A characteristic feature of tumor invasion is the destruction of the healthy tissue surrounding it. Open space is generated, which invasive tumor cells can move into. One such mechanism is the urokinase plasminogen system (uPS), which is found in many processes of tissue reorganization. Lolas, Chaplain and collaborators have developed a series of mathematical models for the uPS and tumor invasion. These models are based upon degradation of the extracellular material through plasmid plus chemotaxis and haptotaxis. In this paper we consider the uPS invasion models in one-space dimension and we identify a condition under which this cancer invasion model converges to a chemotaxis model with logistic growth. This condition assumes that the density of the extracellular material is not too large. Our result shows that the complicated spatio-temporal patterns, which were observed by Lolas and Chaplain et al. are organized by the chaotic attractor of the logistic chemotaxis system. Our methods are based on energy estimates, where, for convergence, we needed to find lower estimates in Lγ for 0 < γ < 1. This is a new method for these types of PDE.
On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms