Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models Export

@article{Champagnat:2006, abstract = {A distinctive signature of living systems is Darwinian evolution, that is, a propensity to generate as 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 a mutation step to new trait values; selection follows from ecological interactions among individuals. Here we present a 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, mutation 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 a common evolutionary framework.}, author = {Champagnat, N. and Ferriere, R. and Meleard, S.}, citeulike-article-id = {5662130}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.tpb.2005.10.004}, citeulike-linkout-1 = {http://linkinghub.elsevier.com/retrieve/pii/S0040580905001632}, comment = {- Aim -> bridge the gap between ecological and evolutionary processes: * ecology -> birth and death processes, evolution -> mutation process; * tools -> formal model and computational simulations - Microscopic model: stochastic process describing a finite population of discrete, interacting individuals characterized by several adaptive phenotypic traits. Focus on haploid individuals with asexual reproduction. Describe dynamics, over continuous time, of birth, mutation and death, as influenced by the trait values of each individual and ecological interactions among individuals. - Macroscopic models: * derive deterministic equations to describe moments of trajectories (statistics of a large nb of independent realizations of the process) -> not successful * renormalize the individual-based process (dividing by the population size) -> ok - Timescales of individual processes, compare to each other, can lead to very different qualitative behaviours. Especially true in terms of diversification.}, doi = {10.1016/j.tpb.2005.10.004}, issn = {00405809}, journal = {Theoretical Population Biology}, keywords = {evolution, modeling}, month = {May}, number = {3}, pages = {297--321}, posted-at = {2009-11-10 22:05:55}, priority = {0}, title = {Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models}, url = {http://dx.doi.org/10.1016/j.tpb.2005.10.004}, volume = {69}, year = {2006} }

TY - JOUR ID - Champagnat:2006 L3 - citeulike-article-id:5662130 N2 - A distinctive signature of living systems is Darwinian evolution, that is, a propensity to generate as 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 a mutation step to new trait values; selection follows from ecological interactions among individuals. Here we present a 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, mutation 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 a common evolutionary framework. IS - 3 JF - Theoretical Population Biology SN - 00405809 EP - 321 TI - Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models VL - 69 SP - 297 KW - evolution KW - modeling N1 - - Aim -> bridge the gap between ecological and evolutionary processes: * ecology -> birth and death processes, evolution -> mutation process; * tools -> formal model and computational simulations - Microscopic model: stochastic process describing a finite population of discrete, interacting individuals characterized by several adaptive phenotypic traits. Focus on haploid individuals with asexual reproduction. Describe dynamics, over continuous time, of birth, mutation and death, as influenced by the trait values of each individual and ecological interactions among individuals. - Macroscopic models: * derive deterministic equations to describe moments of trajectories (statistics of a large nb of independent realizations of the process) -> not successful * renormalize the individual-based process (dividing by the population size) -> ok - Timescales of individual processes, compare to each other, can lead to very different qualitative behaviours. Especially true in terms of diversification. AU - Champagnat, N AU - Ferriere, R AU - Meleard, S PY - 2006/05// UR - http://dx.doi.org/10.1016/j.tpb.2005.10.004 ER -