Non-Gaussian noise-weakened stability in a foraging colony system with time delay

In this paper, the dynamical properties in a foraging colony system with time delay and non-Gaussian noise were investigated. Using delay Fokker-Planck approach, the stationary probability distribution (SPD), the associated relaxation time (ART) and normalization correlation function (NCF) are obtained, respectively. The results show that: (i) the time delay and non-Gaussian noise can induce transition from a single peak to double peaks in the SPD, i.e., a type of bistability occurring in a foraging colony system where time delay and non-Gaussian noise not only cause transitions between stable states, but also construct the states themselves. Numerical simulations are presented and are in good agreement with the approximate theoretical results; (ii) there exists a maximum in the ART as a function of the noise intensity, this maximum for ART is identified as the characteristic of the non-Gaussian noise-weakened stability of the foraging colonies in the steady state; (iii) the ART as a function of the noise correlation time exhibits a maximum and a minimum, where the minimum for ART is identified as the signature of the non-Gaussian noise-enhanced stability of the foraging colonies; and (iv) the time delay can enhance the stability of the foraging colonies in the steady state, while the departure from Gaussian noise can weaken it, namely, the time delay and departure from Gaussian noise play opposite roles in ART or NCF.