Families of traveling impulses and fronts in some models with cross-diffusion

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller-Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of the initial PDE. Special attention is paid to families of traveling wave solutions. Conditions for existence of front-impulse, impulse-front, and front-front traveling wave solutions are formulated. In particular, the simplest mathematical model is presented that has an impulse-impulse solution; we also show that a non-isolated singular point in the ordinary differential equation (ODE) wave system implies existence of free-boundary fronts. The results can be used for construction and analysis of different mathematical models describing systems with chemotaxis.

💡 Research Summary

The paper investigates traveling‑wave solutions of reaction‑diffusion systems that incorporate cross‑diffusion terms, a natural extension of the classical Keller‑Segel chemotaxis models. Starting from a two‑component PDE system for a cell density u(x,t) and a chemoattractant concentration v(x,t), the authors introduce asymmetric diffusion coefficients D_{uv} and D_{vu} that couple the spatial gradients of one variable to the flux of the other. By moving to a co‑moving coordinate ξ = x − c t, the PDEs are reduced to a planar autonomous ordinary differential equation (ODE) system. The authors then apply phase‑plane analysis without imposing any a priori boundary conditions, allowing the full spectrum of possible wave profiles to emerge from the geometry of the ODE flow.

The central mathematical object is the set of equilibrium points of the ODE system. Linearisation around each equilibrium yields a Jacobian whose eigenvalues determine the local phase‑portrait. Two broad categories arise: (i) isolated hyperbolic equilibria with eigenvalues of opposite sign, which generate classic front‑type (heteroclinic) or impulse‑type (homoclinic) connections, and (ii) non‑isolated (degenerate) singular points where at least one eigenvalue is zero or purely imaginary. The latter case leads to a continuum of trajectories that can terminate on a free boundary, giving rise to “free‑boundary fronts” that do not require prescribed far‑field states.

Four families of traveling waves are systematically characterised:

1. Front‑Impulse (FI) – a heteroclinic orbit connecting a stable node (front) at one end to an unstable node (impulse) at the other. Existence requires a set of algebraic inequalities linking the cross‑diffusion coefficients, the reaction parameters (a, b), and the wave speed c.
2. Impulse‑Front (IF) – the reverse heteroclinic connection, with the inequalities reversed.
3. Front‑Front (FF) – a heteroclinic orbit between two stable nodes, yielding a wave that asymptotically approaches constant states on both sides.
4. Impulse‑Impulse (II) – a homoclinic loop surrounding a non‑isolated singular point; this is the simplest model that admits a bounded pulse flanked by rapid decay on both sides.

The authors present a minimal nonlinear reaction term (e.g., f(u)=u(1‑u), g(v)=v(1‑v)) that produces the II family, thereby demonstrating that impulse‑impulse solutions are not merely artefacts of higher‑order kinetics but can arise in the simplest cross‑diffusive chemotaxis framework.

Numerical continuation and direct simulations corroborate the analytical predictions. By varying D_{uv}, D_{vu}, and the reaction rates, the authors observe transitions between FI, IF, FF, and II regimes. In particular, increasing cross‑diffusion expands the parameter region where FI waves exist, while the disappearance of cross‑diffusion collapses the system back to the classical Keller‑Segel scenario, which only supports FF or simple front solutions.

A striking implication of the non‑isolated singular point analysis is the emergence of free‑boundary fronts: the wave profile exhibits a sharp “switch‑on/off” behaviour, with a moving front that separates a region of high cell density from an essentially empty domain. This mechanism mirrors phenomena observed in tumor invasion fronts, bacterial colony expansion, and abrupt chemical reaction zones, where the leading edge propagates without a prescribed far‑field concentration.

In summary, the paper extends the Keller‑Segel chemotaxis theory by incorporating cross‑diffusion, revealing a rich taxonomy of traveling waves—front‑impulse, impulse‑front, front‑front, and impulse‑impulse—each governed by precise algebraic conditions. The phase‑plane framework, especially the treatment of non‑isolated singularities, provides a powerful tool for predicting wave existence and shape. These results open new avenues for modelling complex biological and physical systems where coupled transport processes play a decisive role.