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Existence of periodic solutions for a periodic mutualism model on time scales. (English) Zbl 1146.34326

Summary: By using Mawhin’s continuation theorem of coincidence degree theory, sufficient criteria are obtained for the existence of periodic solutions of the mutualism model

x Δ (t)=r 1 (t)K 1 (t)+α 1 (t)exp{y(t-τ 2 (t,y(t)))} 1+exp{y(t-τ 2 (t,y(t)))}-exp{x(t-σ 1 (t,x(t)))},y Δ (t)=r 2 (t)K 2 (t)+α 2 (t)exp{x(t-τ 1 (t,x(t)))} 1+exp{x(t-τ 1 (t,x(t)))}-exp{y(t-σ 2 (t,y(t)))},

where r i , K i , α i C(𝕋, + ), α i >K i , i=1,2,τ i , σ i C(𝕋×,𝕋 + ), i=1,2,r i ,K i ,α i ,τ i ,σ i (i=1,2) are functions of period ω>0.

MSC:
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
39A10Additive difference equations
References:
[1]Agarwal, R. P.: Difference equations and inequalities: theory, method and applications, Monogr. textbooks pure appl. Math. 228 (2000) · Zbl 0952.39001
[2]Agarwal, R. P.; Wong, P. J. Y.: Advanced topics in difference equations, (1997)
[3]Bohner, M.; Peterson, A.: Dynamic equations on time scales, an introduction with applications, (2001)
[4]Bohner, M.; Peterson, A.: Advances in dynamic equations on time scales, (2003)
[5]Boucher, D. H.; James, S.; Keeler, K. H.: The ecology of mutualism, Ann. rev. Syst. 13, 315-347 (1982)
[6]Dean, A. M.: A simple model of mutualism, Amer. nat. 121, 409-417 (1983)
[7]Freedman, H. I.: Deterministic mathematics models in population ecology, (1980)
[8]Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations, Lecture notes in math. 568 (1977) · Zbl 0339.47031
[9]Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[10]Hilger, S.: Analysis on measure chains — a unified approach to continuous and discrete calculus, Results math. 18, 18-56 (1990) · Zbl 0722.39001
[11]Kaufmann, E. R.; Raffoul, Y. N.: Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. math. Anal. appl. 319, 315-325 (2006) · Zbl 1096.34057 · doi:10.1016/j.jmaa.2006.01.063
[12]Li, Y. K.: On a periodic mutualism model, Anziam j. 42, 569-580 (2001)
[13]Murry, J. D.: Mathematical biology, (1989)
[14]Pianka, E. R.: Evolutionary ecology, (1974)
[15]Wolin, C. L.; Lawlor, L. R.: Model of facultative mutualism: density effects, Amer. nat. 144, 843-862 (1984)