Warning signs for wave speed transitions of noisy Fisher–KPP invasion fronts

Kuehn, Christian

doi:10.1007/s12080-013-0189-1

Cited by 15 publications

(13 citation statements)

References 85 publications

(136 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[18,19,3,20] and the references therein. Some recent papers also explore how the noise changes the dynamics in the context of stochastic PDE's [21,22,23]. However much work remains to be done in this direction.…”

Section: Discussionmentioning

confidence: 99%

Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction–diffusion systems

Tzou

Ward

Kolokolnikov

2015

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We present three examples of delayed bifurcations for spike solutions of reaction-diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analysed in the context of ODE's in [P.Mandel and T.Erneux, J.Stat.Phys 48(5-6) pp.1059-1070, 1987. It was found that the instability would not be fully realized until the system had entered well into the unstable regime. The bifurcation is said to have been "delayed" relative to the threshold value computed directly from a linear stability analysis. In contrast to the study of Mandel and Erneux, we analyze the delay effect in systems of partial differential equations (PDE's). In particular, for spike solutions of singularly perturbed generalized Gierer-Meinhardt and Gray-Scott models, we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability. In the first example, for the Gierer-Meinhardt model on the infinite real line, we analyze the delay resulting from slowly tuning a control parameter through a Hopf bifurcation. In the second example, we consider a Hopf bifurcation of the Gierer-Meinhardt model on a finite one-dimensional domain. In this scenario, as opposed to the extrinsic tuning of a system parameter through a bifurcation value, we analyze the delay of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the third example, we consider competition instabilities triggered by the extrinsic tuning of a feed rate parameter. In all three cases, we find that the system must pass well into the unstable regime before the onset of instability is fully observed, indicating delay. We also find that delay has an important effect on the eventual dynamics of the system in the unstable regime. We give analytic predictions for the magnitude of the delays as obtained through the analysis of certain explicitly solvable nonlocal eigenvalue problems (NLEP's). The theory is confirmed by numerical solutions of the full PDE systems.

show abstract

Section: Discussionmentioning

confidence: 99%

Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction–diffusion systems

Tzou

Ward

Kolokolnikov

2015

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…From a biological and epidemic-modelling perspective, it would be interesting to compare different classes of models to the fast-slow heterogeneous stochastic SIS model we considered with a view towards heterogeneity, epidemic thresholds and warning signs for critical transitions. For example, this could include SIR models [30,52], adaptive network dynamics [25,49,57], and stochastic partial differential equations [1,46].…”

Section: Discussionmentioning

confidence: 99%

Heterogeneous population dynamics and scaling laws near epidemic outbreaks

Widder

Kuehn

2016

Mathematical Biosciences and Engineering

Self Cite

View full text Add to dashboard Cite

In this paper, we focus on the influence of heterogeneity and stochasticity of the population on the dynamical structure of a basic susceptible-infected-susceptible (SIS) model. First we prove that, upon a suitable mathematical reformulation of the basic reproduction number, the homogeneous system and the heterogeneous system exhibit a completely analogous global behaviour. Then we consider noise terms to incorporate the fluctuation effects and the random import of the disease into the population and analyse the influence of heterogeneity on warning signs for critical transitions (or tipping points). This theory shows that one may be able to anticipate whether a bifurcation point is close before it happens. We use numerical simulations of a stochastic fast-slow heterogeneous population SIS model and show various aspects of heterogeneity have crucial influences on the scaling laws that are used as early-warning signs for the homogeneous system. Thus, although the basic structural qualitative dynamical properties are the same for both systems, the quantitative features for epidemic prediction are expected to change and care has to be taken to interpret potential warning signs for disease outbreaks correctly.

show abstract

“…. , 11 = K},σ = 5, (M, N) = (50,45), and the eigenfunctions (43), as before. To carry out the Monte-Carlo simulation, we integrate the full nonlinear system of SODEs derived in Section 7 using a standard Euler-Maruyama method as described in [40].…”

Section: A1 Comparison To Monte-carlo Simulationmentioning

confidence: 99%

“…Stochastic partial differential equations (SPDEs) have emerged as one of the main examples for spatio-temporal stochastic systems, e.g. in ecological invasion waves [25,50,58], voter models [59,20], surface and thin film growth models [17,14], statistical mechanics [43,37], fluid dynamics [41,10], neuroscience [73,67], and many general pattern formation problems [13,29]. In this paper, we study as a prototypical model a version of the Allen-Cahn SPDE, which can formally be written as…”

Section: Introductionmentioning

confidence: 99%

Numerical Continuation and SPDE Stability for the 2D Cubic-Quintic Allen--Cahn Equation

Kuehn¹

2015

SIAM/ASA J. Uncertainty Quantification

Self Cite

View full text Add to dashboard Cite

We study the Allen-Cahn equation with a cubic-quintic nonlinear term and a Q-traceclass stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand, how numerical continuation methods can be carried over to the SPDE setting. First, we compute the deterministic bifurcation diagram for the PDE, i.e. without stochastic forcing. In this case, two locally asymptotically stable steady state solution branches exist upon variation of the linear damping term. Then we consider the Lyapunov operator equation for the locally linearized system around steady states for the SPDE. We discretize the full SPDE using a combination of finite-differences and spectral noise approximation obtaining a finite-dimensional system of stochastic ordinary differential equations (SODEs). The large system of SODEs is used to approximate the Lyapunov operator equation via covariance matrices. The covariance matrices are numerically continued along the two bifurcation branches. We show that we can quantify the stochastic fluctuations along the branches. We also demonstrate scaling laws near branch and fold bifurcation points. Furthermore, we perform computational tests to show that, even with a sub-optimal computational setup, we can quantify the subexponential-timescale fluctuations near the deterministic steady states upon stochastic forcing on a standard desktop computer setup. Hence, the proposed method for numerical continuation of SPDEs has the potential to allow for rapid parametric uncertainty quantification of spatio-temporal stochastic systems.

show abstract

Warning signs for wave speed transitions of noisy Fisher–KPP invasion fronts

Cited by 15 publications

References 85 publications

Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction–diffusion systems

Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction–diffusion systems

Heterogeneous population dynamics and scaling laws near epidemic outbreaks

Numerical Continuation and SPDE Stability for the 2D Cubic-Quintic Allen--Cahn Equation

Contact Info

Product

Resources

About