting that ab θa + b θ θ >0from. for sufficiently small constant ε 0 >0there is an integer T >0suchthatift > T b t<ε 0 it follows that α Bt H θ + θ H H c H + η 3α H θ + η 3αθ H H η 3 eh H r c α η 3α H a + ε 0 a θ θ η 3 e η 3αθ αθ H H. a Denote δ = c a +ε 0 α a θ η 3α θ and δ = η 3 c η 3αθ αθ a.thentaking gives η 3 = a θ = θ = αea +αeε 0 a c c + a + α ε 0 δ = ea c e a α α ε 0 a a c e + a α + α ε 0 δ = a c e a α α ε 0. a ea + ε 0 From B 5 we know that for sufficiently small constant ε 0 >0δ i >0i =.Itiseasyto see that Dt 0. Obviously dl <0forallH >0H >0 > 0 except the equilibrium point 0 H where dl = 0. According to the Lyapunov asymptotic stability theorem [9] the equilibrium point 0 H is globally asymptotically stable in the interior of R 3 +.Thiscompletesthe proof. 5 Example and numeric simulation Consider the following system: d =3 + H 0 dh d = = H H By calculation we have M = αd e =0.9>0α =0.09<8< a c e a +b H s anditiseasytosee that the conditions B andb 4 are verified. It follows from Theorem 4. that there is a unique positive equilibrium point H H = of system 5. and it is globally asymptotically stable. Our numerical simulation supports our result see Fig..

11 Zhao et al. Advances in Difference Equations 08 08:7 age of 3 Figure Dynamical behavior of system 5. with initial values T T and T Consider the following system: d = H 3 dh d = = H H By calculation we have M = αd e = 0.<0α = 0.89 < a c e a +b H s =itiseasyto see that the conditions B 3 andb 5 are verified. It follows from Theorem 3. that there is a unique positive equilibrium point H = 00.5 of system 5. anditis globally asymptotically stable. Our numerical simulation supports our result see Fig.. 6 Discussion In this paper a May cooperative system with strong and weak cooperative partners is studied. We obtained the sufficient conditions that guarantee the permanence nonpermanence and the global stability of the equilibrium points. By comparing the conditions of B andb 3 we found that as α becomes larger and larger the strong partner changes from persistent to extinct. The ecological explanation is that more and more the strong partner become a weak partner thus we have extinction of the strong. The above numerical simulations also supports this conclusion. At the end of this paper we point out that conditions B andb 3 can be weakened or even are unnecessary. This problem is very interesting and worthy of further study in the future.

12 Zhao et al. Advances in Difference Equations 08 08:7 age of 3 Figure Dynamical behavior of system 5. with initial values T T and T Acknowledgements The research was supported by Guangxi College Enhancing Youth s Capacity roject 07KY0599 and the Scientific Research Development Fund of Young Researchers of Guangxi University of Finance and Economics 07QNB8. Competing interests The authors declare that there is no conflict of interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details College of Information and Statistics Guangxi University of Finance and Economics Nanning.R. China. College of Mathematics and Computer Science Fuzhou University Fuzhou.R. China. ublisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 9 January 08 Accepted: May 08 References. May R.M.: Theoretical Ecology rinciples and Applications. Sounders hiladelphia 976. Cui J.A. Chen L.S.: Global asymptotic stability in a nonautonomous cooperative system. Syst. Sci. Math. Sci Chen F.D. Liao X.Y. Huang Z.K.: The dynamic behavior of N-species cooperation system with continuous time delays and feedback controls. Appl. Math. Comput Chen L.J. Xie X.D.: ermanence of an N-species cooperation system with continuous time delays and feedback controls. Nonlinear Anal. Real World Appl Nasertayoob. Vaezpour S.M.: ermanence and existence of positive periodic solution for a multi-species cooperation system with continuous time delays feedback control and periodic external source. Adv. Differ. Equ. 0 Article ID Muhammadhaji A. Teng Z.D. Abdurahman X.: ermanence and extinction analysis for a delayed ratio-dependent cooperative system with stage structure. Afr. Math Li Z. Han M.A. Chen F.D.: Influence of feedback controls on an autonomous Lotka Volterra competitive system with infinite delays. Nonlinear Anal. Real World Appl Leon C.V.D.: Lyapunov functions for two-species cooperative systems. Appl. Math. Comput Yang K. Miao Z.S. Chen F.D. Xie X.D.: Influence of single feedback control variable on an autonomous Holling-II type cooperative system. J. Math. Anal. Appl Miao Z.S. Xie X.D. u L.Q.: Dynamic behaviors of a periodic Lotka Volterra commensal symbiosis model with impulsive. Commun. Math. Biol. Neurosci. 05 Article ID 3 05

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