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We study dynamics of a locally conserved energy in ergodic, local many-body quantum systems on a lattice with no additional symmetry. The resulting dynamics is well approximated by a coarse grained, classical linear functional diffusion…
Many systems involve numerous interacting parts and the whole system can have properties that the individual parts do not. I take this novelty as the defining characteristic of an emergent property. Other characteristics associated with…
In this paper, we study some qualitative properties for an evolution problem that combines local and nonlocal diffusion operators acting in two different subdomains and, coupled in such a way that, the resulting evolution problem is the…
The motivation of this work is to produce an integrated formulation for material response due to detonation wave loading. Here, we focus on elastoplastic structural response. In particular, we are interested to capture miscible and…
Nonlocal aggregation-diffusion models, when coupled with a spatial map, can capture cognitive and memory-based influences on animal movement and population-level patterns. In this work, we study a one-dimensional…
Vegetation in semi-arid ecosystems frequently organizes into spatially heterogeneous mosaics that regulate ecosystem functioning, productivity, and resilience. These patterns arise from local biological interactions, including facilitation…
We are concerned with the persistence of both predator and prey in a diffusive predator-prey system with a climate change effect, which is modeled by a spatial-temporal heterogeneity depending on a moving variable. Moreover, we consider…
Spatial patterns are widely observed in numerous nonequilibrium natural systems, often undergoing complex transitions and bifurcations, thereby exhibiting significant importance in many physical and biological systems such as embryonic…
We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially…
How diversity is maintained in natural ecosystems is a long-standing question in Theoretical Ecology. By studying a system that combines ecological dynamics, heterogeneous interactions and spatial structure, we uncover a new mechanism for…
Spatially explicit models have been widely used in today's mathematical ecology and epidemiology to study persistence and extinction of populations as well as their spatial patterns. Here we extend the earlier work--static dispersal between…
This paper investigates pattern formation in reaction--diffusion systems with both diffusive and nondiffusive components, providing necessary and sufficient conditions for diffusion-driven instability (DDI) and establishing the existence of…
Neurons are often connected, spatially and temporally, in phenomenal ways that promote wave propagation. Therefore, it is essential to analyze the emergent spatiotemporal patterns to understand the working mechanism of brain activity,…
A one dimensional stochastic exclusion process with two species of particles, $+$ and $-$, is studied where density of each species can fluctuate but the total particle density is conserved. From the exact stationary state weights we show…
We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of…
Dynamic fragmentation simulations are essential for predicting material response at high strain rates, yet explicit dynamic simulations that combine an extrinsic cohesive-zone model (CZM) with penalty-based contact often exhibit severe…
Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution.…
We introduce a new class of models for emergent dynamics. It is based on a new communication protocol which incorporates two main features: short-range kernels which restrict the communication to local geometric balls, and anisotropic…
In this paper, we study multi-species stochastic interacting particle systems and their mean-field McKean-Vlasov partial differential equations (PDEs) in non-convex landscapes. We discuss the well-posedness of the multi-species SDE system,…
This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which describe the macroscopic dynamics of a variety of large systems of interacting particles.…