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

Trait evolution in two-sex populations

We present an individual-based model of phenotypic trait evolution in two-sex populations, which includes semi-random mating of individuals of the opposite sex, natural death and intra-specific competition. By passing the number of individuals to infinity, we derive the macroscopic system of nonlinear differential equations describing the evolution of trait distributions in male and female subpopulations. We study solutions, give criteria for persistence or extinction, and state theorem on asymptotic stability, which we apply later to particular examples of trait inheritance

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is in particular crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction-diffusion equations with Neumann's boundary conditions. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction-diffusion equations, with Neumann's boundary conditions. Finally, simulations based on the microscopic individual-based model are given, illustrating the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions

Differential evolution with an evolution path: a DEEP evolutionary algorithm

Utilizing cumulative correlation information already existing in an evolutionary process, this paper proposes a predictive approach to the reproduction mechanism of new individuals for differential evolution (DE) algorithms. DE uses a distributed model (DM) to generate new individuals, which is relatively explorative, whilst evolution strategy (ES) uses a centralized model (CM) to generate offspring, which through adaptation retains a convergence momentum. This paper adopts a key feature in the CM of a covariance matrix adaptation ES, the cumulatively learned evolution path (EP), to formulate a new evolutionary algorithm (EA) framework, termed DEEP, standing for DE with an EP. Without mechanistically combining two CM and DM based algorithms together, the DEEP framework offers advantages of both a DM and a CM and hence substantially enhances performance. Under this architecture, a self-adaptation mechanism can be built inherently in a DEEP algorithm, easing the task of predetermining algorithm control parameters. Two DEEP variants are developed and illustrated in the paper. Experiments on the CEC'13 test suites and two practical problems demonstrate that the DEEP algorithms offer promising results, compared with the original DEs and other relevant state-of-the-art EAs

Model of phenotypic evolution in hermaphroditic populations

We consider an individual based model of phenotypic evolution in hermaphroditic populations which includes random and assortative mating of individuals. By increasing the number of individuals to infinity we obtain a nonlinear transport equation, which describes the evolution of distribution of phenotypic traits. Existence of an one-dimensional attractor is proved and the formula for the density of phenotypic traits in the limiting (asymptotic) population is derived in some particular case

Metabolic flexibility as a major predictor of spatial distribution in microbial communities

A better understand the ecology of microbes and their role in the global ecosystem could be achieved if traditional ecological theories can be applied to microbes. In ecology organisms are defined as specialists or generalists according to the breadth of their niche. Spatial distribution is often used as a proxy measure of niche breadth; generalists have broad niches and a wide spatial distribution and specialists a narrow niche and spatial distribution. Previous studies suggest that microbial distribution patterns are contrary to this idea; a microbial generalist genus (Desulfobulbus) has a limited spatial distribution while a specialist genus (Methanosaeta) has a cosmopolitan distribution. Therefore, we hypothesise that this counter-intuitive distribution within generalist and specialist microbial genera is a common microbial characteristic. Using molecular fingerprinting the distribution of four microbial genera, two generalists, Desulfobulbus and the methanogenic archaea Methanosarcina, and two specialists, Methanosaeta and the sulfate-reducing bacteria Desulfobacter were analysed in sediment samples from along a UK estuary. Detected genotypes of both generalist genera showed a distinct spatial distribution, significantly correlated with geographic distance between sites. Genotypes of both specialist genera showed no significant differential spatial distribution. These data support the hypothesis that the spatial distribution of specialist and generalist microbes does not match that seen with specialist and generalist large organisms. It may be that generalist microbes, while having a wider potential niche, are constrained, possibly by intrageneric competition, to exploit only a small part of that potential niche while specialists, with far fewer constraints to their niche, are more capable of filling their potential niche more effectively, perhaps by avoiding intrageneric competition. We suggest that these counter-intuitive distribution patterns may be a common feature of microbes in general and represent a distinct microbial principle in ecology, which is a real challenge if we are to develop a truly inclusive ecology