Cell Segregation by Differential Attraction and Repulsion

Definitions and mathematical statement

The supplied question is: For a two-species model with differential attraction, repulsion, and diffusion, can these mechanisms lead to cell sorting and segregation?

To explain one possible mathematical meaning of segregation, suppose that the two species are represented by nonnegative measurable densities $\rho_1,\rho_2$ on a measure space $(\Omega,\mu)$. Full segregation can then mean \[ \rho_1(x)\rho_2(x)=0\qquad\text{for }\mu\text{-almost every }x\in\Omega. \] This is an explanatory convention, not a definition specified by the supplied source. It neither defines cell sorting nor determines whether segregation is required at a stationary state, at a finite time, or in a long-time limit.

The mathematical task indicated by the record is to determine whether differential attraction, repulsion, and diffusion can produce sorting or segregation in the intended two-species model. A precise quantified assertion cannot be recovered from the supplied record: it specifies no evolution equations, interaction kernels, diffusion laws, parameter ranges, spatial domain or dimension, initial or boundary conditions, or solution concept. It also does not state whether the target is existence of segregated stationary states, their stability, convergence toward them, or another sorting property. These alternatives remain unresolved and must not be identified with one another.

Short English statement

The question asks whether differential attraction, repulsion, and diffusion can sort two cell species into separate regions. The supplied record does not specify the model or the intended mathematical criterion for sorting, so a precise theorem statement requires further source information.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30003941$, title Cell Segregation by Differential Attraction and Repulsion, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The record gives proposed year $2018$.
• Supplied report attribution: Reactive Flows in Deformable, Complex Media (2019), Oberwolfach Reports, pp. 2385--2473, DOI 10.4171/owr/2018/39. https://doi.org/10.4171/owr/2018/39. The collection describes these reports as records of mathematical workshops and their open-problem discussions. The supplied page range covers the report; no precise location of this question is given.
• Supplied literature reference: Sorting Phenomena in a Mathematical Model for Two Species. https://doi.org/10.48550/arXiv.1704.04179.
• Supplied literature reference: Zoology of a Nonlocal Cross-Diffusion Model for Two Species. https://doi.org/10.1137/17M1128782.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved and states that rigorous one-dimensional results prove fully segregated stationary states for broad attractive-repulsive kernels, while later work studies additional steady states and dynamics; it describes the general sorting question as model-dependent and open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The source explicitly calls for independent verification of time-sensitive status and cited claims.