Modeling Networks of Evolving Populations

Abstract

The goal of this research is to devise a method of differential equation based modeling of evolution that can scale up to capture complex dynamics by enabling the inclusion of many—potentially thousands—of biological characteristics. Towards that goal, a mathematical model for evolution based on the well-established Fisher-Eigen process is built with a unique and efficient structure. The Fisher-Eigen partial differential equation (PDE) describes the evolution of a probability density function representing the distribution of a population over a phenotype space. This equation depends on the choice of a fitness function representing the likelihood of reproductive success at each point in the phenotype space. The Fisher-Eigen model has been studied analytically for simple fitness functions, but in general no analytic solution is known. Furthermore, with traditional numerical methods, the equation becomes exponentially complex to simulate as the dimensionality of the problem expands to include more phenotypes. For this research, a network model is synthesized and a set of ordinary differential equations (ODEs) is extracted based on the Fisher-Eigen PDE to describe the dynamic behavior of the system. It is demonstrated that, when juxtaposed with full numerical PDE simulations, this ODE model finds well-matched transient and precise equilibrium solutions. This prototype method makes modeling of high-dimensional data possible, allowing researchers to examine and even predict complex dynamic behavior based on a snapshot of a population. KEYWORDS: Evolutionary Modeling; Mathematical Biology; Network Dynamics; Ordinary Differential Equations; Partial Differential Equations; Fisher-Eigen model; Phenotype; Fitness Function