"Two phase Stefan problems as the singular limit ofcompetition-diffusion systems arising in population dynamics"
Danielle Hilhorst (Université Paris-Saclay, France)
Competition-diffusion systems are coupled systems of nonlinear parabolic equations, where theunknown functions represent the densities of interacting biological populations. We will first study
the singular limit of of a two-component competition-diffusion system in population dynamics when
the interspecific competition rate tends to infinity [7], [8]. Using energy estimates, we will prove thatthe solution converges to the weak solution of a problem with a free boundary, which Mayan Mimuraused to call a Stefan problem with zero latent heat [1], [2], [3]. In biological terms, this amounts toproving that the habitats of two interacting populations become completely disjoint in the fast
reaction limit. We will then consider a three component competition-diffusion system and prove thatits solution converges to a Stefan problem with positive latent heat [4], [6].
Another question involves the limit of the Stefan problem as the latent heat coefficient tends to zero;
we will show that it converges to the Stefan problem with zero latent heat [5]. A question which wehave started to study is then the following : can we prove a similar result in the case that the partial
differential equations in the Stefan problems are perturbed by a white noise in time [9]References[1] Dancer, E. N.; Hilhorst, D.; Mimura, M.; Peletier, L. A. Spatial segregation limit of a competitiondiffusion system. European J. Appl. Math. 10 (1999), no. 2, 97-115.
[2] Crooks, E. C. M.; Dancer, E. N.; Hilhorst, D.; Mimura, M.; Ninomiya, H. Spatial segregation
limit of a competition-diffusion system with Dirichlet boundary conditions. Nonlinear Anal. RealWorld Appl. 5 (2004), no. 4, 645-665.
[3] Hilhorst, Danielle; Martin, Sébastien; Mimura, Masayasu Singular limit of a competitiondiffusion system with large interspecific interaction. J. Math. Anal. Appl. 390 (2012), no. 2, 488513.
[4] Hilhorst, Danielle; Iida, Masato; Mimura, Masayasu; Ninomiya, Hirokazu A competitiondiffusion system approximation to the classical two-phase Stefan problem. Recent topics inmathematics moving toward science and engineering. Japan J. Indust. Appl. Math. 18 (2001), no.
2, 161-180.
[5] Hilhorst, Danielle; Mimura, Masayasu; Schätzle, Reiner, Vanishing latent heat limit in a Stefanlike problem arising in biology, Nonlinear Anal. Real World Appl. 4 (2003), no. 2, 261–285.
[6] Hilhorst, Danielle; Iida, Masato; Mimura, Masayasu; Ninomiya, Hirokazu Relative compactness
in Lp of solutions of some 2m components competitiondiffusion systems. Discrete Contin. Dyn.
Syst. 21 (2008), no. 1, 233-244.
[7] Hilhorst, Danielle; Mimura, Masayasu; Ninomiya, Hirokazu Fast reaction limit of competitiondiffusion systems. Handbook of differential equations: evolutionary equations. Vol. V, 105-168,Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2009.
[8] Hilhorst, Danielle; Salin, Florian; Schneider, Victor; Gao, Yueyuan, Lecture notes on the singular
limit of reaction-diffusion systems dedicated to the Memory of Masayasu Mimura, Interdiscip.
Inform. Sci. 29 (2023), no. 1, 1–53.
[9] Ciotir, Ioana; El Kettani, Perla; Goreac, Dan; Hilhorst, Danielle, The vanishing latent heat limitof stochastic Stefan problems in population dynamics, in preparation.