Foliation in Dynamical Systems: Two models of competition

Speaker: Gaurav Mendiratta

Title: Foliation in Dynamical Systems: Two models of competition

Abstract:

Foliation in dynamical systems is a global property where the solution set
of trajectories span a subspace, parametrized by one or more parameters.
We start with the example of a specific model where tumor cells and normal
cells compete to survive. In the case of mathematical models of a disease
such as cancer, immunity is an important feature of the competitive
dynamics without which cancer almost always wins. The tumor growth model
we consider is a template model for a general system with two outcomes.
Such a dynamical system has two attractors and a saddle in between. We
find the separatrix that divides the basins of trajectories which limit to
two attractors. The separatrix acts as a prognosis function for tumor in
the space of tumor cells, cancer cells and food availability. We then
extend to four dimensions including immunity and show that the separatrix
are foliated as a function of immunity and therefore track the prognosis
as the patient improves or declines. Next we simplify to a larger class of
models where competitive dynamics are modeled by logistic equations. In
this system, we show that the presence of a continuous line of attractors
provides a foliated sub-space for the solutions. The foliation in this
system is particularly interesting as it entails a memory of initial
conditions.