The critical phenomenon and the re-entrance phenomenon in the anti-tumor model induced by the time delay

The effects of time delay τ on an anti-tumor model driven by a multiplicative noise and a periodic signal are investigated. The results obtained from the small delay approximation and numerical simulations indicate: (i) For the absence of the periodic signal in the system, the two-peak structure of the stationary probability distribution transforms into the single-peak structure with the increasing τ, and τ exists   a critical value τcτc. For τ<τcτ<τc, the stationary mean value 〈x〉st〈x〉st of the cell population decreases as the noise intensity D   increases, however, for τ>τcτ>τc, the 〈x〉st〈x〉st increases as the D increases; (ii) For the presence of the periodic signal in the system, the structure of the signal-to-noise ratio with changes of the D exhibits the transitions of one peak → two peaks → one peak as τ increases.