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We study the possibility of chaotic dynamics in the externally driven Droop model. This model describes a phytoplankton population in a chemostat under periodic supply of nutrients. Previously it has been proven under very general assumptions that such systems are not able to exhibit chaotic dynamics. Here we show that the simple introduction of algal mortality may lead to chaotic oscillations of algal density in the forced chemostat. Our numerical simulations show that the existence of chaos is intimately related to plankton overshooting in the unforced model. We provide a simple measure, based on stability analysis, for estimating the amount of overshooting. These findings are not restricted to the Droop model but hold also for other chemostat models with mortality. Our results suggest periodically driven chemostats as a simple model system for the experimental verification of chaos in ecology.
In this paper, we demonstrate that it is possible to control the hyperchaos into the Rossler hyperchaotic system (RHS) by linear feedback of own signals. After introducing of the parameter "b" in the z-equation (b --> b + b(1)x(t) + b(2)y(t) + b(3)z(t) + b(4)w(t)), we study how the global dynamics can be altered in a desired direction (b(n) are considered as free parameters). We make a detailed bifurcation investigation of the modified Rossler hyperchaotic systems by varying the parameters b,. Finally, we calculate the Lyapunov exponents and the information dimension, where the regular, chaotic and hyperchaotic motion of modified RHS exist. (C) 2004 Published by Elsevier Ltd
One of the most striking features of ecological systems is their ability to undergo sudden outbreaks in the population numbers of one or a small number of species. The similarity of outbreak characteristics, which is exhibited in totally different and unrelated (ecological) systems naturally leads to the question whether there are universal mechanisms underlying outbreak dynamics in Ecology. It will be shown into two case studies (dynamics of phytoplankton blooms under variable nutrients supply and spread of epidemics in networks of cities) that one explanation for the regular recurrence of outbreaks stems from the interaction of the natural systems with periodical variations of their environment. Natural aquatic systems like lakes offer very good examples for the annual recurrence of outbreaks in Ecology. The idea whether chaos is responsible for the irregular heights of outbreaks is central in the domain of ecological modeling. This question is investigated in the context of phytoplankton blooms. The dynamics of epidemics in networks of cities is a problem which offers many ecological and theoretical aspects. The coupling between the cities is introduced through their sizes and gives rise to a weighted network which topology is generated from the distribution of the city sizes. We examine the dynamics in this network and classified the different possible regimes. It could be shown that a single epidemiological model can be reduced to a one-dimensional map. We analyze in this context the dynamics in networks of weighted maps. The coupling is a saturation function which possess a parameter which can be interpreted as an effective temperature for the network. This parameter allows to vary continously the network topology from global coupling to hierarchical network. We perform bifurcation analysis of the global dynamics and succeed to construct an effective theory explaining very well the behavior of the system.