Derivation of stochastic partial differential equations for correlated random walk models and stochastic differential equation models for phylogenetic trees

This work has two parts. In the first part, stochastic partial differential equations for the one-dimensional telegraph equation and the two-dimensional linear transport equation are derived from basic principles. The telegraph equation and the linear transport equation are well-known correlated random walk (CRW) models, i.e., transport models characterized by correlated successive-step orientations. In the present investigation, these deterministic CRW equations are generalized to stochastic CRW equations. To derive the stochastic CRW equations, the possible changes in direction and particle movement for a small time interval are carefully determined to obtain discrete stochastic models. As the time interval decreases, the discrete stochastic models lead to systems of Itˆo stochastic differential equations. As the position intervals decrease, stochastic partial differential equations are derived for the telegraph and transport equations. Comparisons between numerical solutions of the stochastic partial differential equations and independently formulated Monte Carlo calculations support the accuracy of the derivations. In the second part of this work, two mathematical models for development of phylogenetic trees for families of genera and species are derived and studied. In one model, the rate of formation of new genera in a family is assumed to be proportional to the number of genera in the family while in the other model, the rate of formation of new genera is assumed to be proportional to the number of species in the family. Each of these models has a deterministic formulation and stochastic formulation. For both models, the number of genera with k species for large time is approximated. In addition, the exact number of genera with k species is determined. Calculational results of the two models are compared with data for three families of animals.