In the present paper, noise-induced escape from the domain of attraction of a stable fixed point of a fast-slow insect outbreak system is investigated. According to Dannenberg's theory(Dannenberg PH, Neu JC, 2014)[1], different noise amplitude ratios μ lead to the change of the Most Probable Escape Path(MPEP). Therefore, the research emphasis of this paper is to extend their study and discuss the changes of the MPEPs in more detail. Firstly, the case for μ=1, wherein the MPEP almost traces out the critical manifold, is considered. Via projecting the full system onto the critical manifold, a reduced system is obtained and the quasi-potential of the full system can be partly evaluated by that of this reduced system. In order to test the accuracy of the computed MPEP, a new relaxation method is then presented. Then, as μ converges to zero, an improved analytical method is given, through which a better approximation for the MPEP at the turning point is obtained. And then, in the case that the value of μ is moderate, wherein the MPEP will peel off the critical manifold, to determine the changing point of the MPEP on the critical manifold, an effective numerical algorithm is given. In brief, in this paper, a complete investigation on the structural changes of the MPEPs of a fast-slow insect outbreak system under different values of μ is given, and the results of the numerical simulations match well with the analytical ones.
The noise-induced transition of the Zeldovich–Semenov model in a continuous stirred tank reactor is investigated under small random perturbations. The deterministic model will exhibit mono- and bistable characteristics via local and global bifurcations. In the bistable zone, based on the Freidlin–Wentzell large deviation theory, the stochastic preference is explained by analyzing the required action of the fluctuational path. For the case of monostability, in the weak noise limit, the emergence of the switching line gives rise to the sudden switch of the optimal path and the sliding cycle will appear via the sliding bifurcation, which is verified by numerical methods. In addition, when there is no saddle in phase space, stochastic excitation with large-amplitude spikes is studied. On the quasi-threshold manifold, the point with the minimum quasi-potential plays the same role as the saddle, which means that the optimal path will undergo a large excursion by crossing this special point. These phenomena are verified by employing stochastic simulations.
Noise-induced escape in a 2D generalized Maier–Stein model with two parameters μ and α is investigated in the weak noise limit. With the WKB approximation, the patterns of extreme paths and singularities are displayed. By employing the Freidlin–Wentzell action functional and the asymptotic series, critical parameters α inducing singularity bifurcation are determined analytically for μ=1. The switching line will appear with singularities and is equivalent to the sliding set in the Filippov system. The pseudo-saddle-node bifurcation on the switching line is found. Then, when −1<μ<1, it is found that all bifurcation values α will decrease as μ decreases and the second-order bifurcation values are bigger than all first-order ones. In addition, the variation of the switching line is also analyzed and a new switching line will emerge when the location of the minimum quasi-potential on the boundary changes. At last, when the noise is anisotropic, only the noise intensity ratio will affect the bifurcation value α.
We present a new method for calculation of quasi-potential, which is a key concept in the large deviation theory. This method adopts the “ordered” idea in the ordered upwind algorithm and different from the finite difference upwind scheme, the first-order line integral is used as its update rule. With sufficient accuracy, the new simplified method can greatly speed up the computational time. Once the quasi-potential has been computed, the minimum action path (MAP) can also be obtained. Since the MAP is of concernin most stochastic situations, the effectiveness of this new method is checked by analyzing the accuracy of the MAP. Two cases of isotropic diffusion and anisotropic diffusion are considered. It is found that this new method can both effectively compute the MAPs for systems with isotropic diffusion and reduce the computational time. Meanwhile anisotropy will affect the accuracy of the computed MAP.
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