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Populations interact non-linearly and are influenced by environmental fluctuations. In order to have realistic mathematical models, one needs to take into account that the environmental fluctuations are inherently stochastic. Often,…
The observation that phenotypic variability is ubiquitous in isogenic populations has led to a multitude of experimental and theoretical studies seeking to probe the causes and consequences of this variability. Whether it be in the context…
We consider a model for a population in a heterogeneous environment, with logistic type local population dynamics, under the assumption that individuals can switch between two different nonzero rates of diffusion. Such switching behavior…
Recently, a first step was made by the authors towards a systematic investigation of the effect of reaction-step-size noise - uncertainty in the step size of the reaction - on the dynamics of stochastic populations. This was done by…
The maintenance of diversity, the `commonness of rarity', and compositional turnover are ubiquitous features of species-rich communities. Through a minimal model, we consider how these features reflect the interplay between environmental…
In its simplest form, the competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. However, it has been observed empirically that in some settings it is possible to have…
Species coexistence is a complex, multifaceted problem. At an equilibrium, coexistence requires two conditions: stability under small perturbations; and feasibility, meaning all species abundances are positive. Which of these two conditions…
Rock-paper-scissors games metaphorically model cyclic dominance in ecology and microbiology. In a static environment, these models are characterized by fixation probabilities obeying two different "laws" in large and small well-mixed…
Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and…
We consider the classical two-dimensional Rosenzweig-MacArthur prey-predator model with a degenerate noise, whereby only the prey variable is subject to small environmental fluctuations. This model has already been introduced in…
Resource are often not uniformly distributed within a population. Spatial variations of concentration of a resource, change the fitness of competing strategies locally. The notion of fitness varying with respect to both genotype and…
We consider a Lotka-Volterra food chain model with possibly intra-specific competition in a stochastic environment represented by stochastic differential equations. In the non-degenerate setting, this model has already been studied by A.…
Populations are made up of an integer number of individuals and are subject to stochastic birth-death processes whose rates may vary in time. Useful quantities, like the chance of ultimate fixation, satisfy an appropriate difference…
In this paper we introduce a class of stochastic population models based on "patch dynamics". The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean field theories,…
We study the stability of non-conservative deterministic cross diffusion models and prove that they are approximated by stochastic population models when the populations become locally large. In this model, the individuals of two species…
Dispersal is an important strategy that allows organisms to locate and exploit favorable habitats. The question arises: given competition in a spatially heterogeneous landscape, what is the optimal rate of dispersal? Continuous population…
In this kind of model, the main characteristic that determines population viability in the long term is the stochastic growth rate (SGR) denoted $\lambda_S$. When $\lambda_S$ is larger than one, the population grows exponentially with…
The role of the selection pressure and mutation amplitude on the behavior of a single-species population evolving on a two-dimensional lattice, in a periodically changing environment, is studied both analytically and numerically. The…
Many socio-economic and biological processes can be modeled as systems of interacting individuals. The behaviour of such systems can be often described within game-theoretic models. In these lecture notes, we introduce fundamental concepts…
Living species, ranging from bacteria to animals, exist in environmental conditions that exhibit spatial and temporal heterogeneity which requires them to adapt. Risk-spreading through spontaneous phenotypic variations is a known concept in…