Research

 
 

How to define efficient models to describe emergence in systems composed of many interacting agents? Mesoscopic and macroscopic models involving partial differential equations (PDEs) are a crucial tool for answering this question: they have been widely used in Physics, and more recently they are expanding rapidly in other contexts such as Ecology, Ethology and Cell Biology.

My research aims at developing and analyzing mathematical models, mostly at the PDE level, to describe emerging phenomena in such systems. The phenomena I work on include the self-aggregation of a self-interacting population (observed in colonies of bacteria, as well as in colonies of ants, …); on the opposite the segregation between several groups (observed in the territorial occupancy of competing bird species or mammal groups, as well as in the cell sorting arising in vertebrate embryonic development such as quail, …); the synchronization or harmonization of a large number of agents (observed in bird flocks and fish schools, but also related to the ram’s fertility at a cellular level, …); and equilibration or stabilization of the system in specific (but various) senses. To develop such models I frequently interact with Biologists, and particularly specialists of Developmental Biology and Cell Biology.

My main PDE tools are reaction-cross-diffusion systems and kinetic theory. Interestingly, these models often display singular behaviour, such as sharp transitions, discontinuities, and blow-ups. My research particularly aims at developing specific mathematical techniques to handle such singularities.

Cross-diffusion systems

  1. •M. Herda, A. Trescases, A. Zurek A finite volume scheme for the local sensing chemotaxis model, SMAI Journal of Computational Mathematics 11 (2025) pp.637–676

  2. •Ph. Laurençot, A. Trescases No pattern formation in a quasilinear chemotaxis model with local sensing, SIAM Journal on Mathematical Analysis 56 (2024) no. 5 pp.6861–6884

  3. •J. Guerand, A. Menegaki, A. Trescases Global smooth solutions for triangular reaction-cross diffusion systems, Bulletin des Sciences Mathématiques 189 (2023) pp.103342

  4. •L. Desvillettes, Ph. Laurençot, A. Trescases, M. Winkler Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing, Nonlinear Analysis: Theory, Methods and Applications 226 (2023) pp.113–153

  5. •M. Burger, Ph. Laurençot, A. Trescases Delayed blow-up for chemotaxis models with local sensing Journal of the London Mathematical Society 103 (2021) no. 2 pp.1596–1617

  6. •L. Desvillettes, Y.-J. Kim, A. Trescases, C. Yoon A logarithmic chemotaxis model featuring global existence and aggregation, Nonlinear Analysis: Real World Applications 50 (2019) pp.562-582

  7. •A. Trescases. On triangular reaction cross-diffusion systems with possible self-diffusion Bulletin des Sciences Mathématiques 140 (2016) no. 7 pp.796–829

  8. •L. Desvillettes, Th. Lepoutre, A. Moussa, A. Trescases On the entropic structure of reaction-cross diffusion systems Communications in Partial Differential Equations 40 (2015) 9 pp.1705-1747

  9. •L. Desvillettes, A. Trescases New results for triangular reaction cross diffusion system Journal of Mathematical Analysis and Applications 430 (2015) pp.32-59


Kinetic theory and collective dynamics

  1. •P. Degond, A. Frouvelle, S. Merino-Aceituno, A. Trescases Hyperbolicity and non-conservativity of a hydrodynamic model of swarming rigid bodies Quarterly of Applied Mathematics 82 (2024) 1 pp.35-64

  2. •P. Degond, A. Frouvelle, S. Merino-Aceituno, A. Trescases Alignment of self-propelled rigid bodies: from particle systems to macroscopic equations in Stochastic Dynamics out of Equilibrium, Springer Proceedings in Mathematics & Statistics 282 (2019)

  3. •P. Degond, A. Frouvelle, S. Merino-Aceituno, A. Trescases Quaternions in collective dynamics Multiscale Modeling and Simulation 16 (2018) 1 pp.28-77

  4. •Y. Guo , C. Kim, D. Tonon, A. Trescases Regularity of the Boltzmann equation in convex domains (116 pages) Inventiones Mathematicae 207 (2017) 1 pp. 115-290

  5. •Y. Guo , C. Kim, D. Tonon, A. Trescases BV-regularity of the Boltzmann equation in non-convex domains Archive for Rationale Mechanics and Analysis 220 (2015) 3 pp.1045-1093


Modeling and applications in Cell biology and Morphogenesis

  1. •M. Romanos, T. Salisbury, S. Stephan, R. Lansford, P. Degond, A. Trescases, B. Bénazéraf Differential proliferation regulates multi-tissue morphogenesis during embryonic axial extension: Integrating viscous modeling and experimental approaches Development (2024) 151 (13): dev202836

  2. •P. Degond, S. Hecht, A. Trescases, M. Romanos Multi-species viscous models for tissue growth: incompressible limit and qualitative behaviour Journal of Mathematical Biology 85:16 (2022)

  3. •M. Romanos, G. Allio, M. Roussigné, L. Combres, N. Escalas, C. Soula, F. Médevielle, B. Steventon, A. Trescases, B. Bénazéraf Cell-to-cell heterogeneity in Sox2 and Bra expression guides progenitor motility and destiny eLife (2021) 10:e66588

  4. •F. Hubert, I. Khames, M. Jedouaa, J. Olivier, O. Theodoly, A. Trescases Cell motility in confinement: a computational model for the shape of the cell ESAIM: procS 55 (2016) pp. 148-166 (project report for the Cemracs 2015)

Articles

Research interests

  1. -Nonlinear partial differential equations

  2. -Nonlinear diffusion, cross-diffusion, strongly coupled parabolic systems

  3. -Kinetic theory

  4. -Modeling and applications in Life sciences: embryology, population dynamics, ecology

  1. •HDR: A. Trescases Agrégation, ségrégation, conformisation dans les systèmes vivants (2025)

  2. •PhD: A. Trescases Modélisation et Analyse Mathématique d'Equations aux Dérivées Partielles Issues de la Physique et de la Biologie (2015)

Thesis and Habilitation (in French)