University of Cambridge > > Isaac Newton Institute Seminar Series > INI-RIMS joint seminar: The multi-target problem on Cartesian, hexagonal and triangular lattices in homogeneous and heterogeneous environments

INI-RIMS joint seminar: The multi-target problem on Cartesian, hexagonal and triangular lattices in homogeneous and heterogeneous environments

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MMV - Mathematics of movement: an interdisciplinary approach to mutual challenges in animal ecology and cell biology

In many complex systems the emergence of spatio-temporal patterns depends on the interaction between pairs of individuals, agents or subunits comprising the whole system. Theoretical predictions of such patterns rely upon quantifying when and where interaction events might occur. Even in simple scenarios when the dynamics are Markovian, it has been challenging to obtain estimates of encounter statistics between individuals due to the lack of a mathematical formalism to represent the occurrence of multiple random processes at the same time, the so-called splitting probability. With the help of a resolution of a hundred year old  problem on lattice random walks, I will present such formalism and develop a general theory that allows to quantify the spatio-temporal dynamics of interactions when a token of information is transferred upon co-location or proximity. Spatial discretisation is key to develop such a theory bypassing the need to solve unwieldy boundary value problems, giving predictions that are either fully analytical or obtained through the simple inversion of a generating function. The formalism is also applied to hexagonal and triangular lattices for which an analytical exact description of the occupation probability has recently been derived. In heterogeneous environments, a modification of the original formalism has been developed to quantify the interactions of a lattice random walk with the so-called inert heterogeneities. Such heterogeneities may represent locations with long-range connections to distant sites, areas with different diffusivity, or when permeable or impenetrable barriers are present. In this latter context, in the space-time continuous limit, a new fundamental equation that go beyond the diffusion and the Smoluchowski equation in the presence of permeable barriers has been found. Some applications of the formalism such as animal thigmotaxis, i.e. the tendency of an animal to remain close to the boundaries of a confining domain, and the search of a promoter region on DNA by transcription factors, a prototypical two-particle coalescent process, will also be presented.

This talk is part of the Isaac Newton Institute Seminar Series series.

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