Nonlinear Dynamics of Interacting Populations
World Scientific, 1998 - Science - 193 pages
This book contains a systematic study of ecological communities of two or three interacting populations. Starting from the Lotka-Volterra system, various regulating factors are considered, such as rates of birth and death, predation and competition. The different factors can have a stabilizing or a destabilizing effect on the community, and their interplay leads to increasingly complicated behavior. Studying and understanding this path to greater dynamical complexity of ecological systems constitutes the backbone of this book. On the mathematical side, the tool of choice is the qualitative theory of dynamical systems ? most importantly bifurcation theory, which describes the dependence of a system on the parameters. This approach allows one to find general patterns of behavior that are expected to be observed in ecological models. Of special interest is the reaction of a given model to disturbances of its present state, as well as to changes in the external conditions. This leads to the general idea of ?dangerous boundaries? in the state and parameter space of an ecological system. The study of these boundaries allows one to analyze and predict qualitative and often sudden changes of the dynamics ? a much-needed tool, given the increasing antropogenic load on the biosphere.As a spin-off from this approach, the book can be used as a guided tour of bifurcation theory from the viewpoint of application. The interested reader will find a wealth of intriguing examples of how known bifurcations occur in applications. The book can in fact be seen as bridging the gap between mathematical biology and bifurcation theory.
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A1 and A2 analyse Andronov-Hopf bifurcation Andronov-Hopf curve attractors basin of attraction Bazykin behavior biangle bifurcation curves bifurcation diagram bifurcation theory bilocal biological biotic potential changes codimension codimension-two bifurcation consider curve of saddle-nodes decrease depending described diagram of system differential equations dissipative structure ecological niche ecosystem equilibrium A2 equilibrium density excitation of oscillations existence harvesting intensity heteroclinic cycle homoclinic loop curve interacting populations intersection intraspecies competition Khibnik locally stable Lyapunov quantity neighborhood nontrivial equilibrium nullclines ö)-plane octant oscillatory regime parameter regions parameter space parameters from region perturbation phase portraits phase space plane points A1 portraits of Eq portraits of system possible relative positions predator and prey predator competition predator population density predator saturation predator-prey system prey population density producer population protocooperation saddle saddle-node bifurcation saddle-nodes of limit shown in Fig species stable equilibrium stable limit cycle stable manifold threshold trajectories trophic function unstable Volterra
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