Studying the Logistic Model

Abstract

Several studies have used the logistic equation to model the growth of cancer cell populations1 as seen in Eq. (1). This has included correlated multiplicative, [Formula: see text] and additive, [Formula: see text], noise terms. These noise terms can affect the growth rate, [Formula: see text], and death rate, [Formula: see text], of tumor cells and can be induced from factors such as radiotherapy or other cancer treatments. Depending on the intensity of the noise the terms, the fluctuations can induce a phase transition. Noise-induced transitions of nonlinear stochastic systems have applications in the fields of physics, chemistry and biology. (1)dxdt=ax−bx2+xϵ(x)−Γ(t). We study the logistic differential equation with a multiplicative noise term before and at phase transition. Computational methods used to investigate this cancer cell model include a Diffusion Entropy Analysis method and a waiting time distribution method.2,3,4 DEA will establish the scaling of a simulated series without altering the data through detrending. We hypothesize the treatment that causes a phase transition in the logistic model will induce tumor extinction and management. Understanding how to better evaluate and study cancer cell growth models will assist in assessing the efficacy of cancer treatments. Future work will include running simulations with a modified DEA method that includes the use of stripes.2 For better statistics, the code will be adopted to run ensembles of simulated data instead of a single series. Generating and analyzing these large datasets can be computationally expensive. Through multiprocessing and the use of a supercomputer, we believe these computational limitations can be overcome.