The stationary properties and the state transition of the tumor cell growth mode

We study the stationary properties and the state transition of the tumor cell growth model (the logistic model) in presence of correlated noises for the case of nonzero correlation time. We derived an approximative Fokker-Planck equation and the stationary probability distribution (SPD) of the model. Based the SPD, we investigated the effects of both correlation strength (<InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\lambda$</EquationSource> </InlineEquation>) and correlation time (<InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\tau$</EquationSource> </InlineEquation>) of cross-correlated noises on the SPD, the mean of the tumor cell population and the normalized variance (<InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\lambda_2$</EquationSource> </InlineEquation>) of the system, and calculated the state transition rate of the system between two stable states. Our results indicate that: (i) <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\lambda$</EquationSource> </InlineEquation> and <InlineEquation ID="Equ5"> <EquationSource Format="TEX">$\tau$</EquationSource> </InlineEquation> play opposite roles in the stationary properties and the state transition of the system, i.e. increase of <InlineEquation ID="Equ6"> <EquationSource Format="TEX">$\lambda$</EquationSource> </InlineEquation> can produce a smaller mean value of the cell population and slow down the state transition, but increase of <InlineEquation ID="Equ7"> <EquationSource Format="TEX">$\tau$</EquationSource> </InlineEquation> can produce a larger mean value of the cell population and enhance state transition; (ii) For large <InlineEquation ID="Equ8"> <EquationSource Format="TEX">$\lambda$</EquationSource> </InlineEquation>, there a peak structure on both <InlineEquation ID="Equ9"> <EquationSource Format="TEX">$\lambda_2$</EquationSource> </InlineEquation>-<InlineEquation ID="Equ10"> <EquationSource Format="TEX">$\lambda$</EquationSource> </InlineEquation> plot and <InlineEquation ID="Equ11"> <EquationSource Format="TEX">$\lambda_2$</EquationSource> </InlineEquation>-<InlineEquation ID="Equ12"> <EquationSource Format="TEX">$\tau$</EquationSource> </InlineEquation> plot. For the small <InlineEquation ID="Equ13"> <EquationSource Format="TEX">$\lambda$</EquationSource> </InlineEquation>, <InlineEquation ID="Equ14"> <EquationSource Format="TEX">$\lambda_2$</EquationSource> </InlineEquation> increases with increasing <InlineEquation ID="Equ15"> <EquationSource Format="TEX">$\lambda$</EquationSource> </InlineEquation>, but <InlineEquation ID="Equ16"> <EquationSource Format="TEX">$\lambda_2$</EquationSource> </InlineEquation> increases with decreasing <InlineEquation ID="Equ17"> <EquationSource Format="TEX">$\tau$</EquationSource> </InlineEquation>. Copyright Springer-Verlag Berlin/Heidelberg 2004