Lotka-Volterra equations with chemotaxis: walls, barriers and travelling waves

In this paper we consider a simple two species model for thegrowth of new blood vessels. The model is based upon the Lotka-Volterrasystem of predator and prey interaction, where we identify newlydeveloped capillary tips as the predator species and a chemoattractantwhich directs their motion as the prey. We extend the Lotka-Volterrasystem to include a one-dimensional spatial dependence, by allowingthe predators to migrate in a manner modelled on the phenomenonof chemotaxis. A feature of this model is its potential to supporttravelling wave solutions. We emphasize that in order to determinethe existence of such travelling waves it is essential thatthe global relationships of a number of phase plane featuresother than the equilibria be investigated.