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On the stable periodic solutions of a delayed two-species model of facultative mutualism. (English) Zbl 1143.34054
Let \(a_i(\cdot), b_i(\cdot)\in C(R,[0,+\infty));\) \(\tau_i(\cdot),\rho_i(\cdot)\in C^1(R,[0,+\infty));\) \(c_i(\cdot)\in C(R,(0,+\infty)); r_i(\cdot)\in C(R,R)\, (i=1,2)\) be \(\omega\)-periodic functions. The authors establish sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions for the following two-species system modelling “facultative mutualism”: \[ y'_1(t)=y_1(t)[r_1(t)-a_1(t)y_1(t)-b_1(t)y_1(t-\tau_1(t))+c_1(t)y_2(t-\rho_1(t)))], \] \[ y'_2(t)=y_2(t)[r_2(t)-a_2(t)y_2(t)-b_2(t)y_2(t-\tau_2(t))+c_2(t)y_1(t-\rho_2(t)))], \] where \(\int_0^\omega r_i(t)dt>0\,(i=1,2).\) These results are extended to a more general two-species facultative mutualism system involving multiple delays. Some applications and biological interpretations are discussed.

MSC:
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
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[1] Odum, E.P., Fundamentals of ecology, (1971), Saunders Philadelphia
[2] Gilbert, L.E., Ecological consequences of a coevolved mutualism between butterflies and plants, (), 210-240
[3] Handel, S.N., The competitive relationship of three woodland sedges and its bearing on the evolution of ant-dispersal of carex pedunculata, Evolution, 32, 151-163, (1978)
[4] Hale, M.E., The biology of lichens, (1974), Arnold London
[5] Tinker, P.B.H., Effects of vesicular-arbuscular mycorrhizas on higher plants, (), 325-349
[6] Burns, R.C.; Hardy, R.W., Nitrogen fixation in bacteria and higher plants, (1975), Springer-Verlag New York
[7] Batra, L.R., Insect – fungus symbiosis: nutrition, mutualism and commensalism, Proc. symp. second international mycological congress, (1979), Wiley New York
[8] Dustan, P., Distribution of zooxanthellae and photosynthetic chlorophyll pigments of the reef-building coral monastrea annularis Ellis and solander in relation to depth on a west Indian coral reef, Bull. mar. sci., 29, 79-95, (1979)
[9] Roughgarden, J., Evolution of marine symbiosis - A simple cost-benefit model, Ecology, 56, 1201-1208, (1975)
[10] Janzen, D.H., Coevolution of mutualism between ants and acacias in central America, Evolution, 20, 249-275, (1966)
[11] Porter, K.G., Enhancement of algal growth and productivity by grazing zooplankton, Science, 192, 1332-1334, (1976)
[12] May, R.M., Stability and complexity in model ecosystems, (1974), Princeton University Press Princeton, NJ
[13] Fan, M.; Wang, K., Periodic solutions of single population model with hereditary effect, Math. appl., 13, 58-61, (2000), (in Chinese) · Zbl 1008.92028
[14] Miler, R.K., On voterra’s population equation, SIAM J. appl. math., 14, 446-452, (1996)
[15] Seifert, G., On delay-differential equation for single species population variations, Nonlin. anal. TMA, 9, 1051-1059, (1987) · Zbl 0629.92019
[16] Freedman, H.I.; Xia, H., Periodic solution of single species models with delay, differential equations, dynamical systems and control science, Lecture note in pure appl. math., 152, 55-74, (1994) · Zbl 0794.34056
[17] Fujimoto, H., Dynamical behaviours for population growth equations with delays, Nonlin. anal. TMA, 31, 549-558, (1998) · Zbl 0887.34071
[18] Chen, B.S.; Liu, Y.Q., On the stable periodic solutions of single species molds with hereditary effects, Math. appl., 12, 42-46, (1999), (in Chinese) · Zbl 0948.34047
[19] Kuang, Y.; Smith, H.L., Global stability for infinite delay lotka – volterra type systems, J. differen. equat., 103, 221-246, (1993) · Zbl 0786.34077
[20] Goh, B.S., Stability in models of mutualism, Am. natural., 113, 216-275, (1979)
[21] Travis, C.C.; Post, W.M., Dynamics and comparative status of mutualistic communities, J. theoret. biol., 78, 553-571, (1979)
[22] Wolin, C.; Lawlor, L., Models of facultative mutualism: density effect, Am. natural., 124, 843-862, (1984)
[23] Boucher, D.H.; James, S.; Keeler, K.H., The ecology of mutualism, Ann. rev. ecol. syst., 13, 315-347, (1985)
[24] Dean, A.M., A simple model of mutualism, Am. natural., 12, 409-417, (1983)
[25] He, X.Z.; Gopalsamy, K., Persistence, attractivity, and delay in facultative mutualism, J. math. anal. appl., 215, 154-173, (1997) · Zbl 0893.34036
[26] Hirsch, M.W., Systems of differential equations which are competitive or cooperative. I. limit sets, SIAM J. math. anal., 13, 167-179, (1982) · Zbl 0494.34017
[27] Smith, H.L., On the asymptotic behaviour of a class of deterministic models of cooperating species, SIAM J. appl. math., 46, 368-375, (1986) · Zbl 0607.92023
[28] Mukherjee, D., Permanence and global attractivity for facultative mutualism system with delay, Math. methods appl. sci., 26, 1-9, (2003) · Zbl 1028.34069
[29] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Berlin · Zbl 0326.34021
[30] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Boston · Zbl 0752.34039
[31] Zaghrout, A.S., Stability and persistence of facultative mutualism with populations interacting in a food chain, Appl. math. comput., 45, 1-15, (1991) · Zbl 0727.92030
[32] Hui, H.H.; Xia, Y.H.; Lin, M.R., Existence of positive periodic solution of mutualism system with several delays, Chaos, solitons fractals, (2006)
[33] Yang, F.; Jiang, D.Q.; Wan, A.Y., Existence of positive solution of multidelays facultative mutualism system, J. eng. math., 19, 64-68, (2002), (in Chinese) · Zbl 1055.34137
[34] Chen, F.D.; Shi, J.L.; Chen, X.X., Periodicity in lotka – volterra facultative mutualism system with several delays, J. eng. math., 21, 403-409, (2004), (in Chinese)
[35] Kumar, R.; Freedman, H.I., A mathematical model of facultative mutualism with populations interacting in a food chain, Math. biosci., 97, 235-261, (1989) · Zbl 0695.92016
[36] Liu, Z.J.; Tan, R.H.; Chen, L.S., Global stability in a periodic delayed predator – prey system, Appl. math. comput., (2006)
[37] Alvarez, C.; Lazer, A., An application of topological degree to the periodic competing species problem, J. aust. math. soc., 28, 202-219, (1986) · Zbl 0625.92018
[38] Ahmad, S., Convergence and ultimate bounds of solutions of the nonautonomous volterra – lotka competition equations, J. math. anal. appl., 127, 377-387, (1987) · Zbl 0648.34037
[39] Amine, Z.; Ortega, R., A periodic prey – predator system, J. math. anal. appl., 185, 477-489, (1994) · Zbl 0808.34043
[40] Cushing, J.M., Periodic time-dependent predator – prey systems, SIAM J. appl. math., 34, 82-95, (1997) · Zbl 0348.34031
[41] Liu, Z.J.; Chen, L.S., Periodic solutions of a discrete time nonautonomous two-species mutualistic system with delays, Adv. complex syst., 9, 87-98, (2006) · Zbl 1107.92053
[42] Ye, D.; Fan, M., Periodicity in mutualism systems with impulse, Taiwanese J. math., 10, 723-737, (2006) · Zbl 1105.34026
[43] Wang, W.D.; Ma, Z.E., Harmless delays for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085
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