Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect

Sex, ducks, and rock "n" roll: Mathematical model of sexual response

In this paper, we derive and analyze a mathematical model of a sexual response. As a starting point, we discuss two studies that proposed a connection between a sexual response cycle and a cusp catastrophe and explain why that connection is incorrect but suggests an analogy with excitable systems. This then serves as a basis for derivation of a phenomenological mathematical model of a sexual response, in which the variables represent levels of physiological and psychological arousal. Bifurcation analysis is performed to identify stability properties of the model's steady state, and numerical simulations are performed to illustrate different types of behavior that can be observed in the model. Solutions corresponding to the dynamics associated with the Masters-Johnson sexual response cycle are represented by "canard"-like trajectories that follow an unstable slow manifold before making a large excursion in the phase space. We also consider a stochastic version of the model, for which spectrum, variance, and coherence of stochastic oscillations around a deterministically stable steady state are found analytically, and confidence regions are computed. Large deviation theory is used to explore the possibility of stochastic escape from the neighborhood of the deterministically stable steady state, and the methods of an action plot and quasi-potential are employed to compute most probable escape paths. We discuss implications of the results for facilitating better quantitative understanding of the dynamics of a human sexual response and for improving clinical practice.

Bistability and noise-induced transient behaviour of steady states in a cancer network with the regulation of microRNA

MicroRNAs (miRs) regulatory network models are highly non-linear due to the negative regulation of gene expression at the post-transcriptional level by miRs and can produce interesting dynamics of the system such as bistability for miR-transcriptional factors interactions. In this article, we focus on the miR-17-92 cluster and its interaction with transcriptional factors (proteins) E2F and Myc. Environmental fluctuations (noise) and randomness in the bio-chemical reactions can be very important and change the dynamical role of miR-17-92 in the regulatory network. We have formulated a stochastically forced miR-17-92 and E2F-Myc interaction model and study the phenomena of intrinsic and extrinsic noise which can induce random switching between steady states or the destruction of the bistability. Using a method that employs stochastic sensitivity functions we have constructed confidence ellipses to determine the configurational arrangements of equilibrium and spatial arrangements of random states near stable equilibria. Simulations are carried out to numerically show the flow of the solution trajectories under noise. Finally, we summarize the simulation results and the impact of noise on the dual non-linear role of miR-17-92 cluster to act as an oncogene or as a tumour suppressor gene.

Stochastic sensitivity analysis and early warning signals of critical transitions in a tri-stable prey-predator system with noise

Near a tipping point, small changes in a certain parameter cause an irreversible shift in the behavior of a system, called critical transitions. Critical transitions can be observed in a variety of complex dynamical systems, ranging from ecology to financial markets, climate change, molecular bio-systems, health, and disease. As critical transitions can occur suddenly and are hard to manage, it is important to predict their occurrence. Although it is very tough to predict such critical transitions, various recent works suggest that generic early warning signals can detect the situation when systems approach a critical point. The most important indicator that predicts the risk of an upcoming critical transition is critical slowing down (CSD). CSD indicates a slow recovery rate from external perturbations of the stable state close to a bifurcation point. In this contribution, we study a two dimensional prey-predator model. Without any noise, the prey-predator model shows bistability and tri-stability due to the Allee effect in predators. We explore the critical transitions when external noise is added to the prey-predator system. We investigate early warning indicators, e.g., recovery rate, lag-1 autocorrelation, variance, and skewness to predict the critical transition. We explore the confidence domain method using the stochastic sensitivity function (SSF) technique near a stable equilibrium point to find a threshold value of noise intensity for a transition. The SSF technique in a two stage transition through confidence ellipse is described. We also show that the possibility of a transition to the predator-free state is independent of initial conditions. Our result may serve as a paradigm to understand and predict the critical transition in a two dimensional system.

The influences of correlated spatially random perturbations on first passage time in a linear-cubic potential

The influences of correlated spatially random perturbations (SRPs) on the first passage problem are studied in a linear-cubic potential with a time-changing external force driven by a Gaussian white noise. First, the escape rate in the absence of SRPs is obtained by Kramers' theory. For the random potential case, we simplify the escape rate by multiplying the escape rate of smooth potentials with a specific coefficient, which is to evaluate the influences of randomness. Based on this assumption, the escape rates are derived in two scenarios, i.e., small/large correlation lengths. Consequently, the first passage time distributions (FPTDs) are generated for both smooth and random potential cases. We find that the position of the maximal FPTD has a very good agreement with that of numerical results, which verifies the validity of the proposed approximations. Besides, with increasing the correlation length, the FPTD shifts to the left gradually and tends to the smooth potential case. Second, we investigate the most probable passage time (MPPT) and mean first passage time (MFPT), which decrease with increasing the correlation length. We also find that the variation ranges of both MPPT and MFPT increase nonlinearly with increasing the intensity. Besides, we briefly give constraint conditions to guarantee the validity of our approximations. This work enables us to approximately evaluate the influences of the correlation length of SRPs in detail, which was always ignored previously.

Sensitivity analysis of the noise-induced oscillatory multistability in Higgins model of glycolysis

A phenomenon of the noise-induced oscillatory multistability in glycolysis is studied. As a basic deterministic skeleton, we consider the two-dimensional Higgins model. The noise-induced generation of mixed-mode stochastic oscillations is studied in various parametric zones. Probabilistic mechanisms of the stochastic excitability of equilibria and noise-induced splitting of randomly forced cycles are analysed by the stochastic sensitivity function technique. A parametric zone of supersensitive Canard-type cycles is localized and studied in detail. It is shown that the generation of mixed-mode stochastic oscillations is accompanied by the noise-induced transitions from order to chaos.

Stochastic sensitivity technique in a persistence analysis of randomly forced population systems with multiple trophic levels

Motivated by important ecological applications we study how noise can reduce a number of trophic levels in hierarchically related multidimensional population systems. A nonlinear model with three trophic levels under the influence of external stochastic forcing is considered as a basic conceptual example. We analyze a probabilistic mechanism of noise-induced extinction of separate populations in this "prey-predator-top predator" system. We propose a new general mathematical approach for the estimation of the proximity of equilibrium regimes of this stochastic model to hazardous borders where abrupt changes in dynamics of ecological systems can occur. Our method is based on the stochastic sensitivity function technique and visualization method of confidence domains. Constructive abilities of this mathematical approach are demonstrated in the analysis of different scenaria of noise-induced reducing of the number of trophic levels.

How environmental noise can contract and destroy a persistence zone in population models with Allee effect

A problem of the analysis of the noise-induced extinction in population models with Allee effect is considered. To clarify mechanisms of the extinction, we suggest a new technique combining an analysis of the geometry of attractors and their stochastic sensitivity. For the conceptual one-dimensional discrete Ricker-type model, on the base of the bifurcation analysis, deterministic persistence zones are constructed in the space of initial states and biological parameters. It is shown that the random environmental noise can contract, and even destroy these persistence zones. A parametric analysis of the probabilistic mechanism of the noise-induced extinction in regular and chaotic zones is carried out with the help of the unified approach based on the sensitivity functions technique and confidence domains method.

Analysis of the noise-induced regimes in Ricker population model with Allee effect via confidence domains technique

We consider a discrete-time Ricker population model with the Allee effect under the random disturbances. It is shown that noise can cause various dynamic regimes, such as stable stochastic oscillations around the equilibrium, noise-induced extinction, and a stochastic trigger. For the parametric analysis of these regimes, we develop a method based on the investigation of the dispersions and arrangement of confidence domains. Using this method, we estimate threshold values of the noise generating such regimes.

Convergence time and speed of multi-agent systems in noisy environments

In this paper, the finite-time consensus problem of noise-perturbed multi-agent systems with fixed and switching undirected topologies is investigated. A continuous non-Lipschitz protocol for realizing stochastic consensus in a finite time is proposed. Based on the finite-time stability theory of stochastic differential equations, sufficient conditions are obtained to ensure finite-time stochastic consensus of multi-agent systems. An analytical upper bound for the convergence time is given. The effects of control parameters and noise intensity on convergence speed and time are also analyzed. Furthermore, numerical examples are provided to illustrate the effectiveness of the theoretical results.

Introduction to Focus Issue: nonlinear and stochastic physics in biology

Frank Moss was a leading figure in the study of nonlinear and stochastic processes in biological systems. His work, particularly in the area of stochastic resonance, has been highly influential to the interdisciplinary scientific community. This Focus Issue pays tribute to Moss with articles that describe the most recent advances in the field he helped to create. In this Introduction, we review Moss's seminal scientific contributions and introduce the articles that make up this Focus Issue.