Related papers: Derivations of Animal Movement Models with Explici…
Thanks to recent technological advances, it is now possible to track with an unprecedented precision and for long periods of time the movement patterns of many living organisms in their habitat. The increasing amount of data available on…
Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction-diffusion equations. While continuum models based on linear diffusion…
Human trajectory forecasting is a critical challenge in fields such as robotics and autonomous driving. Due to the inherent uncertainty of human actions and intentions in real-world scenarios, various unexpected occurrences may arise. To…
Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great…
We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field,…
Unraveling patterns of animals' movements is important for understanding the fundamental basics of biogeography, tracking range shifts resulting from climate change, predicting and preventing biological invansions. Many researchers have…
Aggregation-diffusion equations are foundational tools for modelling biological aggregations. Their principal use is to link the collective movement mechanisms of organisms to their emergent space use patterns in a concrete mathematical…
The inclusion of cognitive processes, such as perception, learning and memory, are inevitable in mechanistic animal movement modelling. Cognition is the unique feature that distinguishes animal movement from mere particle movement in…
We present a random walk model that exhibits asymptotic subdiffusive, diffusive, and superdiffusive behavior in different parameter regimes. This appears to be the first instance of a single random walk model leading to all three forms of…
Ensuring safe interactions between autonomous vehicles (AVs) and human drivers in mixed traffic systems remains a major challenge, particularly in complex, high-risk scenarios. This paper presents a cognition-decision framework that…
Many diffusion processes in nature and society were found to be anomalous, in the sense of being fundamentally different from conventional Brownian motion. An important example is the migration of biological cells, which exhibits…
We present a new method to derive kinetic equations for systems undergoing non-linear transport in the presence of memory effects. In the framework of mesoscopic nonequilibrium thermodynamics, we derive a generalized Fokker-Planck equation…
We propose a new evolutionary dynamics for population games with a discrete strategy set, inspired by the theory of optimal transport and Mean field games. The dynamics can be described as a Fokker-Planck equation on a discrete strategy…
Despite their empirical success, how diffusion models generalize remains poorly understood from a mechanistic perspective. We demonstrate that diffusion models trained with flow-matching objectives exhibit grokking--delayed generalization…
Safe and effective motion planning is crucial for autonomous robots. Diffusion models excel at capturing complex agent interactions, a fundamental aspect of decision-making in dynamic environments. Recent studies have successfully applied…
A canonical foraging task is the patch-leaving problem, in which a forager must decide to leave a current resource in search for another. Theoretical work has derived optimal strategies for when to leave a patch, and experiments have tested…
A discrete-time model of reacting evolving fields, transported by a bidimensional chaotic fluid flow, is studied. Our approach is based on the use of a Lagrangian scheme where {\it fluid particles} are advected by a $2d$ symplectic map…
Uncovering the mechanisms behind long-term memory is one of the most fascinating open problems in neuroscience and artificial intelligence. Artificial associative memory networks have been used to formalize important aspects of biological…
Mathematical models of spatial population dynamics typically focus on the interplay between dispersal events and birth/death processes. However, for many animal communities, significant arrangement in space can occur on shorter timescales,…
Diffusion of colloidal particles in a complex environment such as polymer networks or biological cells is a topic of high complexity with significant biological and medical relevance. In such situations, the interaction between the…