Topic of Research Seminars
1. Seminar (i): Resolution of a century old problem on lattice random walks and its applications on to random search in a multi-target environment: from foraging processes to infection transmission
2. Seminar (ii): Stigmergy and the emergence of territoriality in scent-marking animals
Abstracts
1. Seminar (i): A random or stochastic search is a general mathematical representation of the so-called reactive interactions, that is those processes whereby an agent, being a particle, an animal or a human, moving with some degree of randomness, at specified locations may collect a resource or may get trapped or cease to exist. Examples include the absorption of an exciton in a semiconductor, the collection by a forager of food items, or the passing of a pathogen between an infected and a susceptible individual. When multiple target locations or reactive centres exist, it has been challenging to obtain theoretical predictions of when and where reactive interaction events occur, even in simple scenarios when the movement dynamics are Markovian. The challenge can be ascribed to the lack of a generalisation to multiple targets of the so-called renewal formalism for the first-passage probability, that is the probability to reach a specific location for the first time.
With the help of a resolution of a hundred year old problem on lattice random walks, such formalism has now been developed and has allowed the construction of a general theory to quantify the spatio-temporal dynamics of reactive interactions in a multi-target environment. Spatial discretisation is key to develop such a theory bypassing the need to solve unwieldy boundary value problems with spatially continuous variables, giving predictions that are either fully analytical or obtained through the simple inversion of a generating function. The formalism is valid independently of the topology and I will show results for hypercubic, hexagonal and triangular lattices.
The theory has also been extended to spatially heterogeneous environments that is in the presence of inert interactions (probability conserving) as opposed to reactive interactions (probability non-conserving). In this context, the discrete formulation has allowed to derive, in the space-time continuous limit, a new fundamental equation to represent diffusion across thin permeable barriers, supplementing the diffusion and Smoluchowski equation.
If time allows, I will also show a generalisation of the renewal equation to a single or multiple targets when the movement statistics is correlated, that is for the one-step non-Markov lattice random walk.
2. Seminar (ii): Various animals, mammals in particular, display some form of territorial behaviour for which they make their presence conspicuous to others claiming exclusive ownership of regions of space. The signals employed to perform this form of spatial exclusion may be visual, auditory or olfactory depending on the species and the environment. When the mechanism of territorial exclusion occurs via marks deposited on the terrain (olfactory cues), one talks about stigmergy, a form of environment-mediated interaction often encountered in social insect societies.
To study theoretically the emergence of spatial segregation in stigmergic systems I have introduced a new type of collective animal movement model where alignment of the agents does not play any role. It is called the territorial random walk model as agents move freely as random walkers on a lattice, scent-marking the terrain wherever they go. As deposited marks remain active for a finite amount of time, each walker retreats upon encountering an active foreign scent. The emerging spatio-temporal dynamics of the system can be quite rich and can be studied at the meso-scale (the territories) as well as at the micro-scale (the agents).
At the meso-scale short-lived marks produce rapidly morphing and highly mobile territories, while long-lived marks yield slow territories with a narrowly defined shape distribution. More importantly the full dependence in territory mobility as a function of the time for which individual marks remain active appears accompanied by a liquid-hexatic-solid transition akin to the Kosterlitz-Thouless melting scenario.
The dynamics at the micro-scale is in general non-Markovian, but when population density is sufficiently large some mean-field analytic approaches have proved useful. By considering localized walls to mimic the sharp (retreat) interaction when an animal encounters a foreign scent, it is possible to represent via a Fokker-Planck formalism an animal roaming within neighbouring territorial boundaries. Application of this analytic model to movement data from a red fox population in Bristol, UK, is also shown.
Subject Field of Topic: Stochastic Processes
Name of Speaker: Luca Giuggioli
Professorial Rank of Speaker: Professor
Email of Speaker: [email protected]
Date and Venue: (Thursday) 28 November 2024, 11:00 am, & 03:30 pm at SMME Seminar Hall, School of Mechanical and Manufacturing Engineering (SMME), NUST Islamabad