Related papers: Slow-fast dynamics in a planar parasite--host mode…
While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological…
Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a…
The evolution of dispersal is a classical question in evolutionary ecology, which has been widely studied with several mathematical models. The main question is to define the fittest dispersal rate for a population in a bounded domain, and,…
Motivated by a real problem in steel production, we introduce and analyze a general class of singularly perturbed linear hybrid systems with both switches and impulses, in which the slow or fast nature of the variables can be…
Hydrodynamic systems arising in swarming modelling include nonlocal forces in the form of attractive-repulsive potentials as well as pressure terms modelling strong local repulsion. We focus on the case where there is a balance between…
The main objective of this paper is a study of the asymptotic behavior of distributional solutions to the one-dimensional repulsive pressureless Euler-Poisson system. The system is a model for the dynamics of a mass distribution evolving on…
The symbiotic branching model in $\mathbb{R}$ describes the behavior of two branching populations migrating in space $\mathbb{R}$ in terms of a corresponding system of stochastic partial differential equations. The system is parametrized…
We performed a comprehensive study of the spike autosolitons: self-sustained solitary inhomogeneous states, in the classical reaction-diffusion system --- the Gray-Scott model. We developed singular perturbation techniques based on the…
Species populations often modify their environment as they grow. When environmental feedback operates more slowly than population growth, the system can undergo boom-bust dynamics, where the population overshoots its carrying capacity and…
We present a novel model that describes the within-host evolutionary dynamics of parasites undergoing antigenic variation. The approach uses a multi-type branching process with two types of entities defined according to their relationship…
This study is motivated by the question of how singularity formation and other forms of extreme behavior in nonlinear dissipative partial differential equations are affected by stochastic excitations. To address this question we consider…
Pancreatic $\beta-$cells regulate insulin secretion through complex oscillations, which are vital for glucose control and diabetes research. In this paper, an existing mathematical model of $\beta-$cell dynamics is analyzed using a…
Stochastic chemical reaction or population dynamics in finite systems often terminates in an absorbing state. Yet in large spatially extended systems, the time to reach species extinction (or fixation) becomes exceedingly long. Tuning…
In this paper we study the dynamics of a fast-slow Fokker-Planck partial differential equation (PDE) viewed as the evolution equation for the density of a multiscale planar stochastic differential equation (SDE). Our key focus is on the…
This paper deals with a new epidemiological model of SIRS with stochastic perturbations. The primary objective is to establish the existence of a unique non-negative nonlocal solution. Using the basic reproduction number $\mathscr{R}_0$…
We study the large space and time scale behavior of a totally asymmetric, nearest-neighbor exclusion process in one dimension with random jump rates attached to the particles. When slow particles are sufficiently rare the system has a phase…
We study the exchange of stability in scalar reaction-diffusion equations which feature a slow passage through transcritical and pitchfork type singularities in the reaction term, using a novel adaptation of the geometric blow-up method.…
In sustained growth with random dynamics stationary distributions can exist without detailed balance. This suggests thermodynamical behavior in fast growing complex systems. In order to model such phenomena we apply both a discrete and a…
These lectures present the analysis of stability and control of long time behavior of PDE models described by nonlinear evolutions of hyperbolic type. Specific examples of the models under consideration include: (i) nonlinear systems of…
Human mobility is a key factor in spatial disease dynamics and related phenomena. In computational models host mobility is typically modelled by diffusion in space or on metapolulation networks. Alternatively, an effective force of…