Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models
Authored by N Champagnat, R Ferriere, S Meleard
Date Published: 2006
DOI: 10.1016/j.tpb.2005.10.004
Sponsors:
European Union
French Ministries
Platforms:
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Model Documentation:
Other Narrative
Mathematical description
Model Code URLs:
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Abstract
A distinctive signature of living systems is Darwinian evolution, that
is, a propensity to generate its well as self-select individual
diversity. To capture this essential feature of life while describing
the dynamics Of Populations, mathematical models must be rooted in the
microscopic, stochastic description of discrete individuals
characterized by one or several adaptive traits and interacting with
each other. The simplest models assume asexual reproduction and haploid
genetics: an offspring usually inherits the trait values of her
progenitor, except when a Mutation causes the offspring to take it
mutation step to new trait values; selection follows from ecological
interactions among individuals. Here we present it rigorous construction
of the microscopic population process that captures the probabilistic
dynamics over continuous time of birth, mutation, and death, as
influenced by the trait values of each individual, and interactions
between individuals. A by-product of this formal construction is a
general algorithm for efficient numerical simulation of the
individual-level model. Once the microscopic process is in place, we
derive different macroscopic models of adaptive evolution. These models
differ in the renormalization they assume, i.e. in the limits taken, in
specific orders, on population size, initiation rate, mutation step, while rescaling time accordingly. The macroscopic models also differ in
their mathematical nature: deterministic, in the form of ordinary, integro-, or partial differential equations, or probabilistic, like
stochastic partial differential equations or superprocesses. These
models include extensions of Kimura's equation (and of its approximation
for small mutation effects) to frequency- and density-dependent
selection. A novel class of macroscopic models obtains when assuming
that individual birth and death occur on a short timescale compared with
the timescale of typical Population growth. On a timescale of very rare
mutations, we establish rigorously the models of ``trait substitution
sequences{''} and their approximation known as the ``canonical equation
of adaptive dynamics{''}. We extend these models to account for mutation
bias and random drift between multiple evolutionary attractors. The
renormalization approach used in this Study also opens promising avenues
to Study and predict patterns of life-history allometries, thereby
bridging individual physiology, genetic variation, and ecological
interactions in it common evolutionary framework. (c) 2006 Elsevier Inc.
All rights reserved.
Tags
ecology
Mutation
Rapid evolution
Adaptive dynamics
Intraspecific
competition
Weak-convergence
Moment equations
Locally regulated population
Density-dependent selection
Stabilizing selection