Generalized Lotka Volterra equations and maximal independent sets

Abstract

In applied mathematics and theoretical biology pattern formation is often modelled by partial differential equations, especially of the reaction-diffusion type. In this thesis however we will study pattern formation on discrete latices, formed by generalized Lotka Volterra (LV) equations. In [MK24] it is shown that for certain types of the LV equations the system converges asymptotically to vectors indicating maximal independent sets (MIS) of the underlying graph. We will reprove this result, using Lyapunov functions. Furthermore we will study probability distributions of resulting MISs. We will do this for a stochastic version of the LV equations, for which a steady state solution exists[BBC18], and with numerical simulations performed on a few chosen graphs. Our findings indicate that the LV system exhibits a bias towards large MISs and a dependency on the shape of the MIS. Lastly, we demonstrate how an LV system can be reformulated to include a graph Laplacian, which in the case of a lattice can be interpreted as exhibiting negative non-linear diffusion. Similar to the approach in [BV11], we transform the system into a non-homogeneous lattice differential equation (LDE) with positive diffusion. We postulate the existence of periodic travelling waves, akin to those observed in some reaction-diffusion systems.