  Chemotaxis-driven invasions have been proposed across a broad spectrum of
biological processes, from cancer to ecology. The influential system of
equations introduced by Keller and Segel has proven a popular choice in the
modelling of such phenomena, but in its original form restricts to a
homogeneous population. To account for the possibility of phenotypic
heterogeneity, we extend to the case of a population continuously structured
across space, time and phenotype, where the latter determines variation in
chemotactic responsiveness, proliferation rate, and the level of chemical
environment modulation. The extended model considered here comprises a
non-local partial differential equation for the local phenotype distribution of
cells which is coupled, through an integral term, with a differential equation
for the concentration of an attractant, which is sensed and degraded by the
cells. In the framework of this model, we concentrate on a
chemotaxis/proliferation trade-off scenario, where the cell phenotypes span a
spectrum of states from highly-chemotactic but minimally-proliferative to
minimally-chemotactic but highly-proliferative. Using a combination of
numerical simulation and formal asymptotic analysis, we explore the properties
of travelling-wave solutions. The results of our study demonstrate how
incorporating phenotypic heterogeneity may lead to a highly-structured wave
profile, where cells in different phenotypic states dominate different spatial
positions across the invading wave, and clarify how the phenotypic structuring
of the wave can be shaped by trade-offs between chemotaxis and proliferation.

Populations and Evolution