Expansion rates and Lyapunov exponents
Sebastian J. Schreiber
Discrete & Continuous Dynamical Systems - A 1997, 3(3): 433-438 doi: 10.3934/dcds.1997.3.433
The logarithmic expansion rate of a positively invariant set for a $C^1$ endomorphism is shown to equal the infimum of the Lyapunov exponents for ergodic measures with support in the invariant set. Using this result, aperiodic flows of the two torus are shown to have an expansion rate of zero and the effects of conjugacies on expansion rates are investigated.
keywords: expansion rates. Lyapunov exponents
The Dynamics of the Schoener-Polis-Holt model of Intra-Guild Predation
Eric Ruggieri Sebastian J. Schreiber
Mathematical Biosciences & Engineering 2005, 2(2): 279-288 doi: 10.3934/mbe.2005.2.279
Intraguild predation occurs when one species (the intraguild predator) predates on and competes with another species (the intraguild prey). A classic model of this interaction was introduced by Gary Polis and Robert Holt building on a model of competing species by Thomas Schoener. A global analysis reveals that this model exhibits generically six dynamics: extinction of one or both species; coexistence about a globally stable equilibrium; contingent exclusion in which the first established species prevents the establishment of the other species; contingent coexistence in which coexistence or displacement of the intraguild prey depend on initial conditions; exclusion of the intraguild prey; and exclusion of the intraguild predator. Implications for biological control and community ecology are discussed.
keywords: ecology nonlinear dynamics intraguild predation.
On persistence and extinction for randomly perturbed dynamical systems
Sebastian J. Schreiber
Discrete & Continuous Dynamical Systems - B 2007, 7(2): 457-463 doi: 10.3934/dcdsb.2007.7.457
Let $f:\M\to\M$ be a continuous map of a locally compact metric space. Models of interacting populations often have a closed invariant set $\partial \M$ that corresponds to the loss or extinction of one or more populations. The dynamics of $f$ subject to bounded random perturbations for which $\partial \M$ is absorbing are studied. When these random perturbations are sufficiently small, almost sure absorbtion (i.e. extinction) for all initial conditions is shown to occur if and only if $M\setminus \partial M$ contains no attractors for $f$. Applications to evolutionary bimatrix games and uniform persistence are given. In particular, it shown that random perturbations of evolutionary bimatrix game dynamics result in almost sure extinction of one or more strategies.
keywords: absorbing sets persistence. Random perturbations of dynamical systems

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