Chemotaxis-driven aggregation in Keller-Segel-type models rests on a positive feedback between cell density and production of a chemoattractant that the cells themselves sense. Natural populations are phenotypically heterogeneous: individual cells differ in chemotactic sensitivity and in attractant secretion rate. The problem, posed by Lorenzi and Painter in their study of a phenotype-structured chemotaxis model with attractant consumption, asks how such trait-structured heterogeneity modifies the classical chemotactic instability: which phenotypic (trait) distributions of chemotactic sensitivity and attractant production promote or inhibit the onset of aggregation, and under what conditions solutions of the phenotype-structured system exist globally in time rather than blow up in finite time. The binary two-phenotype version is now rigorously settled (global bounded solutions, no finite-time blow-up), and formal linear-stability thresholds are known in asymptotic switching regimes for the continuum-trait model, but a rigorous global-existence/blow-up theory and a complete instability characterization at generic switching rates for the continuously structured model remain open.

Background

Chemotaxis is the directed movement of cells or organisms along the gradient of a chemical signal; when the signal is attractive, movement is up the gradient. Keller and Segel (1971) formalized the minimal autoaggregation mechanism: a population of density nn secretes a chemoattractant of concentration ss, moves randomly (diffusion, rate DD) and chemotactically (sensitivity χ\chi) up gradients of ss, and the attractant diffuses and decays. The dimensionless minimal parabolic-parabolic system (both equations of diffusion type, one for the cells and one for the signal) reads ∂tn=∇⋅(D∇n−χn∇s)\partial_t n=\nabla\cdot(D\nabla n-\chi n\nabla s), ∂ts=Δs+n−s\partial_t s=\Delta s+n-s on a bounded domain with zero-flux boundary conditions. Linear Turing-type (i.e. diffusion-driven) stability analysis of the spatially uniform steady state predicts aggregation when the autoattraction potential χα\chi\alpha times the mean density overcomes diffusion and attractant decay; rigorously, in one space dimension solutions exist globally, while in dimensions d≥2d\geq 2 finite-time blow-up (unbounded density concentration) can occur, making the global-existence versus blow-up dichotomy a central question of the field.

This classical theory assumes a phenotypically homogeneous population: all cells have the same chemotactic sensitivity and the same attractant secretion rate. Experiments (e.g. bacterial populations that spatially sort by chemotactic ability) and trade-off arguments (chemotaxis and secretion both cost energy) show that real populations span a spectrum of phenotypes. Lorenzi and Painter (2022) studied a population continuously structured by a phenotype variable yy governing chemotactic responsiveness, proliferation, and modulation of the chemical environment, in which the attractant is sensed and degraded (consumed) by cells; they analysed invasion travelling waves (moving fronts separating invaded from uninvaded regions) under a chemotaxis/proliferation trade-off. In their discussion they posed the extension in which cells also produce the attractant: under phenotypic variation in chemotactic sensitivity and attractant production rates, do certain phenotypic distributions limit or promote pattern formation, and when does global existence or finite-time blow-up occur?

Subsequent work has developed and partially analysed such models. Macfarlane, Lorenzi and Painter (2022) introduced a binary-phenotype model in which cells switch between a chemotaxing state (performing chemotaxis only) and a secreting state (producing the attractant only), derived linear-stability patterning conditions depending on the switching rates, observed numerically that switching appears to curb blow-up, and wrote down the discrete multi-phenotype and continuous-trait generalizations as open targets. The continuous-trait (phenotype-structured) autoattraction model couples a nonlocal equation for the phenotype-space density n(t,x,y)n(t,x,y) — diffusing in space, chemotaxing with trait-dependent sensitivity χ(y)\chi(y), and switching between trait states through a kernel — to an attractant equation sourced by the trait integral of α(y)n\alpha(y)n, where α(y)\alpha(y) is the trait-dependent secretion rate. Formally, for negligible switching the instability threshold involves the trait average of the product, ⟨αχ⟩\langle\alpha\chi\rangle, while for infinitely fast switching it involves the product of trait averages, ⟨α⟩⟨χ⟩\langle\alpha\rangle\langle\chi\rangle (Lorenzi and Painter, 2025, who model switching by trait diffusion β ∂yy2n\beta\,\partial^2_{yy}n; for a general switching kernel the fast-switching trait equilibrium is instead the kernel's stationary distribution), so positively and negatively correlated trait distributions affect aggregation onset differently; the rigorous well-posedness theory for this nonlocal system is, however, still missing. For the binary two-phenotype version, Painter and Winkler (2023) proved that in dimensions d≤3d\leq 3 global bounded classical solutions exist for all positive switching rates, so no finite-time blow-up occurs there, although arbitrarily large densities emerge when both switching rates grow simultaneously.

Problem Statement

Consider the phenotype-structured Keller-Segel autoattraction system on a bounded smooth domain Ω⊂Rd\Omega\subset\mathbb{R}^d with zero-flux (no-net-flux) boundary conditions, in which the cell density n(t,x,y)n(t,x,y) is structured by a phenotype (trait) variable yy in a phenotype space Y\mathcal{Y} (a compact interval, e.g. [0,1][0,1], with Lebesgue measure):

∂tn=∇x⋅(Dn∇xn−χ(y) n ∇xs)+∫Yγ(θ,y,ρ,s) n(t,x,θ) dθ−n∫Yγ(y,θ,ρ,s) dθ,\partial_t n=\nabla_x\cdot\Big(D_n\nabla_x n-\chi(y)\,n\,\nabla_x s\Big)+\int_{\mathcal{Y}}\gamma(\theta,y,\rho,s)\,n(t,x,\theta)\,d\theta-n\int_{\mathcal{Y}}\gamma(y,\theta,\rho,s)\,d\theta,

ρ(t,x):=∫Yn(t,x,y) dy,∂ts=DsΔs+∫Yα(y) n(t,x,y) dy−ηs,\rho(t,x):=\int_{\mathcal{Y}}n(t,x,y)\,dy,\qquad \partial_t s=D_s\Delta s+\int_{\mathcal{Y}}\alpha(y)\,n(t,x,y)\,dy-\eta s,

where s(t,x)s(t,x) is the chemoattractant concentration, ρ(t,x)\rho(t,x) the total (phenotype-integrated) population density, χ(y)>0\chi(y)>0 the trait-dependent chemotactic sensitivity, α(y)>0\alpha(y)>0 the trait-dependent attractant production (secretion) rate, γ(θ,y,ρ,s)≥0\gamma(\theta,y,\rho,s)\geq 0 the phenotypic switching kernel (evaluated at the local values ρ(t,x)\rho(t,x) and s(t,x)s(t,x)), and Dn,Ds,η>0D_n,D_s,\eta>0 the cell diffusivity, attractant diffusivity, and attractant decay rate. The switching terms are in gain-loss form and conserve the total population mass. The spatially uniform steady state whose instability is characterized in item 1 below is an xx-independent solution of the system: a trait profile n⋆(y)n^{\star}(y) stationary for the switching kernel at the corresponding (ρ⋆,s⋆)(\rho^{\star},s^{\star}), with s⋆=η−1∫Yα(y) n⋆(y) dys^{\star}=\eta^{-1}\int_{\mathcal{Y}}\alpha(y)\,n^{\star}(y)\,dy; the formal thresholds recalled in Background use the phenotype-homogeneous convention n⋆=ρm/∣Y∣n^{\star}=\rho_m/|\mathcal{Y}|, s⋆=⟨α⟩ρm/ηs^{\star}=\langle\alpha\rangle\rho_m/\eta, where ρm\rho_m is the conserved uniform total density and ⟨⋅⟩\langle\cdot\rangle denotes the trait average over Y\mathcal{Y}.

The problem is to determine how phenotypic heterogeneity modifies the classical chemotactic instability, specifically:

  1. Pattern onset: characterize, in terms of the trait distributions χ(⋅)\chi(\cdot), α(⋅)\alpha(\cdot) and the switching structure (i.e. the switching rate β>0\beta>0 for kernels of the scaled form γ=β κ(θ,y,ρ,s)\gamma=\beta\,\kappa(\theta,y,\rho,s) with fixed shape κ\kappa, and the general kernel γ\gamma itself otherwise), when the spatially uniform steady state is unstable to inhomogeneous perturbations and aggregation ensues — in particular, which trait distributions promote and which inhibit the onset relative to a phenotypically homogeneous population with the same trait averages, for generic switching rates (not only the negligible-switching and fast-switching asymptotic regimes), including the dependence of the instability threshold on β\beta in the scaled case and the criterion in terms of γ\gamma for a general kernel.

  2. Well-posedness dichotomy: determine under what conditions on the space dimension, the total population mass, the trait distributions, and the switching structure the classical solution exists globally in time and remains bounded, versus blows up in finite time (or exhibits unbounded growth of densities), for the phenotype-structured system above — thereby establishing the analogue, for phenotype-structured populations, of the classical Keller-Segel global-existence/blow-up dichotomy.

References

  1. Tommaso Lorenzi, Kevin J. Painter, 'Trade-offs between chemotaxis and proliferation shape the phenotypic structuring of invading waves', International Journal of Non-Linear Mechanics 139 (2022), article 103885, DOI 10.1016/j.ijnonlinmec.2021.103885, arXiv:2109.06633, https://arxiv.org/abs/2109.06633
  2. Fiona R. Macfarlane, Tommaso Lorenzi, Kevin J. Painter, 'The impact of phenotypic heterogeneity on chemotactic self-organisation', Bulletin of Mathematical Biology 84 (2022), no. 12, article 143, DOI 10.1007/s11538-022-01099-z, arXiv:2206.14448, https://arxiv.org/abs/2206.14448
  3. Kevin J. Painter, Michael Winkler, 'Phenotype switching in chemotaxis aggregation models controls the spontaneous emergence of large densities', SIAM Journal on Applied Mathematics 83 (2023), no. 5, pp. 2096-2117, DOI 10.1137/22m1539393, arXiv:2210.04805, https://arxiv.org/abs/2210.04805
  4. Tommaso Lorenzi, Kevin J. Painter, 'Pattern formation within phenotype-structured chemotactic populations', Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 481 (2025), article 20250483, DOI 10.1098/rspa.2025.0483, arXiv:2506.03389, https://arxiv.org/abs/2506.03389
  5. Philippe Laurençot, Christian Stinner, 'Singular limit of a chemotaxis model with indirect signal production and phenotype switching', Nonlinearity 37 (2024), no. 10, article 105007, DOI 10.1088/1361-6544/ad6bdf, arXiv:2403.13402, https://arxiv.org/abs/2403.13402
  6. Viktoria Freingruber, Tommaso Lorenzi, Kevin J. Painter, Mariya Ptashnyk, 'Trait-structured chemotaxis: exploring ligand-receptor dynamics and travelling wave properties in a Keller-Segel model', Nonlinearity 38 (2025), no. 10, article 105006, DOI 10.1088/1361-6544/ae03ff, arXiv:2502.18947, https://arxiv.org/abs/2502.18947
  7. Evelyn F. Keller, Lee A. Segel, 'Model for chemotaxis', Journal of Theoretical Biology 30 (1971), no. 2, pp. 225-234, DOI 10.1016/0022-5193(71)90050-6, https://doi.org/10.1016/0022-5193(71)90050-6