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Abstract: In order to understand the origin of long jumps in the motion of bacteria, we introduce an individual-based model and its associated kinetic equation which relate the intra-cellular reaction with the run-and-tumble process. Due to the randomness of receptor methylation or intra-cellular chemical reactions, noise occurs in the signaling pathways and affects the tumbling rate. Then, comes the question to understand the role of an internal noise on the behavior of the full population. We use biologically meaningful pathways and tumbling kernels and under proper scaling and conditions on the tumbling frequency as well as the form of noise, Lévy-type motions are observed numerically at the population level. We also rigorously recover the fractional diffusion equation as the macroscopic limits of the kinetic model, which gives a new mathematical theory about how long jumps can be due to the internal noise of the bacteria.

Abstract: The spread of invasive species is a serious problem for ecosystem integrity in many regions of the world. Mathematical theory has made great contributions to understanding the mechanisms behind such invasions and to estimate spread rates in homogeneous landscapes. However, most landscapes are heterogeneous at many levels, and individuals adjust their movement behaviour to local environmental conditions. Including such movement details into tractable mathematical models is a challenge. In this talk, I will present some background on models for spatial spread and a relatively recent approach to model movement in ``patchy'' landscapes. I will show that these models allow for some explicit calculations of spread rates, and that including the individual level details is crucial for obtaining biologically reasonable results. I will also point out mechanisms by which management strategies aimed at reducing the spread rate of invasives could backfire and increase the spread rate.

Abstract: The TFDW (Thomas-Fermi-Dirac-Weizacker) model has been used to describe electron configurations of molecules. On the other hand, the Ohta-Kawasaki model arises in the context of diblock copolymer melts. In this talk, we discuss results concerning compactness of minimizing sequences of the TFDW model and a variant of the Ohta-Kawasaki model.