Abstract
In this paper, we introduce and study a model of Tessiet type food chain chemostat with periodically varying substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Furthermore, we numerically simulate a model with sinusoidal dilution rate, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halving.
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Schaffer, W.M.: Can nonlinear dynamics clucidate mechanisms in ecology and epidemiology?. IMA J. Math. Appl. Med. Biol. 2, 221–252 (1985)
Cushing, J.M.: Two species competition in a periodic environment. J. Math. Biol. 10, 348–400 (1980)
Hale, J.K., Somolinos, A.S.: Competition for fluctuating nutrient. J. Math. Biol. 18, 255–280 (1983)
Hsu, S.B.: A competition model for a seasonally fluctuating nutrient. J. Math. Biol. 18, 115–132 (1980)
Alessandra, G., Oscar, D.F., Sergio, R.: Food chains in the chemostat: relationships between mean yield and complex dynamics. Bull. Math. Biol. 60, 703–719 (1998)
Mark, K., Sayler, G.S., Waltman, T.W.: Complex dynamics in a model microbial system. Bull. Math. Biol. 54, 619–648 (1992)
Eric, F., Mark, K.: Invasion and chaos in periodically pulsed mass-action chemostat. Theor. Popul. Biol. 44, 203–224 (1993)
Butler, G.J., Hsu, S.B., Waltman, P.: A mathematical model of the chemostat with periodic washout rate. SIAM J. Appl. Math. 45, 435–449 (1985)
Lenas, P., Pavlou, S.: Coexistence of three competing microbial populations in a chemostat with periodically varying dilution rate. Math. Biosci. 129, 111–142 (1995)
Pilyugin, S.S., Waltman, P.: Competition in the unstirred chemostat with periodic input and washout. SIAM J. Appl. Math. 59, 1157–1177 (1999)
Cushing, J.M.: Periodic time-dependent predator–prey systems. SIAM J. Appl. Math. 10, 384–400 (1977)
May, R.M.: Biological populations with non-overlapping generations: stable points, stable cycles, and chaos. Science 186, 645–647 (1974)
Gakkhar, S., Naji, M.A.: Order and chaos in predator to prey ratio-dependent food chain. Chaos Solitons Fractal 18, 229–239 (2003)
Klebanoff, A., Hastings, A.: Chaos in three species food chains. J. Math. Biol. 32, 427–451 (1994)
Grebogi, C., Ott, E., York, J.A.: Crises, Sudden Changes in Chaotic attractors and Choatic transients. Physica D 7, 181–200 (1983)
Neubert, M.G., Caswell, H.: Density-dependent vital rates and their population dynamic consequences. J. Math. Biol. 41, 103–121 (2000)
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Pang, G., Liang, Y. & Wang, F. Bifurcation and chaos of Tessiet type food chain chemostat with periodically varying substrate. J Math Chem 44, 674–690 (2008). https://doi.org/10.1007/s10910-008-9347-x
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DOI: https://doi.org/10.1007/s10910-008-9347-x