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Robust permanence for ecological equations with internal and external feedbacks.
Species experience both internal feedbacks with endogenous factors such as trait evolution and external feedbacks with exogenous factors such as weather. These feedbacks can play an important role in determining whether populations persist or communities of species coexist. To provide a general mathematical framework for studying these effects, we develop a theorem for coexistence for ecological models accounting for internal and external feedbacks. Specifically, we use average Lyapunov functions and Morse decompositions to develop sufficient and necessary conditions for robust permanence, a form of coexistence robust to large perturbations of the population densities and small structural perturbations of the models. We illustrate how our results can be applied to verify permanence in non-autonomous models, structured population models, including those with frequency-dependent feedbacks, and models of eco-evolutionary dynamics. In these applications, we discuss how our results relate to previous results for models with particular types of feedbacks
Robust permanence for interacting structured populations
The dynamics of interacting structured populations can be modeled by
where , , and
are matrices with non-negative off-diagonal entries. These models are
permanent if there exists a positive global attractor and are robustly
permanent if they remain permanent following perturbations of .
Necessary and sufficient conditions for robust permanence are derived using
dominant Lyapunov exponents of the with respect to
invariant measures . The necessary condition requires for all ergodic measures with support in the boundary of the
non-negative cone. The sufficient condition requires that the boundary admits a
Morse decomposition such that for all invariant
measures supported by a component of the Morse decomposition. When the
Morse components are Axiom A, uniquely ergodic, or support all but one
population, the necessary and sufficient conditions are equivalent.
Applications to spatial ecology, epidemiology, and gene networks are given
Generalized Urn Models of Evolutionary Processes
Generalized Polya urn models can describe the dynamics of finite populations
of interacting genotypes. Three basic questions these models can address are:
Under what conditions does a population exhibit growth? On the event of growth,
at what rate does the population increase? What is the long-term behavior of
the distribution of genotypes? To address these questions, we associate a mean
limit ordinary differential equation (ODE) with the urn model. Previously, it
has been shown that on the event of population growth, the limiting
distribution of genotypes is a connected internally chain recurrent set for the
mean limit ODE. To determine when growth and convergence occurs with positive
probability, we prove two results. First, if the mean limit ODE has an
``attainable'' attractor at which growth is expected, then growth and
convergence toward this attractor occurs with positive probability. Second, the
population distribution almost surely does not converge to sets where growth is
not expecte
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