Impulsive diffusion in single species model. (English) Zbl 1131.92071

Summary: In most population models, diffusion between patches is assumed to be continuous or discrete, but in practice many species diffuse only during a single period. We propose a single species model with impulsive diffusion between two patches, which provides a more natural description of population dynamics. By using the discrete dynamical system generated by a monotone, concave map for the population, and \(\varepsilon _{1} - \varepsilon _{2}\) variation, we prove that the map always has a globally stable positive fixed point. This means that a single species system with impulsive diffusion always has a globally stable positive periodic solution. This result is further substantiated by numerical simulations.


92D40 Ecology
37N25 Dynamical systems in biology
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
92D25 Population dynamics (general)
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[1] Bainov, D.; Simeonov, P., Impulsive differential equations: periodic solution and applications, (1993), Longman London · Zbl 0815.34001
[2] Beretta, E.; Takeuchi, Y., Global asymptotic stability of Lotka-Volterra diffusion models with continuous time delays, SIAM J appl math, 48, 627-651, (1998) · Zbl 0661.92018
[3] Beretta, E.; Takeuchi, Y., Global stability of single species diffusion Volterra models with continuous time delays, Bull math biol, 49, 431-448, (1987) · Zbl 0627.92021
[4] Hui, J.; Chen, L-s., A single species model with impulsive diffusion, Acta mathematicae applicatae sinica, English series, 21, 1, 43-48, (2005) · Zbl 1180.92072
[5] Freedman, H.I.; Takeuchi, Y., Global stability and predator dynamics in a model of prey dispersal in a patchy environment, Nonlinear anal, 13, 993-1002, (1989) · Zbl 0685.92018
[6] Freedman, H.I., Single species migration in two habitats: persistence and extinction, Math model, 8, 778-780, (1987)
[7] Freedman, H.I.; Rai, B.; Waltman, P., Mathematical models of population interactions with dispersal ii: differential survival in a change of habitat, J math anal appl, 115, 140-154, (1986) · Zbl 0588.92020
[8] Freedman, H.I.; Takeuchi, Y., Predator survival versus extinction as a function of dispersal in a predatorprey model with patchy environment, Appl anal, 31, 247-266, (1989) · Zbl 0641.92016
[9] Freedman, H.I.; Waltman, P., Mathematical models of population interaction with dispersal I: stability of two habitats with and without a predator, SIAM J appl math, 32, 631-648, (1977) · Zbl 0362.92006
[10] Hastings, A., Dynamics of a single species in a spatially varying environment: the stability role of high dispersal rates, J math biol, 16, 49-55, (1982) · Zbl 0496.92010
[11] Holt, R.D., Population dynamics in two patch environments:some anomalous consequences of optional habitat selection, Theor pop biol, 28, 181-208, (1985) · Zbl 0584.92022
[12] Funasaki, E.; Kot, M., Invasion and chaos in a periodically pulsed mass-action chemostat, Theor pop biol, 44, 203-224, (1993) · Zbl 0782.92020
[13] Kuang, Y.; Takeuchi, Y., Predator-prey dynamics in models of prey dispersal in two patch environments, Math biosci, 120, 77-98, (1994) · Zbl 0793.92014
[14] Levin, S.A., Dispersion and population interactions, Amer natur, 108, 207-228, (1974)
[15] Lakmeche, A.; Arino, O., Bifurcation of nontrivial periodic solution of impulsive differential equations arising chemotherapeutic treatment, Dyn cont dis impulsive syst, 7, 265-287, (2000) · Zbl 1011.34031
[16] Levin, S.A., Spatial patterning and the structure of ecological communities,in some mathematical questions in biology. VII, vol. 8, (1976), Amer. Math. Soc. Providence, RI
[17] Liu, X., Impulsive stabilization and applications to population growth models, J math, 25, 1, 381-395, (1995) · Zbl 0832.34039
[18] Liu, X.; Zhang, S., A cell population model described by impulsive pdes-existence and numerical approximation, Comput math appl, 36, 8, 1-11, (1998) · Zbl 0962.35181
[19] Liu, X.; Rohof, K., Impulsive control of a Lotka-bolterra system, IMA J math cont inform, 15, 269-284, (1998)
[20] Lu, Z.Y.; Takeuchi, Y., Permanence and global stability for cooperative Lotka-Volterra diffusion systems, Nonlinear anal, 10, 963-975, (1992) · Zbl 0784.93092
[21] Shulgin, B.; Stone, L., Pulse vaccination strategy in the sir epidemic model, Bull math biol, 60, 1-26, (1998)
[22] Amith, H.L., Cooperative systems of differential equations with concave nonlinearities, Nonlinear anal TMA, 10, 1037-1052, (1986) · Zbl 0612.34035
[23] Liu, X.; Chen, L., Complex dynamics of Holling II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, solitons & fractals, 16, 311-320, (2003) · Zbl 1085.34529
[24] Vandermeer, J.; Stone, L.; Blasius, B., Categeories of chaos and fractal basin boundaries in forced predator-prey models, Chaos, solitons & fractals, 12, 265-276, (2001) · Zbl 0976.92033
[25] Tang, S.; Chen, L., Chaos in functional response host-parasitoid ecosystem models, Chaos, solitons & fractals, 13, 875-884, (2002) · Zbl 1022.92042
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