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Balls, cups, and quasi-potentials: quantifying stability in stochastic systems

Ben C. Nolting, Karen C. Abbott

arXiv:1508.02088v3q-bio.PE

TL;DR

Comparing stability among coexisting ecological states matters because their relative stability bears on regime shifts, yet stability concepts are ambiguous and ordinary potentials apply only to gradient systems. The paper generalizes the potential using the quasi-potential and demonstrates its use in three ecological models. In the examples, its stability rankings can differ from linear stability analysis and align with stochastic dynamics.

Abstract

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When a system has more than one stable state, how can the stability of these states be compared? This deceptively simple question has important consequences for ecosystems, because systems with alternative stable states can undergo dramatic regime shifts. The probability, frequency, duration, and dynamics of these shifts will all depend on the relative stability of the stable states. Unfortunately, the concept of stability in ecology has suffered from substantial confusion and this is particularly problematic for systems where stochastic perturbations can cause shifts between coexisting alternative stable states. A useful way to visualize stable states in stochastic systems is with a ball-in-cup diagram, in which the state of the system is represented as the position of a ball rolling on a surface, and the random perturbations can push the ball from one basin of attraction to another. The surface is determined by a potential function, which provides a natural stability metric. However, systems amenable to this representation, called gradient systems, are quite rare. As a result, the potential function is not widely used and other approaches based on linear stability analysis have become standard. Linear stability analysis is designed for local analysis of deterministic systems and, as we show, can produce a highly misleading picture of how the system will behave under continual, stochastic perturbations. In this paper, we show how the potential function can be generalized so that it can be applied broadly, employing a concept from stochastic analysis called the quasi-potential. Using three classic ecological models, we demonstrate that the quasi-potential provides a useful way to quantify stability in stochastic systems.

Introduction

The paper presents the quasi-potential as a way to clarify stability in stochastic ecological systems and compare alternative stable states. It introduces lake eutrophication as an example where stochastic perturbations can move a system between coexisting states.

Introduction

  • The quasi-potential can quantify stability across alternative states and reveal dynamics that standard linear stability analysis may misrepresent.The paper also applies a numerical algorithm to systems with multiple stable states and examines likely transition paths.Source: Introduction · pp. 3–4
  • The paper illustrates its framework with three ecological models, including a lake eutrophication model and consumer-resource systems.The examples include a model with alternative stable states and one with a stable limit cycle.Source: Introduction · p. 4

Example 1: Lake Eutrophication

  • In the lake model, nutrient concentration has low-phosphorus oligotrophic and high-phosphorus eutrophic stable states, separated by a saddle.At the stated parameter values, the equilibria are xL = 0.537, xS = 0.971, and xH = 1.491.Source: Example 1: Lake Eutrophication · p. 4
  • Linear stability analysis classifies both lake equilibria as stable but indicates that the oligotrophic state is more stable because its eigenvalue is more negative.The reported eigenvalues are λL = −0.899 and λH = −0.797.Source: Example 1: Lake Eutrophication · p. 4
  • The ball-in-cup heuristic represents system state as a ball on a surface, with valleys corresponding to stable equilibria and basins to their domains of attraction.For the lake system, the surface is specified by a potential function.Source: Ball-in-cup · pp. 4–5
  • Stochastic perturbations are needed for transitions between coexisting stable states; without them, the deterministic system remains at an equilibrium unless its stable state is destroyed.The authors therefore frame transitions between coexisting states as a stochastic-system problem.Source: Ball-in-cup · p. 5

Stochastic Differential Equation Model

The model adds random perturbations to nutrient input and treats nutrient concentration as a stochastic process. It uses Gaussian white noise as a simple model of variable input.

Stochastic Differential Equation Model

  • The lake model replaces constant nutrient input with a random input, allowing nutrient concentration to vary stochastically over time.The continuous-time formulation is a stochastic differential equation.Source: Stochastic Differential Equation Model · pp. 5–6
  • The stochastic state X(t) is a random variable at each time, while a realization is a deterministic time series associated with particular random events.A realization can be viewed as an observed time series or one simulation run.Source: Stochastic Differential Equation Model · pp. 5–6
  • Gaussian white noise has independent, mean-zero increments with variance dt and no temporal autocorrelation.The accumulated changes form a Wiener process, also called Brownian motion.Source: Stochastic Differential Equation Model · p. 6

Utility of the potential for understanding the stochastic lake eutrophication model

In the lake model, the potential ranks the eutrophic state as more stable than the oligotrophic state, unlike linear stability analysis. In the consumer-resource model, the quasi-potential similarly favors the state that stochastic simulations visit more often, again differing from the linear ranking.

Utility of the potential for understanding the stochastic lake eutrophication model

  • U(xL) = 0.011, U(xS) = 0.047, and U(xH) = 0; the potential therefore ranks the eutrophic state as more stable, contrary to linear stability analysis.The authors attribute the discrepancy to linear analysis being local: it captures curvature near an equilibrium but not the broader basin geometry under continual perturbations.Source: Utility of the potential for… · p. 7
  • The steady-state probability density is maximized where U is minimized, linking potential minima to the most likely system states.The probability density is the solution to the Fokker–Planck equation.Source: Utility of the potential for… · p. 7
  • Mean first passage time depends largely on basin depth, the potential difference between the saddle and stable equilibrium; curvature also affects the approximation’s prefactor.At fixed valley depth, increased curvature is associated with decreased mean first passage time.Source: Utility of the potential for… · pp. 7–8

Example 2: Consumer and Resource With Alternative Stable States

Generalizing The Potential

The paper generalizes the potential to non-gradient systems using a function derived from a small-noise approximation to the steady-state distribution. This function retains potential-like properties and decomposes deterministic motion into downhill and circulatory components.

Generalizing The Potential

  • For an n-dimensional stochastic system with additive noise, the state variables form a vector X and the deterministic skeleton is a vector field f.The noise is represented by independent Wiener processes and an intensity σ.Source: Generalizing The Potential · p. 9
  • A small-noise approximation to the steady-state distribution yields V0, which satisfies the static Hamilton–Jacobi equation and is independent of σ.The paper treats V0 as an analogue of the potential in gradient systems.Source: Generalizing The Potential · p. 10
  • V0 is a Lyapunov function: along deterministic trajectories it decreases, with its time derivative equal to zero only where its gradient vanishes.The deterministic vector field decomposes into a downhill component −∇V0 and a remaining circulatory component Q.Source: Generalizing The Potential · p. 10
  • The circulatory component Q is perpendicular to ∇V0 and would move trajectories around V0 level sets in the absence of other forces.The function V0 generalizes the potential to non-gradient systems and is proportional to the Freidlin–Wentzell quasi-potential.Source: Generalizing The Potential · pp. 10–11

The Freidlin-Wentzell Quasi-potential

The quasi-potential is the minimum action required to reach a state from a stable equilibrium, extending potential-based analysis to non-gradient systems. In the examples, it changes the stability comparison for a consumer-resource system and separates restoring forces from cycling around a predator-prey limit cycle.

The Freidlin-Wentzell Quasi-potential

  • The quasi-potential minimizes action over paths and durations from a stable equilibrium; multiple base-point quasi-potentials can be stitched into a global function.The paper uses V0 = Φ/2 because V0 agrees with the true potential in gradient systems.Source: The Freidlin-Wentzell Quasi-potential · p. 11
  • The Hamilton-Jacobi equation yields the quasi-potential, while Cameron’s enhanced ordered upwind method computes it efficiently for two-dimensional stochastic differential equation systems.The method expands a front of known solution values and accepts adjacent values in ascending order; example 2 took less than ten minutes on an average personal computer.Source: The Freidlin-Wentzell Quasi-potential · pp. 11–12

Example 2 Continued

  • For the consumer-resource system, V0(eA)=0, V0(eS)=0.007, and V0(eB)=0.006, indicating eA is more stable than eB, unlike linear stability analysis.The quasi-potential also places eB and eS closer in stability than either is to eA, a more graded comparison than the stable/unstable classification.Source: Example 2 Continued · p. 12
  • In the consumer-resource system, V0 decomposes the deterministic skeleton into a component moving the system toward stable states and a component causing consumer-resource cycling.The decomposition is f = −∇V0 + Q.Source: Example 2 Continued · p. 12

Example 3: Predator and Prey With A Limit Cycle

  • For the predator-prey limit cycle, −∇V0 attracts trajectories to the cycle’s trough, while the circulatory component drives cycling within it.The quasi-potential can describe attractors such as limit cycles; linear stability analysis cannot provide a comparison here because it applies only to equilibrium points.Source: Example 3: Predator and Prey With A… · pp. 12–13

Limitations and Generalizations

The paper focuses on continuous-state stochastic differential equation models with additive, equal-intensity, white noise applied continually at low intensity. Some departures can be accommodated, but extreme shocks, small-noise assumptions, and the need to fit and validate a model constrain applications.

Scope

  • The framework is applied to stochastic differential equation models with continuous time and states, equal-intensity additive white noise acting directly on state variables, and continual low-intensity perturbations.Discrete-state models require different approaches in large deviation theory, while discrete-time Markov chains have analogous concepts.Source: Limitations and Generalizations · p. 13

Generalizations

  • Unequal or population-dependent noise, parameter noise, and colored noise can be accommodated through variable transformations or by modeling the noise or parameter as additional state variables.Increasing state-space dimensionality makes numerical quasi-potential calculations more challenging.Source: Limitations and Generalizations · pp. 13–14

Small noise and shocks

  • The small-noise assumption is central: approximations for steady-state probabilities and mean first passage times are useful only when σ2 is much less than 2 ∆V0.Large shocks can make probabilistic predictions from the quasi-potential unavailable; changing the potential surface dynamically is difficult to analyze with these methods.Source: Limitations and Generalizations · p. 14

Empirical application

  • Quasi-potential calculation requires fitting and validating a model to observed data, although this requirement is shared by other model-based approaches to alternative stable states.The authors point to controlled experiments and advances in model-fitting as promising for empirical study of shifts.Source: Limitations and Generalizations · p. 14

A Path Through the Quagmire of Stability Concepts

The quasi-potential gives a mathematical interpretation of the ball-in-cup metaphor for stochastic systems, clarifying how several distinct stability concepts relate. Its basin-height metric is suited to continual stochastic perturbations, while local and basin-width metrics address different aspects of stability.

Quasi-potential and the ball-in-cup metaphor

  • The quasi-potential gives the ball-in-cup metaphor rigorous meaning beyond gradient systems, where the translation from a mathematical model to a potential surface is less obvious.In one dimension, every system is a gradient system.Source: A Path Through the Quagmire of… · p. 15

Asymptotic stability

  • Asymptotic stability is local and assumes an isolated perturbation followed by deterministic dynamics, so it can mislead when perturbations are continual and stochastic.Nonlinear effects can make the local approximation uninformative once a trajectory leaves a small neighborhood of equilibrium.Source: A Path Through the Quagmire of… · p. 15

Basin width

  • Basin width measures the perturbation magnitude a system can sustain and still return to equilibrium, but it considers basin boundaries rather than potential shape or height.Like asymptotic stability, it treats perturbations as isolated events.Source: A Path Through the Quagmire of… · pp. 15–16

Potential height and resilience

  • Potential height, which Holling called resilience, addresses a distinct stability property and can be quantified using the quasi-potential.Unlike local stability and basin width, this metric is appropriate under continuous stochastic perturbations and relates to mean first passage times and steady-state probability densities.Source: A Path Through the Quagmire of… · p. 16

Comparing stability metrics

  • For equilibria eA and eB, basin width and quasi-potential rank eA as more stable, while asymptotic stability ranks eB higher; the three metrics can yield any combination.The metrics therefore distinguish different aspects rather than providing interchangeable rankings.Source: A Path Through the Quagmire of… · p. 16

Related stability concepts

  • Ecological uses of resilience vary, but the quasi-potential framework relates basin width and asymptotic stability to components discussed in the resilience literature.Harrison’s persistence—remaining in a specified range under continual perturbations—can be defined as the first passage time out of a domain and is directly related to the quasi-potential.Source: A Path Through the Quagmire of… · pp. 16–17

A Stochastic Differential Equations

The section develops the stochastic and action-based tools used to define quasi-potentials, then explains how local quasi-potentials can be combined to describe switching between stable states.

A Stochastic Differential Equations

  • With small random perturbations, the accumulated change in a stochastic process is modeled as a normally distributed increment, and the paper uses the Itô integral, which cannot depend on future values.The Itô integral evaluates the integrand at left endpoints of partition intervals; the corresponding stochastic differential equation notation represents an integral equation, since dW/dt does not exist.Source: A Stochastic Differential Equations · pp. 26–27

B Freidlin-Wentzell quasi-potential

  • Because most systems lack a potential function, the quasi-potential provides a conceptually analogous way to quantify stability in non-gradient systems.The section first treats constant-intensity isotropic noise as a useful starting point.Source: B Freidlin-Wentzell quasi-potential · p. 28
  • The quasi-potential is the minimum action over paths from a stable equilibrium to a target, and its action describes the most likely transition path in the small-noise setting.The minimization considers paths of all durations; the quasi-potential is closely related to first-passage times from attraction domains.Source: B Freidlin-Wentzell quasi-potential · pp. 28–29

C A global quasi-potential

  • When two attraction domains share a single saddle, local quasi-potentials can be shifted to agree there and combined into a global surface describing state switching.More complicated combinations arise when domains are connected by multiple saddles.Source: C A global quasi-potential · p. 30

D Small noise expansion of V

This section relates steady-state probability and an effective potential to noise intensity, then uses a small-noise expansion to obtain a leading-order equation.

D Small noise expansion of V

  • The steady-state distribution for the consumer-resource model varies with noise intensity, limiting its usefulness for describing general system properties.The distributions indicate that trajectories spend more time near eA than eB, but each noise level produces a different distribution.Source: D Small noise expansion of V · pp. 30–31
  • The effective potential is unsuitable as a stability metric because it depends on noise intensity and is not a Lyapunov function for the deterministic system.Its peaks and valleys are inversely related to those of the steady-state distribution.Source: D Small noise expansion of V · p. 31
  • Expanding V in the small parameter ϵ = σ^2 and retaining the lowest-order terms yields the Hamilton-Jacobi equation for V0.The expansion expresses V in coefficient functions associated with successive powers of ϵ.Source: D Small noise expansion of V · p. 31

E Hamilton-Jacobi equation for the quasi-potential

The section connects the quasi-potential to a Hamilton-Jacobi equation and states how the framework extends beyond the paper’s principal noise assumption.

E Hamilton-Jacobi equation for the quasi-potential

  • The quasi-potential and the leading-order effective-potential term satisfy Φ = 2V0, so they share the described properties up to a scalar factor.The Hamilton-Jacobi equation may require viscosity solutions when classical solutions do not exist.Source: E Hamilton-Jacobi equation for the… · pp. 31–32

F Other noise structures

  • The paper focuses on identity noise coupling, but the quasi-potential can also be calculated for more general noise structures with a modified action.A coordinate transform is another stated route for some systems with non-identity noise.Source: F Other noise structures · pp. 32–33

G Curvature

  • For gradient systems, the principal curvatures of V0 at an equilibrium equal the eigenvalues from linear stability analysis with their signs reversed.These curvatures are the eigenvalues of the Hessian of V0.Source: G Curvature · p. 33

H Mean first passage time asymptotics

This section describes small-noise first-passage-time asymptotics and compares quasi-potential estimates with simulations for one example.

H Mean first passage time asymptotics

  • Quasi-potential-based steady-state densities and mean first-passage times are determined only in the small-noise limit, and accurate estimates require prefactors as well as exponential terms.The section gives a gradient-system transition-time expression and notes a similar expression for non-gradient systems.Source: H Mean first passage time asymptotics · pp. 33–34
  • For non-gradient systems, the stated transition-time estimate uses the unstable eigenvalue of the full deterministic skeleton at the saddle.It also involves the least-action path and the divergence of the circulatory component.Source: H Mean first passage time asymptotics · p. 34
  • For the consumer-resource example, quasi-potential approximations closely match the means of simulated first-passage times, although simulated distributions have outliers in their tails.The example drops a term because numerical integration along the least-action path suggests it is approximately zero.Source: H Mean first passage time asymptotics · p. 34

I Further example

Four bistable cases show that linear stability, basin width, and potential depth can rank the same equilibria differently. No single metric captures every aspect of stability.

I Further example

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