Most Probable Dynamics of the Single-Species with Allee Effect under Jump-diffusion Noise

Abstract

For a stochastic single-species model with Allee effect, we compute the most probable phase portrait (MPPP) using the non-local Fokker-Planck equation. This bistable stochastic model is driven by multiplicative Lévy noise as well as white noise. The two fixed points are stable equilibria and one is unstable state between them. In order to study biological behavior of species, we focus on the transition pathways from the extinction state to the upper fixed stable state for the transcription factor activator in a single-species model. By calculating the solution of the non-local Fokker-Planck equation corresponding to the population system of single-species model, the maximum possible path is obtained and corresponding maximum possible stable equilibrium state is determined. We also obtain the Onsager-Machlup (OM) function for the stochastic model, and solve the corresponding most probable paths. Our numerical simulation shows that: (i) The maximum of the stationary density function is located at the most probable stable equilibrium state when non-Gaussian noise is presented in the system; (ii) When the initial value increases from extinction state to the upper stable state, the most probable trajectories converge to the maximal likely equilibrium state (maximizer), in our case lies between 9 and 10; (iii) As time goes on, the most probable paths increase to stable state quickly, and remain a nearly constant level, then approach to the upper stable equilibrium state. These numerical experiment findings help researchers for further experimental study, in order to achieve good knowledge about dynamic systems in biology.

Slow Manifolds for Stochastic Koper Models with Stable Lévy Noises

“…The orbits can escape from the region of the metastable equilibrium if we face random perturbation [35]. It is interesting to characterize the most probable transition from one metastable state to another [36,37]. The random slow manifold still depends on k and λ.…”

The Onsager-Machlup action functional for McKean-Vlasov stochastic differential equations

“…More recently, Γ-convergence of Onsager-Machlup action functional [12,27], the bounds for the most probable transition time when time is not fixed [23] and the connection between OM action functional and information projection problem [39] are also considered. On the other hand, in specific application, climate change for global warming [49], single-species with Allee effect [44], genetic regulatory network [21] and so on. OM action functional has become an important tool for understanding the dynamics of system.…”