613 research outputs found
Convergence of generalized urn models to non-equilibrium attractors
Generalized Polya urn models have been used to model the establishment
dynamics of a small founding population consisting of k different genotypes or
strategies. As population sizes get large, these population processes are
well-approximated by a mean limit ordinary differential equation whose state
space is the k simplex. We prove that if this mean limit ODE has an attractor
at which the temporal averages of the population growth rate is positive, then
there is a positive probability of the population not going extinct (i.e.
growing without bound) and its distribution converging to the attractor.
Conversely, when the temporal averages of the population growth rate is
negative along this attractor, the population distribution does not converge to
the attractor. For the stochastic analog of the replicator equations which can
exhibit non-equilibrium dynamics, we show that verifying the conditions for
convergence and non-convergence reduces to a simple algebraic problem. We also
apply these results to selection-mutation dynamics to illustrate convergence to
periodic solutions of these population genetics models with positive
probability.Comment: 29 pages, 2 figure
Pushed beyond the brink: Allee effects, environmental stochasticity, and extinction
A demographic Allee effect occurs when individual fitness, at low densities,
increases with population density. Coupled with environmental fluctuations in
demographic rates, Allee effects can have subtle effects on population
persistence and extinction. To understand the interplay between these
deterministic and stochastic forces, we analyze discrete-time single species
models allowing for general forms of density-dependent feedbacks and stochastic
fluctuations in demographic rates. Our analysis provide criteria for stochastic
persistence, asymptotic extinction, and conditional persistence. Stochastic
persistence requires that the geometric mean of fitness at low densities is
greater than one. When this geometric mean is less than one, asymptotic
extinction occurs with a high probability whenever the initial population
density is low. If in addition the population only experiences positive
density-dependent feedbacks, conditional persistence occurs provided the
geometric mean of fitness at high population densities is greater than one.
However, if the population experiences both positive and negative
density-dependent feedbacks, conditional persistence is only possible if
fluctuations in demographic rates are sufficiently small. Applying our results
to stochastic models of mate-limitation, we illustrate counter-intuitively that
the environmental fluctuations can increase the probability of persistence when
populations are initially at low densities, and decrease the likelihood of
persistence when populations are initially at high densities. Alternatively,
for stochastic models accounting for predator saturation and negative
density-dependence, environmental stochasticity can result in asymptotic
extinction at intermediate predation rates despite conditional persistence
occurring at higher predation rates.Comment: 19 pages, 3 figure
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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
Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments
We consider a population living in a patchy environment that varies
stochastically in space and time. The population is composed of two morphs
(that is, individuals of the same species with different genotypes). In terms
of survival and reproductive success, the associated phenotypes differ only in
their habitat selection strategies. We compute invasion rates corresponding to
the rates at which the abundance of an initially rare morph increases in the
presence of the other morph established at equilibrium. If both morphs have
positive invasion rates when rare, then there is an equilibrium distribution
such that the two morphs coexist; that is, there is a protected polymorphism
for habitat selection. Alternatively, if one morph has a negative invasion rate
when rare, then it is asymptotically displaced by the other morph under all
initial conditions where both morphs are present. We refine the
characterization of an evolutionary stable strategy for habitat selection from
[Schreiber, 2012] in a mathematically rigorous manner. We provide a necessary
and sufficient condition for the existence of an ESS that uses all patches and
determine when using a single patch is an ESS. We also provide an explicit
formula for the ESS when there are two habitat types. We show that adding
environmental stochasticity results in an ESS that, when compared to the ESS
for the corresponding model without stochasticity, spends less time in patches
with larger carrying capacities and possibly makes use of sink patches, thereby
practicing a spatial form of bet hedging.Comment: Revised in light of referees' comments, Published on-line Journal of
Mathematical Biology 2014
http://link.springer.com/article/10.1007/s00285-014-0824-
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