Abstract
A system of delay differential equation is proposed to account the effect of delay in the predator–prey model of interacting population. In this article, the modified ratio-dependent Bazykin model with delay in predator equation has been considered. The essential mathematical features of the proposed model are analyzed with the help of equilibria, local and global stability analysis, and bifurcation theory. The parametric space under which the system enters into a Hopf-bifurcation has been investigated. Global stability results are obtained by constructing suitable Lyapunov functions. We derive the explicit formulae for determining the stability, direction, and other properties of bifurcating periodic solutions by using normal form and central manifold theory. Using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc., 350:4799–4838, 1998) for functional differential equations, the global existence of periodic solutions has been established. Our analytical findings are supported by numerical experiments. Biological implication of the analytical findings are discussed in the conclusion section.
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References
Wu, J.: Symmetric functional differential equations and nural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)
Anderson, R.M., May, R.M.: Regulation and stability of host-parasite population interactions I Regulatory Processes. J. Anim. Ecol. 47, 219–247 (1978)
Anderson, R.M., May, R.M.: The population dynamics of microparasites and their invertebrates hosts. Proc. R. Soc. Lond. B 291, 451–524 (1981)
Anderson, R.M., May, R.M.: The invasion persistence and spread of infectious diseases within animal and plant communities. Proc. R. Soc. Lond. B 314, 533–570 (1986)
Anderson, R.M., May, R.M.: Infectious Disease of Humans, Dynamics and Control. Oxford University Press, Oxford (1991)
Bailey, N.J.T.: The Mathematical Theory of Infectious Disease and Its Applications. Griffin, London (1975)
Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: Epidemic Models, Their Structure and Relation to Data. Cambridge University Press, Cambridge (1994)
Kuang, Y.: Delay Differential Equation with Applications in Population Dynamics. Academic Press, New York (1993)
Gopalsamy, K.: Stability and Oscillation in Delay Differential Equation of Population Dynamics. Kluwer Academic, Dordrecht (1992)
MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge (1989)
May, R.M.: Stability and Complexity in Model Ecosystem. Princeton Univ. Press, Princeton (1974)
Curds, C.R., Cockburn, A.: Studies on the growth and feeding of tetrahymena pyriformis in axenic and monoxenic culture. J. Gen. Microbiol. 54, 343–358 (1968)
Hassell, M.P., Varley, G.C.: New inductive population model for insect parasites and its bearing on biological control. Nature 223, 1133–1137 (1969)
Salt, G.W.: Predator and prey densities as controls of the rate of capture by the predator Didinium nasutum. Ecology 55, 434–439 (1974)
Arditi, R., Ginzburg, L.R.: Coupling in predator–prey dynamics: ratiodependence. J. Theor. Biol. 139, 311–326 (1989)
Arditi, R., Berryman, A.A.: The biological control paradox. Trends Ecol. Evol. 6, 32 (1991)
Arditi, R., Saiah, H.: Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73, 1544–1551 (1992)
Akcakaya, H.R., Arditi, R., Ginzburg, L.R.: Ratio dependent predation: an abstraction that works. Ecology 76, 995–1004 (1995)
Conser, C., Angelis, D.L., Ault, J.S., Olson, D.B.: Effects of spatial grouping on functional response of predators. Theor. Popul. Biol. 56, 65–75 (1999)
Gutierrez, A.P.: The physiological basis of ratio-dependent predator–prey theory: a metabolic pool model of Nicholson’s blowflies as an example. Ecology 73, 1552–1563 (1992)
Jost, C., Arino, O., Arditi, R.: About deterministic extinction in ratio-dependent predator–prey models. Bull. Math. Biol. 61, 19–32 (1999)
Kuang, Y.: Rich dynamics of Gause-type ratio-dependent predator–prey system. Fields Inst. Commun. 21, 325–337 (1999)
Thieme, H.: Mathematical biology, an introduction via selected topics. Lecture Note at Arizona State University (1997)
Hsu, S.B., Hwang, T.W., Kuang, Y.: Rich dynamics of ratio-dependent one prey two predators model. J. Math. Biol. 43, 377–396 (2001)
Cosner, C.: Variability vagueness and comparison methods for ecological models. Bull. Math. Biol. 58, 207–246 (1996)
Kuang, Y., Bertta, E.: Global qualitative analysis of a ratio-dependent predator–prey system. J. Math. Biol. 36, 389–406 (1998)
Bazykin, A.D.: Non-linear dynamics of interacting populations. In: Chua, L.O. (ed.) Series on Non Linear Science, Series A, vol. 11. World Scientific, Singapore (1985). Original Russian version: Bazykin, A.D., Nauka, Moscow (1998)
Alekseev, V.V.: Effect of saturation factor on dynamics of predator prey system. Biofizika 18, 922–926 (1973)
Bazykin, A.D.: Volterra’s system and Michaelis–Menten equation. In: Voprosy Matematicheskoi Genetiki, Novosibirsk, pp. 103–143 (1974)
Bazykin, A.D.: Structure and dynamical stability of model predator–prey systems. Laxenburg, IIASA, RM-76-8 (1976)
Bazykin, A.D., Brezovskaya, F.S., Buriev, T.I.: Dynamics of predator–prey system including predator saturation and competition. In: Faktory Raznoobraziya v Matematicheskoi Ekologii i Populyatsionnoi Genetike, Pushchino, pp. 6–33 (1980)
Wang, W.D., Chen, L.S.: A predator–prey system with stage-structure for predator. J. Comput. Appl. Math. 33, 83–101 (1997)
Zhao, T., Kuang, Y., Smith, H.L.: Global existence of periodic solutions in a class of delayed gause-type predator–prey systems. Nonlinear Anal. 28, 1373–1390 (1997)
Freedman, H.I., Hari Rao, V.S.: The trade-off between mutual interference and time lags in predator–prey system. Bull. Math. Biol. 45, 991–1004 (1983)
Song, X.Y., Chen, L.S.: Optimal harvesting and stability with stage-structure for a two species competitive system. Math. Biosci. 170, 173–186 (2001)
Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1996)
Hassard, B.D., Kazarinof, N.D., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)
Wang, W.D., Ma, Z.: Harmless delays for uniform persistence. J. Math. Anal. Appl. 158, 256–268 (1991)
Bhunia, A.B.: Ecology of tidal creeks and mudflats of Sagar Island (Sunderbans) West Bengal. Ph.D. dissertation, Calcutta University (1979)
Roy, M., Mandal, S., Ray, S.: Detrital ontogenic model including decomposer diversity. Ecol. Model. 215, 200–206 (2008)
Mandal, S., Ray, S., Ghosh, P.B.: Modelling of the contribution of dissolved inorganic nitrogen (DIN) from litterfall of adjacent mangrove forest to Hooghly–Matla estuary. India Ecol. Model. 220, 2988–3000 (2009)
Sun, C., Han, M., Lin, Y., Chen, Y.: Global qualitative analysis for a predator–prey system with delay. Chaos Solitons Fractals 32, 1582–1596 (2007)
Song, Y., Han, M., Peng, Y.: Stability and Hopf bifurcations in a competitive Lotka–Volterra system with two delays. Chaos Solitons Fractals 22, 1139–1148 (2004)
Song, Y., Yuan, S.: Bifurcation analysis in a predator–prey system with time delay. Nonlinear Anal. 7, 265–284 (2006)
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Sarwardi, S., Haque, M. & Mandal, P.K. Ratio-dependent predator–prey model of interacting population with delay effect. Nonlinear Dyn 69, 817–836 (2012). https://doi.org/10.1007/s11071-011-0307-9
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DOI: https://doi.org/10.1007/s11071-011-0307-9