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28
The subcritical collapse of predator populations in discretetime predator–prey models
 Math. Biosci
, 1992
"... Many discretetime predatorprey models possess three equilibria, corresponding to (1) extinction of both species, (2) extinction of the predator and survival of the prey at its carrying capacity, or (3) coexistence of both species. For a variety of such models, the equilibrium corresponding to coex ..."
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Cited by 8 (2 self)
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Many discretetime predatorprey models possess three equilibria, corresponding to (1) extinction of both species, (2) extinction of the predator and survival of the prey at its carrying capacity, or (3) coexistence of both species. For a variety of such models, the equilibrium corresponding to coexistence may lose stability via a Hopf bifurcation, in which case trajectories approach an invariant circle. Alternatively, the equilibrium may undergo a subcritical flip bifurcation with a concomitant crash in the predator’s population. We review a technique for distinguishing between subcritical and supercritical flip bifurcations and provide examples of predatorprey systems with a subcritical flip bifurcation.
Hyperbolic Sets for Noninvertible Maps and Relations
 Disc. Cont. Dyn. Systems
, 1996
"... Hyperbolic Sets for Noninvertible Maps and Relations by Evelyn Sander This thesis presents a theory of hyperbolic structures and dynamics of smooth noninvertible maps and relations. In this context, it includes a new proof of the stable manifold theorem for fixed points, the shadowing lemma, and a v ..."
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Hyperbolic Sets for Noninvertible Maps and Relations by Evelyn Sander This thesis presents a theory of hyperbolic structures and dynamics of smooth noninvertible maps and relations. In this context, it includes a new proof of the stable manifold theorem for fixed points, the shadowing lemma, and a version of the stable manifold theorem for hyperbolic sets. It also gives a description of some of the behavior of transverse homoclinic orbits for noninvertible maps and relations. ii Contents 1 Introduction 1 1.1 Preliminary remarks : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Previous theory and applications : : : : : : : : : : : : : : : : : : : 2 1.2.1 The noninvertible stable manifold theorem : : : : : : : : : : 3 1.3 Hyperbolic sets, stable manifolds, and the shadowing lemma : : : : 4 1.4 Transverse homoclinic orbits : : : : : : : : : : : : : : : : : : : : : : 5 2 Applications 7 2.1 Difference methods : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2.2 Adaptive ...
Trace formulas for stochastic evolution operators: Weak noise perturbation theory
 J. Stat. Phys
, 1998
"... Periodic orbit theory is an effective tool for the analysis of classical and quantum chaotic systems. In this paper we extend this approach to stochastic systems, in particular to mappings with additive noise. The theory is cast in the standard field theoretic formalism, and weak noise perturbation ..."
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Periodic orbit theory is an effective tool for the analysis of classical and quantum chaotic systems. In this paper we extend this approach to stochastic systems, in particular to mappings with additive noise. The theory is cast in the standard field theoretic formalism, and weak noise perturbation theory written in terms of Feynman diagrams. The result is a stochastic analog of the nexttoleading ¯h corrections to the Gutzwiller trace formula, with long time averages calculated from periodic orbits of the deterministic system. The perturbative corrections are computed analytically and tested numerically on a simple 1dimensional system. 1
A route to computational chaos revisited: noninvertibility and the breakup of an invariant circle, Phsys
 D
"... the breakup of an invariant circle ..."
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Some properties of the kdimensional Lyness’ map
 J. Phys. A
"... This paper is devoted to study some properties of the kdimensional Lyness ’ map F (x1,..., xk) = (x2,..., xk, (a + ∑ k i=2 xi)/x1). Our main result presentes a rational vector field that gives a Lie symmetry for F. This vector field is used, for k ≤ 5, to give information about the nature of the i ..."
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This paper is devoted to study some properties of the kdimensional Lyness ’ map F (x1,..., xk) = (x2,..., xk, (a + ∑ k i=2 xi)/x1). Our main result presentes a rational vector field that gives a Lie symmetry for F. This vector field is used, for k ≤ 5, to give information about the nature of the invariant sets under F. When k is odd, we also present a new (as far as we know) first integral for F◦F which allows to deduce in a very simple way several properties of the dynamical system generated by F. In particular for this case we prove that, except on a given codimension one algebraic set, none of the positive initial conditions can be a periodic point of odd period.
How to manage nature? Strategies, predatorprey models, and chaos
 Marine Resource Economics
, 1997
"... Abstract The LotkaVolterra predatorprey model exemplifies the implicit and explicit assumptions managers often have regarding species interaction—populations are stable or fluctuate periodically. The reality is often much more complicated and there is overwhelming evidence that many populations fl ..."
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Abstract The LotkaVolterra predatorprey model exemplifies the implicit and explicit assumptions managers often have regarding species interaction—populations are stable or fluctuate periodically. The reality is often much more complicated and there is overwhelming evidence that many populations fluctuate in a nonperiodic way. Using a discrete predatorprey model that generates chaos, it is possible to qualitatively mimic the interaction of some predatorprey populations. The implications of the paper are that managers should place greater emphasis on theoretical modeling and simulations, try to understand ecosystems and broad relationships between species rather than obtain minute details and data on individual populations, make management as flexible as possible to help people adjust to rapid changes in populations, employ mixed strategies so as to give options whatever the underlying dynamics, and, where appropriate, experiment with different strategies for different subpopulations to learn more about the effectiveness of alternative management approaches. Key words Chaos, management strategies, predatorprey models, renewable resources. If, along the road, you chance upon a bird’s nest, in any tree or on the ground, with fledglings or eggs and the mother sitting over the fledglings or on the eggs, do not take the mother together with her young. Let the mother go, and take only the young. Deuteronomy 22:6–7
Computer Simulations on the GumowskiMira Transformation. Forma 15
"... Abstract. We perform a computer simulation on the GumowskiMira transformation (hereafter abbreviated as GM tranformation) and present a variety of 2dimensional patterns obtained from the GM transformation. Among these patterns, there are images which resemble very much “living marine creatures”. 1 ..."
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Abstract. We perform a computer simulation on the GumowskiMira transformation (hereafter abbreviated as GM tranformation) and present a variety of 2dimensional patterns obtained from the GM transformation. Among these patterns, there are images which resemble very much “living marine creatures”. 1. Definition of the GM Transformation One encounters various nonlinear phenomena in nature. Computer simulation provides a powerful tool in understanding such nonlinear phenomena (KINZEL and REENTS, 1998). In this study, we perform a computer simulation on the GM transformation (GUMOWSKI and MIRA, 1980) and present a variety of 2dimensional images obtained from the GM transformation. Among these images, there are patterns which resemble very much (cross sections of) “living marine creatures”. The GM transformation is the 2dimensional discrete dynamic system, which is expressed by the following recurrent formula: x y a y y f xn n n n n+ = + − +1 21 0 05 (.) (), y x f xn n n+ + = − +1 1 (), f x x x x µ µ2 1
Chapter 1. Some methods for the global analysis of closed invariant curves in twodimensional maps
 In Business Cycle Dynamics – Models and
, 2006
"... It is well known that models of nonlinear oscillators applied to the study of the business cycle can be formulated both as continuos or discrete time dynamic models (see e.g. [23], [33], [34]). However, economic time is often discontinuous (discrete) because decisions in economics cannot be con ..."
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It is well known that models of nonlinear oscillators applied to the study of the business cycle can be formulated both as continuos or discrete time dynamic models (see e.g. [23], [33], [34]). However, economic time is often discontinuous (discrete) because decisions in economics cannot be con
Modeling and Analysis of Sustainable Systems
 European Research Consortium in Informatics and Mathematics
, 1999
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Complete Synchronization in Coupled Neuromodules of Different Types
 Theory in Biosciences
, 1999
"... We discuss the parametrized dynamics of two coupled recurrent neural networks comprising either additive sigmoid neurons in discrete time or biologically more plausible timecontinuous leakyintegrateand re cells. General conditions for the existence of synchronized activity in such networks are g ..."
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We discuss the parametrized dynamics of two coupled recurrent neural networks comprising either additive sigmoid neurons in discrete time or biologically more plausible timecontinuous leakyintegrateand re cells. General conditions for the existence of synchronized activity in such networks are given, which guarantee that corresponding neurons in both coupled subnetworks evolve synchronously. It is, in particular, demonstrated that even the coupling of totally di erent network structures can result in complex dynamics constrained to a synchronization manifold M. For additive sigmoid neurons the synchronized dynamics can be periodic, quasiperiodic as well as chaotic, and its stability can be determined by Lyapunov exponent techniques. For leakyintegrateand re cells synchronized orbits are typically periodic, often with an extremely long period duration. In addition to synchronized attractors there often coexist asynchronous periodic, quasiperiodic and even chaotic attractors.