Abstract
This paper examines chaos in some population models for biological processes. In particular, the paper is concerned with chaos produced by discretization of continuous-time population models, and with the question of whether or not this chaotic behavior can occur when a feedback control management strategy is applied to continuous-time population models. Using two classical population models, it is shown that both exponential discretization and discretization associated with numerical simulation can produce chaos in the resulting discrete-time system. For example, a continuous-time Lotka-Volterra system exhibits periodic trajectories, but a particular discretization procedure is shown to yield discrete-time trajectories that all converge to a very complicated chaotic strange attractor, even if the discretization time steps are small. The results in the paper also show that this chaotic behavior can occur even if a feedback control management strategy is applied to the continuous-time system to stabilize the equilibrium point.
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Grantham, W.J., Athalye, A.M. (1990). A Chaotic System: Discretization and Control. In: Vincent, T.L., Mees, A.I., Jennings, L.S. (eds) Dynamics of Complex Interconnected Biological Systems. Mathematical Modelling, vol 6. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-6784-0_9
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DOI: https://doi.org/10.1007/978-1-4684-6784-0_9
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