  Hybrid models of chemotaxis combine agent-based models of cells with partial
differential equation models of extracellular chemical signals. In this paper,
travelling wave properties of hybrid models of bacterial chemotaxis are
investigated. Bacteria are modelled using an agent-based (individual-based)
approach with internal dynamics describing signal transduction. In addition to
the chemotactic behaviour of the bacteria, the individual-based model also
includes cell proliferation and death. Cells consume the extracellular nutrient
field (chemoattractant) which is modelled using a partial differential
equation. Mesoscopic and macroscopic equations representing the behaviour of
the hybrid model are derived and the existence of travelling wave solutions for
these models is established. It is shown that cell proliferation is necessary
for the existence of non-transient (stationary) travelling waves in hybrid
models. Additionally, a numerical comparison between the wave speeds of the
continuum models and the hybrid models shows good agreement in the case of weak
chemotaxis and qualitative agreement for the strong chemotaxis case. In the
case of slow cell adaptation, we detect oscillating behaviour of the wave,
which cannot be explained by mean-field approximations.
