Double time-delays induced stochastic dynamical characteristics for a metapopulation system subjected to the associated noises and a multiplicative periodic signal

In this paper, we aim to investigating in detail the stability of the system, the mean extinction time and stochastic resonance (SR) phenomena caused by a multiplicative periodic signal for a dual time-delayed metapopulation system subjected to cross-correlated noises. By use of the fast descent method, the small time delay approximation method and the SR theory, we obtain the expressions of the steady state probability distribution function, the mean first-passage time and signal-to-noise ratio (SNR). Numerical results indicate that the multiplicative, additive and association noises together with time delay τ can all accelerate the transition from the stable state of big density to the extinction one and play significant roles in weakening the stability and shortening the mean extinction time of the metapopulation. In particular, the additive noise and time delay τ can result in the crash of the system, while another time delay θ can strengthen the biological system stability and extend the declining time for the population. On the other hand, with respect to the SNR, the figures show that time delay τ plays entirely antipodal roles in motivating stochastic resonance (SR) in a variety of different situations. Conversely, the multiplicative noise intensity Q and time delay θ all along produce negative effect on exciting the SR. Meanwhile, the increase of the weak additive noise intensity M can stimulate the SR phenomenon, but the bigger values of M will suppress the SNR and SR phenomenon. The strength of the noise correlation λ plays an important role in restraining the SR in most cases except that it does in the plot of SNR-Q.