Stochastic ecological kinetics of regime shifts in a time‐delayed lake eutrophication ecosystem

Mitigation of rare events in multistable systems driven by correlated noise

We consider rare transitions induced by colored noise excitation in multistable systems. We show that undesirable transitions can be mitigated by a simple time-delay feedback control if the control parameters are judiciously chosen. We devise a parsimonious method for selecting the optimal control parameters, without requiring any Monte Carlo simulations of the system. This method relies on a new nonlinear Fokker-Planck equation whose stationary response distribution is approximated by a rapidly convergent iterative algorithm. In addition, our framework allows us to accurately predict, and subsequently suppress, the modal drift and tail inflation in the controlled stationary distribution. We demonstrate the efficacy of our method on two examples, including an optical laser model perturbed by multiplicative colored noise.

Finite-time stability and optimal control of an impulsive stochastic reaction-diffusion vegetation-water system driven by Lévy process with time-varying delay

In this paper, a reaction-diffusion vegetation-water system with time-varying delay, impulse and Lévy jump is proposed. The existence and uniqueness of the positive solution are proved. Meanwhile, mainly through the principle of comparison, we obtain the sufficient conditions for finite-time stability which reflect the effect of time delay, diffusion, impulse, and noise. Besides, considering the planting, irrigation and other measures, we introduce control variable into the vegetation-water system. In order to save the costs of strategies, the optimal control is analyzed by using the minimum principle. Finally, numerical simulations are shown to illustrate the effectiveness of our theoretical results.

Time delay and cross-correlated Gaussian noises-induced stochastic stability and regime shift between steady states for an insect outbreak system

In this paper, we focus on investigating the stochastic stability and the regime transition between the endangered state and the boom state for a time-delayed insect growth system driven by correlated external and internal noises. By use of the Fokker–Planck equation, the method of small time delay approximation and the fast descent method, we explore in detail the joint action of noise terms and time delay on the mean reproduction and depression time for the insect population. Our investigations indicate that the pseudo-resonance phenomenon of the mean first-passage time (MFPT) occurs because of the impact of different noises and time delay. Through the numerical calculation, it is discovered that multiplicative noise can speed up the shift of the insect population from the boom state to the endangered one, while the noise correlation and time delay can propel the insect system to evolve from the endangered state to the boom state and improve the biological stability. In addition, the impact of the additive noise on the stability of the biological system depends on the positive and negative situation of the noise correlation. On the other hand, during the process of suppressing the insect explosion, it is beneficial to the pest control to amplify the association noise strength and weaken the intensities of the multiplicative, additive noises and time delay. However, during the process of eliminating the pests, it can produce nice effect on the disinsection to increase time delay, the intensities of multiplicative and additive noises and weaken the strength of noise correlation.

A systematic path to non-Markovian dynamics: new response probability density function evolution equations under Gaussian coloured noise excitation

Determining evolution equations governing the probability density function (pdf) of non-Markovian responses to random differential equations (RDEs) excited by coloured noise, is an important issue arising in various problems of stochastic dynamics, advanced statistical physics and uncertainty quantification of macroscopic systems. In the present work, such equations are derived for a scalar, nonlinear RDE under additive coloured Gaussian noise excitation, through the stochastic Liouville equation. The latter is an exact, yet non-closed equation, involving averages over the time history of the non-Markovian response. This non-locality is treated by applying an extension of the Novikov–Furutsu theorem and a novel approximation, employing a stochastic Volterra–Taylor functional expansion around instantaneous response moments, leading to efficient, closed, approximate equations for the response pdf. These equations retain a tractable amount of non-locality and nonlinearity, and they are valid in both the transient and long-time regimes for any correlation function of the excitation. Also, they include as special cases various existing relevant models, and generalize Hänggi's ansatz in a rational way. Numerical results for a bistable nonlinear RDE confirm the accuracy and the efficiency of the new equations. Extension to the multidimensional case (systems of RDEs) is feasible, yet laborious.

Stability and Hopf bifurcation analysis for a Lac operon model with nonlinear degradation rate and time delay

In this paper, we construct a discrete time delay Lac operon model with nonlinear degradation rate for mRNA, resulting from the interaction among several identical mRNA pieces. By taking a discrete time delay as bifurcation parameter, we investigate the nonlinear dynamical behaviour arising from the model, using mathematical tools such as stability and bifurcation theory. Firstly, we discuss the existence and uniqueness of the equilibrium for this system and investigate the effect of discrete delay on its dynamical behaviour. Absence or limited delay causes the system to have a stable equilibrium, which changes into a Hopf point producing oscillations if time delay is increased. These sustained oscillation are shown to be present only if the nonlinear degradation rate for mRNA satisfies specific conditions. The direction of the Hopf bifurcation giving rise to such oscillations is also determined, via the use of the so-called multiple time scales technique. Finally, numerical simulations are shown to validate and expand the theoretical analysis. Overall, our findings suggest that the degree of nonlinearity of the model can be used as a control parameter for the stabilisation of the system.

A theoretical study on the cross-talk of stress regulatory pathways in root cells

The plants developed more dedicated regulatory pathways than the animals did to response various environment stresses, since they could not run away. The cross-talk among the pathways generally introduce non-trivial regulatory behaviors, from which the plants may benefit. For better understanding the regulatory mechanism due to cross-talk, we study in this work two entangled stress regulatory pathways in root cells. A quantitative model of the regulatory network is constructed in the simplest fashion. An analytic parameter-free approach is then employed to analyse the response tendencies. It leads us to a simple constraint on the non-linear regulatory exponents. Under the constraint, a transition to the non-monotonic growth inhibition happens at finite concentration of ABA, due to which the plants could survive from cold/heat stress. The parameter-free tendency analysis would also be applied to further experiments, especially in the case of insufficient data for multi-parameter fitting.