8,792 research outputs found
Stochastic and deterministic models for age-structured populations with genetically variable traits
Understanding how stochastic and non-linear deterministic processes interact
is a major challenge in population dynamics theory. After a short review, we
introduce a stochastic individual-centered particle model to describe the
evolution in continuous time of a population with (continuous) age and trait
structures. The individuals reproduce asexually, age, interact and die. The
'trait' is an individual heritable property (d-dimensional vector) that may
influence birth and death rates and interactions between individuals, and vary
by mutation. In a large population limit, the random process converges to the
solution of a Gurtin-McCamy type PDE. We show that the random model has a long
time behavior that differs from its deterministic limit. However, the results
on the limiting PDE and large deviation techniques \textit{\`a la}
Freidlin-Wentzell provide estimates of the extinction time and a better
understanding of the long time behavior of the stochastic process. This has
applications to the theory of adaptive dynamics used in evolutionary biology.
We present simulations for two biological problems involving life-history trait
evolution when body size is plastic and individual growth is taken into
account.Comment: This work is a proceeding of the CANUM 2008 conferenc
Fisher Waves: an individual based stochastic model
The propagation of a beneficial mutation in a spatially extended population
is usually studied using the phenomenological stochastic Fisher-Kolmogorov
(SFKPP) equation. We derive here an individual based, stochastic model founded
on the spatial Moran process where fluctuations are treated exactly. At high
selection pressure, the results of this model are different from the classical
FKPP. At small selection pressure, the front behavior can be mapped into a
Brownian motion with drift, the properties of which can be derived from
microscopic parameters of the Moran model. Finally, we show that the diffusion
coefficient and the noise amplitude of SFKPP are not independent parameters but
are both determined by the dispersal kernel of individuals
Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence
We consider an individual-based spatially structured population for Darwinian
evolution in an asexual population. The individuals move randomly on a bounded
continuous space according to a reflected brownian motion. The dynamics
involves also a birth rate, a density-dependent logistic death rate and a
probability of mutation at each birth event. We study the convergence of the
microscopic process when the population size grows to and the
mutation probability decreases to . We prove a convergence towards a jump
process that jumps in the infinite dimensional space of the stable spatial
distributions. The proof requires specific studies of the microscopic model.
First, we examine the large deviation principle around the deterministic large
population limit of the microscopic process. Then, we find a lower bound on the
exit time of a neighborhood of a stationary spatial distribution. Finally, we
study the extinction time of the branching diffusion processes that approximate
small size populations
Models of Delay Differential Equations
This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin
A mass-structured individual-based model of the chemostat: convergence and simulation
We propose a model of chemostat where the bacterial population is
individually-based, each bacterium is explicitly represented and has a mass
evolving continuously over time. The substrate concentration is represented as
a conventional ordinary differential equation. These two components are coupled
with the bacterial consumption. Mechanisms acting on the bacteria are
explicitly described (growth, division and up-take). Bacteria interact via
consumption. We set the exact Monte Carlo simulation algorithm of this model
and its mathematical representation as a stochastic process. We prove the
convergence of this process to the solution of an integro-differential equation
when the population size tends to infinity. Finally, we propose several
numerical simulations
An Unstructured Mesh Convergent Reaction-Diffusion Master Equation for Reversible Reactions
The convergent reaction-diffusion master equation (CRDME) was recently
developed to provide a lattice particle-based stochastic reaction-diffusion
model that is a convergent approximation in the lattice spacing to an
underlying spatially-continuous particle dynamics model. The CRDME was designed
to be identical to the popular lattice reaction-diffusion master equation
(RDME) model for systems with only linear reactions, while overcoming the
RDME's loss of bimolecular reaction effects as the lattice spacing is taken to
zero. In our original work we developed the CRDME to handle bimolecular
association reactions on Cartesian grids. In this work we develop several
extensions to the CRDME to facilitate the modeling of cellular processes within
realistic biological domains. Foremost, we extend the CRDME to handle
reversible bimolecular reactions on unstructured grids. Here we develop a
generalized CRDME through discretization of the spatially continuous volume
reactivity model, extending the CRDME to encompass a larger variety of
particle-particle interactions. Finally, we conclude by examining several
numerical examples to demonstrate the convergence and accuracy of the CRDME in
approximating the volume reactivity model.Comment: 35 pages, 9 figures. Accepted, J. Comp. Phys. (2018
Adaptation and migration of a population between patches
A Hamilton-Jacobi formulation has been established previously for
phenotypically structured population models where the solution concentrates as
Dirac masses in the limit of small diffusion. Is it possible to extend this
approach to spatial models? Are the limiting solutions still in the form of
sums of Dirac masses? Does the presence of several habitats lead to polymorphic
situations? We study the stationary solutions of a structured population model,
while the population is structured by continuous phenotypical traits and
discrete positions in space. The growth term varies from one habitable zone to
another, for instance because of a change in the temperature. The individuals
can migrate from one zone to another with a constant rate. The mathematical
modeling of this problem, considering mutations between phenotypical traits and
competitive interaction of individuals within each zone via a single resource,
leads to a system of coupled parabolic integro-differential equations. We study
the asymptotic behavior of the stationary solutions to this model in the limit
of small mutations. The limit, which is a sum of Dirac masses, can be described
with the help of an effective Hamiltonian. The presence of migration can modify
the dominant traits and lead to polymorphic situations
Survival of small populations under demographic stochasticity
We estimate the mean time to extinction of small populations in an environment with constant carrying capacity but under stochastic demography. In particular, we investigate the interaction of stochastic variation in fecundity and sex ratio under several different schemes of density dependent population growth regimes. The methods used include Markov chain theory, Monte Carlo simulations, and numerical simulations based on Markov chain theory. We find a strongly enhanced extinction risk if stochasticity in sex ratio and fluctuating population size act simultaneously as compared to the case where each mechanism acts alone. The distribution of extinction times deviates slightly from a geometric one, in particular for short extinction times. We also find that whether maximization of intrinsic growth rate decreases the risk of extinction or not depends strongly on the population regulation mechanism. If the population growth regime reduces populations above the carrying capacity to a size below the carrying capacity for large r (overshooting) then the extinction risk increases if the growth rate deviates from an optimal r-value
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