Stability and mean extinction time for a metapopulation system subjected to two kinds of time delays and associated multiplicative and additive noises

Abstract

In this paper, the stability and the mean extinction time for a double time-delayed metapopulation system driven by associated multiplicative and additive noises are investigated. By using the Fokker–Planck equation, the fast descent method and the method of small time delay, we obtain the expressions of stationary probability distribution and the mean extinction time. The numerical results demonstrate that the multiplicative, additive noises and time delay θ can destroy the stability of the system. Whereas, the associated noise and time delay τ can consolidate the population system. Meanwhile, the resonant phenomenon of the mean first-passage time (MFPT) occurs in the metapopulation system due to the joint action of different types of noises and time delays. The increase of the strength of associated noise and time delay τ can effectively prolong the extinction time of the metapopulation. Inversely, the additive noise plays a negative role in restraining the collapse of the population. On the other hand, the appropriate multiplicative noise intensity can also extend the life of the metapopulation system. However, the varying of time delay θ cannot change the maximum of the mean extinction time, but it could produce different effects on the MFPT under the conditions of different parameters.