Periodic Reporting for period 1 - WACONDY (Waves and concentration dynamics in biology) Reporting period: 2021-01-01 to 2022-06-30 Summary of the context and overall objectives of the project Mathematical modeling and analysis has shown the ability to decipher complex biological phenomena. We are working in that direction, following new experimental set up, and new scientific challenges.More precisely, we aim at building a theoretical framework to describe propagation of biological waves, and patterns of adaptation of populations, either in a stationary or in a changing environment. The objectives are twofold: (i) consolidate the mathematical background underlying ecology and evolutionary biology by addressing some challenging problems which require modern analytic tools; (ii) make new bridges between mathematics and biology by developing new interdisciplinary collaborations on insightful case studies. The latter include: (a) self-organization at the microscopic scale in bacterial colonies and emergence of waves of population density through the synchronization of individuals; (b) radial expansion of cells as a result of a lack of Oxygen due to consumption by the population; (c) propagation of gene drive in theoretical models of spatial ecology and genetics subject to super-mendelian inheritance. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far -/ We have proposed a minimal model for the radial expansion of amoebae (Dicty) cells following a self-generated gradient of oxygen, with severe hypoxia at the core of the colony. We have derived an explicit formula for the spreading speed, unraveling the nature of the expansion front (either pushed or pulled). -/ We have investigated the impact of population size reduction on the expansion of gene drive. So far, the variations on the population size were mostly ignored in population genetics models. We have been able to cover some interesting cases by the mathematical analysis thanks to a fruitful analogy with some SIR type model.-/ We have proved the uniqueness of the ground state in some quantitative genetics model involving the so-called infinitesimal Fisher model. For this purpose, we identified a seemingly new convexity structure for non-conservative problems. -/ We have described in full generality patterns of local adaptation in heterogeneous environments coupled by migration, in the regime of small variance. -/ We have described the stochastic dynamics of ancestral lineages arising from adaptation to a changing environment.-/ We have designed and analysed Asymptotic-Preserving schemes for some quantitative genetics models with a linear operator for the distribution of the effects of mutations, in the regime of vanishing variance. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) We shall continue to explore the new convexity structure unraveled by our analysis of the Fisher infinitesimal model. We will also tackle Asymptotic-Preserving schemes for non-linear operators. We shall put some effort on the microscopic description of crowding in populations of dense myxobacteria, and its role on the emergence of rippling patterns. Schematic description of the analysis of quantitative genetics models in the regime of small varianc
