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Waves and concentration dynamics in biology

Project description

An evolution in mathematical equations sheds light on ecological and evolutionary dynamics

Mathematics forms the foundation for our description of practically limitless behaviours and phenomena in living and non-living systems. Equations tell us concretely what the relationship is of one or more inputs (variables) to one or more possible outputs – in other words the outcome if we change the input. They provide insight into everything from quantum mechanics and neurotransmitter release to fluctuations in the stock market, and partial differential equations have been particularly useful. However, new theories and experiments in ecological and evolutionary dynamics are currently lacking the mathematical foundations to describe them well; this leaves biologists without the models needed to test hypotheses regarding complex dynamical systems. The EU-funded WACONDY project is filling this gap with an expansion of Hamilton-Jacobi equations and related ones. Outcomes will provide tools scientists need to investigate, explain or discover patterns of change in nature.

Objective

Biology is a source of exciting mathematical challenges. Likewise, there is a strong demand from biologists for rationalizing and quantifying their fascinating observations and for testing hypotheses via theoretical models. PDE have proven powerful for these purposes. The main goal of the WACONDY project is to expand the theory of Hamilton-Jacobi (HJ) equations and related ones to encompass recent investigations in ecological and evolutionary dynamics. The asymptotic analysis of wave propagation in structured populations, along with that of equilibria in quantitative genetics models, have generated new problems that fall outside of the scope of the current theory. These novel HJ equations arise in regimes that are analogous to semi-classical analysis in physics. On the one hand, they are valuable for biology because the associated dynamics can be reduced to simpler rules than the original problem. On the other hand, they do not fit in the classical theory of viscosity solutions. Hence, innovative techniques are needed to achieve their deep understanding.

The envisioned outcomes of the project are: a comprehensive analysis of non-local HJ equations arising in kinetic reaction-transport equations and reaction-diffusion equations for dispersal evolution; the asymptotic analysis of quantitative genetics models balancing diversity among offspring and selection of the fittest individuals, in the regime of small variance. The case of sexual reproduction will be emphasized, as the associated limit problem unveils novel features, beyond the HJ formulation. The design of asymptotic-preserving numerical schemes for this new class of equations will complement the program.

Beyond tackling these fundamental aspects, the project aims to open new interdisciplinary research directions. We anticipate contributions to diverse topics such as collective waves of micro-organisms, propagation of genetically engineered organisms, and patterns of adaptation in changing environments.

Host institution

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Net EU contribution
€ 1 312 963,00
Address
RUE MICHEL ANGE 3
75794 Paris
France

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Region
Ile-de-France Ile-de-France Paris
Activity type
Research Organisations
Links
Total cost
€ 1 378 588,00

Beneficiaries (2)