The relaxation time and the correlation function for a logistic growth system driven by colored cross-correlation noises

The associated relaxation time T-c and the normalized correlation function C(s) are investigated in the logistic growth system, which is used to describe a tumor cell growth process, driven by two Gaussian white noise sources and the correlation between the additive and multiplicative noise. The expression of T-c and C(s), which is the function of noise parameters (additive noise intensity alpha, multiplicative noise intensity D, correlation intensity lambda and correlation time tau), is obtained by using the projection operator method. After introducing noise intensity ratio, a dimensionless parameter R = alpha/D, and performing the numerical computations, the two case are analyzed: (1) In the growth case, lambda and tau play opposite roles on the T-c and the C(s). It must emphasize that there is a minimal evolution velocity to appear and the tumor cell numbers is hard to evolve from an arbitrary initial condition to the maximum. (2) In the decay case, lambda and tau play same roles on the T-c and the C(s). There is a maximal evolution velocity to appear. The noises induce different responses of tumor cells between the growth and decay case.