Related papers: Wave Propagation Dynamics via Lattice Difference E…
Diffusion can be conceptualized, at microscopic scales, as the random hopping of particles between neighboring lattice sites. In the case of diffusion in inhomogeneous media, distinct spatial domains in the system may yield distinct…
Density estimates based on point processes are often restrained to regions with irregular boundaries or holes. We propose a density estimator, the lattice-based density estimator, which produces reasonable density estimates under these…
This paper introduces a novel hybrid model combining Partial Differential Equations (PDEs) and Ordinary Differential Equations (ODEs) to simulate infectious disease dynamics across geographic regions. By leveraging the spatial detail of…
Using a reaction-diffusion model with free boundaries in one space dimension for a single population species with density $u(t,x)$ and population range $[g(t), h(t)]$, we demonstrate that the Allee effects can be eliminated if the species…
L\'{e}vy robotic systems combine superdiffusive random movement with emergent collective behaviour from local communication and alignment in order to find rare targets or track objects. In this article we derive macroscopic fractional PDE…
We investigate spin-wave propagation in magnetic insulators in the presence of lattice dislocations. Within a continuum magnetoelastic framework, we show that the strain fields generated by dislocations induce equilibrium magnetic textures.…
A lattice-based model for continuum percolation is applied to the case of randomly located, partially aligned sticks with unequal lengths in 2D which are allowed to cross each other. Results are obtained for the critical number of sticks…
Models of invasive species spread often assume that landscapes are spatially homogeneous; thus simplifying analysis but potentially reducing accuracy. We extend a recently developed partial differential equation model for invasive conifer…
The motive of this work is to understand the complex spatial characteristics of finite-amplitude elastic wave propagation in periodic structures and leverage the unique opportunities offered by nonlinearity to activate complementary…
We consider Markov models of large-scale networks where nodes are characterized by their local behavior and by a mobility model over a two-dimensional lattice. By assuming random walk, we prove convergence to a system of partial…
Bacterial colonies can form a wide variety of shapes and structures based on ambient and internal conditions. To help understand the mechanisms that determine the structure of and the diversity within these colonies, various numerical…
We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random…
We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment, focusing on whether a population already established in one patch either successfully invades an adjacent empty…
We study the effect of disorder on the particle density evolution in a classical Hamiltonian driven lattice setup. If the disorder is localized within a finite sub-domain of the lattice, the emergence of strong tails in the density…
We investigate predator-prey school interactions in aquatic environments using a stochastic differential equation (SDE)-based, particle-level model that incorporates attraction, repulsion, alignment, and environmental noise. Two predation…
We investigate the dynamic properties of elastic lattices defined by tessellations of a curved hyperbolic space. The lattices are obtained by projecting nodes of a regular hyperbolic tessellation onto a flat disk and then connecting those…
Numerous processes across both the physical and biological sciences are driven by diffusion. Partial differential equations (PDEs) are a popular tool for modelling such phenomena deterministically, but it is often necessary to use…
We analyze the propagation properties of the numerical versions of one and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both…
We study the random walk of a particle in a compartmentalized environment, as realized in biological samples or solid state compounds. Each compartment is characterized by its length $L$ and the boundaries transmittance $T$. We identify two…
We consider the system of reaction-diffusion equations proposed in [8] as a population dynamics model. The first equation stands for the population density and models the ecological effects, namely dispersion and growth with a Allee effect…