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Despite the knowledge that social, economical, and ecological networks are often of a small-world nature with inter-nodal distance growing even slower than logarithmically with system size, we often assume theoretical systems to be outside…
This article is a presentation of specific recent results describing scaling limits of individual-based models. Thanks to them, we wish to relate the time-scales typical of demographic dynamics and natural selection to the parameters of the…
The first chapter concerns monotype population models. We first study general birth and death processes and we give non-explosion and extinction criteria, moment computations and a pathwise representation. We then show how different scales…
Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how 'bottom up', individual-based models can be…
The relationship between the M-species stochastic Lotka-Volterra competition (SLVC) model and the M-allele Moran model of population genetics is explored via timescale separation arguments. When selection for species is weak and the…
The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include…
We consider a couple of models for the dynamics of the populations of two interacting species, inspired by Lotka-Volterra's classical equations. The novelty of this work is that the interaction terms are non local and the interaction occurs…
We study the evolution of a system of $N$ interacting species which mimics the dynamics of a cyclic food chain. On a one-dimensional lattice with N<5 species, spatial inhomogeneities develop spontaneously in initially homogeneous systems.…
May and Leonard (SIAM J. Appl. Math 1975) introduced a three-species Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete…
An analytic theory of species abundance patterns (SAPs) in biological networks is presented. The theory is based on multispecies replicator dynamics equivalent to the Lotka-Volterra equation, with diverse interspecies interactions. Various…
We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large…
Species-rich communities, such as the microbiota or microbial ecosystems, provide key functions for human health and climatic resilience. Increasing effort is being dedicated to design experimental protocols for selecting community-level…
We aim to clarify the relationship between interacting three-species models and the two-species Lotka-Volterra (LV) model. We utilize mean-field theory and Monte Carlo simulations on two-dimensional square lattices to explore the temporal…
Ecosystems with a large number of species are often modelled as Lotka-Volterra dynamical systems built around a large random interaction matrix. Under some known conditions, a global equilibrium exists and is unique. In this article, we…
We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but…
The study of interactions between multiple species in an ecosystem is an active and impactful direction of inquiry. This is true in particular for fragile systems in which even small perturbations of their functional parameters can produce…
We analyze the role of the Allee effect, a positive correlation between population density and mean individual fitness, for ecological communities formed by a large number of species. Our study is performed using the generalized…
From bird flocks to fish schools, animal groups often seem to react to environmental perturbations as if of one mind. Most studies in collective animal behaviour have aimed to understand how a globally ordered state may emerge from simple…
Internal feedbacks are commonly present in biological populations and can play a crucial role in the emergence of collective behavior. We consider a generalization of Fisher-KPP equation to describe the temporal evolution of the…
We investigate the outcome of generalised Lotka-Volterra dynamics of ecological communities with random interaction coefficients and non-linear feedback. We show in simulations that the saturation of non-linear feedback stabilises the…