Motivated by pattern formations and cell movements, many evolution equations incorporating spatial convolution with suitable integral kernel have been proposed. Numerical simulations of these nonlocal evolution equations can reproduce various patterns depending on the shape and form of integral kernel.The solutions to nonlocal evolution equations are similar to the patterns obtained by reaction-diffusion system and Keller-Segel system models. In this talk, we classify nonlocal interactions into two types, and investigate their relationship with reaction-diffusion systems and Keller-Segel systems, respectively. In these partial differential equation systems, we introduce multiple auxiliary diffusive substances and consider the singular limit of the quasi-steady state to approximate nonlocal interactions. In particular, we introduce how the parameters of the partial differential equation system are determined by the given integral kernel. These analyses demonstrate that, under certain conditions, nonlocal interactions and partial differential equation systems can be treated within a unified framework.  
This talk is based on collaborations with Hiroshi Ishii of Hokkaido University and Hideki Murakawa of Ryukoku University. 

Extra-chromosomal DNA (ecDNA) is a genetic error found in more than 30% of tumour samples across various cancer types. It is a key driver of oncogene amplification promoting tumour progression and therapeutic resistance, and is correlated to the worse clinical outcomes. Different from chromosomal DNA where genetic materials are on average equally divided to daughter cells controlled by centromeres during mitosis, the segregation of ecDNA copies is random partition and leads to a fast accumulation of cell-to-cell heterogeneity in copy numbers.  I will present our analytical and computational modeling of ecDNA dynamics under random segregation, examining the impact of copy-number-dependent versus -independent fitness, as well as the maintenance and de-mixing of multiple ecDNA species or variants within single cells. By integrating experimental and clinical data, our results demonstrate that ecDNA is not merely a by-product but a driving force in tumor progression. Intra-tumor heterogeneity exists not only in copy number but also in genetic and phenotypic diversity. Furthermore, ecDNA fitness can be copy-number dependent, which has significant implications for treatment.

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We will discuss local regularity properties for solutions to non-local equations naturally arising in kinetic theory. We will focus on the Strong Harnack inequality for global solutions to a non-local kinetic equation in divergence form. We will explain the connection to the Boltzmann equation and we will mention a few consequences on the asymptotic behaviour of the solutions.