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We consider a class of singularly perturbed 2-component reaction-diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. In many example…
We consider a continuum model for the dynamics of systems of self propelling particles with kinematic constraints on the velocities. The model aims to be analogous to a discrete algorithm used in works by T. Vicsek et al. In this paper we…
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We study a singularly perturbed fast-slow system of two partial differential equations (PDEs) of reaction-diffusion type on a bounded domain via Galerkin discretisation. We assume that the reaction kinetics in the fast variable realise a…
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For a Nicholson's blowflies system with patch structure and multiple discrete delays, we analyze several features of the global asymptotic behavior of its solutions. It is shown that if the spectral bound of the community matrix is…
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In $1993$, Holt and Lawton introduced a stochastic model of two host species parasitized by a common parasitoid species. We introduce and analyze a generalization of these stochastic difference equations with any number of host species,…
As a mathematical theory for the stochasstic, nonlinear dynamics of individuals within a population, Delbr\"{u}ck-Gillespie process (DGP) $n(t)\in\mathbb{Z}^N$, is a birth-death system with state-dependent rates which contain the system…
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We construct solutions to the two dimensional parabolic-elliptic Keller-Segel model for chemotaxis that blow up in finite time $T$. The solution is decomposed as the sum of a stationary state concentrated at scale $\lambda$ and of a…
In this paper we use formal asymptotic arguments to understand the stability proper- ties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. We also analyze both the harmonic map heatflow and Schrodinger map…
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We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the…
Ecosystems can undergo sudden shifts to undesirable states, but recent studies with simple single species ecosystems have demonstrated that advance warning can be provided by the slowing down of population dynamics near a tipping point.…
We examine coarsening of field-excitation patterns of the sine-Gordon (SG) model, in two and three spatial dimensions, identifying it as universal dynamics near non-thermal fixed points. The SG model is relevant in many different contexts,…
Under the effect of strong genetic drift, it is highly probable to observe gene fixation or gene loss in a population, shown by infinite peaks on a coherently constructed potential energy landscape. It is then important to ask what such…