Associated relaxation time and the correlation function for a tumor cell growth system subjected to color noises

The associated relaxation time Tc and the normalized correlation function C(s) for a tumor cell growth system subjected to color noises are investigated. Using the Novikov theorem and Fox approach, the steady probability distribution is obtained. Based on them, the expressions of Tc and C(s) are derived by means of projection operator method, in which the effects of the memory kernels of the correlation function are taken into account. Performing the numerical computations, it is found: (1) With the cross-correlation intensity |λ|, the additive noise intensity α and the multiplicative noise self-correlation time τ1 increasing, the tumor cell numbers can be restrained; And the cross-correlation time τ3, the multiplicative noise intensity D can induce the tumor cell numbers increasing; However, the additive noise self-correlation time τ2 cannot affect the tumor cell numbers; The relaxation time Tc is a stochastic resonant phenomenon, and the distribution curves exhibit a single-maximum structure with D increasing. (2) The cross-correlation strength λ weakens the related activity between two states of the tumor cell numbers at different time, and enhances the stability of the tumor cell growth system in the steady state; On the contrast, τ1 and τ3 enhance the related activity between two states at different time; However, τ2 has no effect on the related activity between two states at different time