Stochastic differential equation for two-phase growth model

Abstract

Most mathematical models to describe natural phenomena in ecology are models with single-phase. The models are created as such to represent the phenomena as realistic as possible such as logistic models with different types. However, several phenomena in population growth such as embryos, cells and human are better approximated by two-phase models because their growth can be divided into two phases, even more, each phase requires different growth models. Most two-phase models are presented in the form of deterministic models, since two-phase models using stochastic approach have not been extensively studied. In previous study, Zheng’s two-phase growth model had been implemented in continuous time Markov chain (CTMC). It assumes that the population growth follows Yule process before the critical size, and the Prendiville process after that. In this research, Zheng’s two-phase growth model has been modified into two new models. Generally, probability distribution of birth and death processes (BDPs) of CTMC is intractable; and even if its first–passage time distribution can be obtained, the conditional distribution for the second-phase is complicated to be determined. Thus, two-phase growth models are often difficult to build. To overcome this problem, stochastic differential equation (SDE) for two-phase growth model is proposed in this study. The SDE for BDPs is derived from CTMC for each phase, via Fokker-Planck equations. The SDE for twophase population growth model developed in this study is intended to be an alternative to the two-phase models of CTMC population model, since the significance of the SDE model is simpler to construct, and it gives closer approximation to real data.