14,659 research outputs found

    Simple Models, Catastrophes and Cycles

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    It is often observed in practice that the essential behavior of mathematical models involving many variables can be captured by a much smaller model involving only a few variables. Further, the simpler model very often displays oscillatory behavior of some sort, especially when critical problem parameters are varied in certain ranges. This paper attempts to supply arguments from the theory of dynamical systems for why oscillatory behavior is so frequently observed and to show how such behavior emerges as a natural consequence of focusing attention upon so-called "essential" variables in the process of model simplification. The relationship of model simplification and oscillatory behavior is shown to be inextricably intertwined with the problems of bifurcation and catastrophe in that the oscillations emerge when critical system parameters, i.e., those retained in the simple model, pass through critical regions. The importance of the simplification, oscillation, bifurcation pattern is demonstrated here by consideration of several examples from the environmental, economic, and urban areas

    Limit Cycles in Slow-Fast Forest-Pest Models

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    Some of the most exciting current work in the environmental sciences involves simplified but analytically tractable versions of a few basic equations. IIASA's Environment Program has developed such an approach in its analysis of forest systems. A number of previous papers (WP-87-70 and WP-87-92) have demonstrated the progress that has been made. In this new work some of the ideas contained in those papers have been further developed. In particular a simple aged-structure forest model is considered to prove that a forest can exhibit periodic behavior even in the case the insect pest is adapted only to mature trees. The insect pest assumed to have a very fast dynamics with respect to trees and the analysis is carried out through singular perturbation arguments. The method is based only upon simple geometric characteristics of the equilibrium manifolds of the fast, intermediate and slow variables of the system and allows one to derive explicit conditions on the parameters that guarantee the existence of a limit cycle in the extreme case of very fast-very slow dynamics

    D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories

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    The spectrum of D2-branes wrapped on an ALE space of general ADE type is determined, by representing them as boundary states of N=2 superconformal minimal models. The stable quantum states have RR charges which precisely represent the gauge fields of the corresponding Lie algebra. This provides a simple and direct physical link between the ADE classification of N=2 superconformal field theories, and the corresponding root systems. An affine extension of this structure is also considered, whose boundary states represent the D2-branes plus additional D0-branes.Comment: 12p, harvmac, minor corrrections and ref adde

    On population extinction risk in the aftermath of a catastrophic event

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    We investigate how a catastrophic event (modeled as a temporary fall of the reproduction rate) increases the extinction probability of an isolated self-regulated stochastic population. Using a variant of the Verhulst logistic model as an example, we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction probability caused by the catastrophe. This quantity is given by the eikonal action computed over "the optimal path" (instanton) of an effective classical Hamiltonian system with a time-dependent Hamiltonian. For a general catastrophe the eikonal equations can be solved numerically. For simple models of catastrophic events analytic solutions can be obtained. One such solution becomes quite simple close to the bifurcation point of the Verhulst model. The eikonal results for the increase in the extinction probability caused by a catastrophe agree well with numerical solutions of the master equation.Comment: 11 pages, 11 figure
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