asymptotic and transient mean-square properties of stochastic systems arising in ecology, fluid...

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SIAM J. APPL. MATH. c© 2014 Society for Industrial and Applied MathematicsVol. 74, No. 2, pp. 411–433

ASYMPTOTIC AND TRANSIENT MEAN-SQUARE PROPERTIESOF STOCHASTIC SYSTEMS ARISING IN ECOLOGY, FLUID

DYNAMICS, AND SYSTEM CONTROL∗

EVELYN BUCKWAR† AND CONALL KELLY‡

Abstract. We examine the role of persistent, state-dependent stochastic perturbations on themean-square properties of nonnormal linear systems arising in three applications. In an example frompopulation biology, we extend to the stochastic case measures of asymptotic and transient responseof a predator-prey system to initial value perturbations and examine the relative effects on thesemeasures of persistent stochastic perturbations of each species. In an example from fluid dynamics,we show how a linear stochastic mixing term may induce a transition-to-turbulence in certain low-dimensional models of plane Couette flow. Finally, we look at the role of drift-diffusion interactioneffects in the noise-induced stabilization of a linear system with a single high-gain feedback controlparameter.

Key words. stochastic differential equations, mean-square dynamics, predator-prey, transition-to-turbulence, stochastic control

AMS subject classifications. 60H10, 60H30, 92D25, 96F06, 93E15

DOI. 10.1137/120893859

1. Introduction. We investigate the role of persistent, state-dependent stochas-tic perturbation in three distinct applications: ecology, fluid dynamics, and high-gaincontrol of linear systems. The common property of these applications is that, in eachcase, the unperturbed dynamics may be expressed in terms of the asymptotic and/ortransient properties of the equilibrium of a linear system with nonnormal coefficientmatrix—one which is not diagonalizable by a unitary matrix. This is significant be-cause, if a stable linearized deterministic system has a nonnormal coefficient, then theeigenvalues of the community matrix may not adequately describe the stability of thenonlinear system. The large transients associated with such linear systems render theasymptotic stability of the equilibrium vulnerable to perturbation.

Those perturbations may take the form of the higher-order nonlinear terms dis-carded when extracting the linearization from the original nonlinear model. However,in a stochastic setting, a nonnormal linearization may potentially interact with astochastic perturbation, even if that perturbation is linear.

Consider Fedotov, Bashkirtseva, and Ryashko [12] and Fedotov [13]. In the firstarticle the authors observe that, in a nonlinear low-dimensional model of plane Cou-ette flow with a nonnormal linear term, small state-independent stochastic perturba-tions can move trajectories out of the basin of attraction of the laminar flow equi-librium. In the second article an interaction between linear drift nonnormality andstate-dependent perturbation is shown to induce instability corresponding to a large-scale magnetic field in a low-dimensional dynamo model.

∗Received by the editors October 4, 2012; accepted for publication (in revised form) January 2,2014; published electronically March 27, 2014. This work was supported by an ICMS Research inGroups grant.

http://www.siam.org/journals/siap/74-2/89385.html†Institute for Stochastics, Johannes Kepler University Linz, 4040 Linz, Austria (Evelyn.Buckwar@

jku.at).‡Department of Mathematics, University of the West Indies, Mona, Kingston, Jamaica, W.I.

([email protected]). This author’s work was partially supported by an Exzellenzstipen-dium des Landes Oberosterreich.

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http://www.siam.org/journals/siap/74-2/89385.html

mailto:[email protected]




412 EVELYN BUCKWAR AND CONALL KELLY

Higham and Mao [15] examined the effect of state-dependent stochastic pertur-bation on mean-square asymptotic stability of the equilibrium of a linear system withnonnormal coefficient. They showed that, when the drift coefficient is sufficientlynonnormal, the zero equilibrium may be destabilized in mean-square by a stochas-tic perturbation of arbitrarily small intensity acting orthogonally to the flow of theunperturbed system. After a preliminary numerical investigation of the phenomenonin [7], a more detailed analysis was carried out by Buckwar and Kelly [8], who pre-sented three case studies illustrating the importance of the geometry of the stochasticperturbation and showing how such an analysis could shed light on the numerical dis-cretization of stochastic partial differential equations. The analysis relied upon a tool,developed in Buckwar and Sickenberger [9] and described in section 2.3 of this arti-cle, for investigating mean-square stability properties of systems of linear stochasticdifferential equations (SDEs); we will use this tool again here.

With the exception of these articles, nonnormal drift-diffusion interactions in sto-chastic models have not been adequately explored in the literature, and understandingthem in a linear setting is a necessary prerequisite for further investigation of the re-lationship between a nonlinear stochastic system and its corresponding linearization.

Our first model, contained in section 3, is from ecology. We look at measures,introduced in Neubert and Caswell [23], of asymptotic and transient response to theinitial value perturbation of a species coexistence equilibrium in a predator-prey modeland develop a framework for incorporating persistent stochastic perturbation. Ourmain results show that persistent stochastic perturbations of the predator (ratherthan the prey) render the equilibrium more sensitive in its asymptotic and transientmean-square response to initial value perturbations.

In section 4, we look at low-dimensional models of the transition-to-turbulence inplane Couette flow found in Trefethen et al. [5, 30]. Following the investigation of therole of noise in such systems in [12], we propose a mechanism whereby the transition-to-turbulence observed in such flows at high Reynolds numbers (interpreted as achange in the mean-square asymptotic stability of an equilibrium corresponding tothe laminar flow state) may be influenced by linear stochastic mixing.

In section 5, we investigate noise-assisted high-gain stabilization in mean-square.An analysis in Crauel, Matsikis, and Townley [11] showed that the inclusion of astochastic mixing term (with the stochastic integral interpreted in the Stratonovichsense) in a linear model with a single high-gain feedback control parameter k allowsthe equilibrium to be stabilized in mean-square when k is sufficiently large. We showthat the analytic approach developed in [9] can be used to investigate the dynamicsof the system for fixed values of k, and hence shed light on certain effects that ariseas a result of interaction between the nonnormality of the drift and the geometry ofthe stochastic perturbation.

2. Mathematical preliminaries.

2.1. Linear systems of SDEs. Consider

(2.1) dX(t) = FX(t) dt+

m∑r=1

GrX(t) dWr(t), t > 0,

where F ∈ Rd×d, G1, . . . , Gm ∈ R

d×d, and W = (W1, . . . ,Wm)T is an m-dimensionalWiener process defined on the filtered probability space (Ω,F , (F(t))t≥0,P), where(F(t))t≥0 is the natural filtration generated by W . The initial value X(0) is anF(0)-measurable random variable with finite second moment. In particular, X(0)

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NONNORMALITY AND STOCHASTIC PERTURBATION 413

may be constant. Since (2.1) is a linear, autonomous SDE with constant coefficients,unique strong global solutions X = {X(t;X(0)), t ≥ 0} exist (see, for example, [20]).Equation (2.1) has equilibrium solution X(t) ≡ 0 when X(0) = 0. In this paper weare concerned with the mean-square stability of this equilibrium.

Definition 2.1. The equilibrium of (2.1) is globally mean-square asymptoticallystable if and only if for all X(0) ∈ R

d,

limt→∞E|X(t)|2 = 0.

2.2. Kronecker sums and products. In the subsequent analysis of the stabil-ity properties of systems of linear stochastic ordinary differential equations (SODEs),we will make use of the special properties of Kronecker sums and products.

Definition 2.2.

1. The Kronecker product of an m × n matrix A and a p × q matrix B is themp× nq matrix defined by

A⊗B =

⎛⎜⎝

a11B . . . a1nB...

. . ....

am1B . . . amnB

⎞⎟⎠ .

2. If A is an n× n matrix, B is an m×m matrix, and Ik is the k × k identitymatrix, then the Kronecker sum of A and B is the mn×mn matrix

A⊕B = A⊗ Im + In ⊗B.

3. The vectorization vec(A) of an m× n matrix A is the mn× 1 column vectorobtained by stacking the columns of the matrix A on top of one another.

4. The spectral abscissa α(A) of a matrix A is defined by α(A) = maxiR(λi),where λi represents the eigenvalues of A.

Kronecker products and matrix vectorizations satisfy the following properties.Lemma 2.3 (see [19, Ch. 2]).1. If the matrices A+B and C +D exist, then

(A+B)⊗ (C +D) = A⊗ C +A⊗D +B ⊗ C +B ⊗D.

2. If the matrices AC and BD exist, then

(A⊗B)(C ⊗D) = AC ⊗BD.

3. If A, B, and C are three matrices such that the matrix product ABC isdefined, then

vec(ABC) = (CT ⊗A)vec(B).

4. A special case of part 3 is given by

vec(AB) = (BT ⊗ Im)vec(A) = (Iq ⊗A)vec(B),

where A is an m× n matrix, B is an n× q matrix, and Ik is again the k× k identitymatrix.

Kronecker sums satisfy the following properties.Lemma 2.4. For A ∈ R

n×n and B ∈ Rm×m we have the following:

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1. If A has eigenvalues λi, i = 1, . . . , n, and B has eigenvalues μj, j = 1, . . . ,m,then A⊕B has mn eigenvalues

λ1 + μ1, . . . , λ1 + μm, λ2 + μ1, . . . , λ2 + μm, . . . , λn + μm.

2. A⊕B satisfies

e(A⊕B) = eA ⊗ eB.

3. A⊕B satisfies

(A⊕B)T = (AT ⊕BT ).

Proof. The first result can be found in [35, section 2.6]. The second result followsfrom part 2 of Definition 2.2, part 2 of Lemma 2.3, and Theorems 2 and 3 in [25], whichsay that exp(Im⊗A+B⊗In) = exp(Im⊗A) exp(B⊗In), and exp(Ip⊗A) = Ip⊗exp(A)and exp(B⊗Iq) = exp(B)⊗Iq (p, q arbitrary), respectively. The third part is a simpleconsequence of part 2 of Definition 2.2 and the rules of transposing matrices.

2.3. Mean-square asymptotic stability analysis for linear stochastic dif-ferential systems. The second moment of the solution of (2.1), that is, the expecta-tion of the matrix-valued process P (t) = X(t)X(t)T , is given by the unique solutionof the matrix ordinary differential equation (ODE) [4, Thm. 8.5.1]

(2.2) dE(P (t)) =(FE(P (t)) + E(P (t))FT +

m∑r=1

GrE(P (t))GTr

)dt

for t ≥ t0 ≥ 0, and with initial value P (t0) = X0XT0 . Clearly, the zero solution

of (2.1) is asymptotically mean-square stable if and only if the zero solution of thedeterministic ODE of the second moments of (2.2) is asymptotically stable; see also[16, Remark VI.2.2].

In Buckwar and Sickenberger [9], the authors showed how the matrix ODE givenby (2.2) could be recast as a d2-dimensional deterministic linear ODE system asfollows: the vectorization of P yields a d2-dimensional process {Y (t)}t≥t0 of the form

vec(P (t)) = Y (t) = (Y1(t), Y2(t), . . . , Yd2(t))T

= (X21 (t), X2(t)X1(t), . . . , Xd(t)X1(t),

X1(t)X2(t), X22 (t), X3(t)X2(t), . . . , Xd(t)X2(t), . . . , X

2d(t))

T .

Applying the vectorization operation on both sides of (2.2), and employing parts 3 and4 of Lemma 2.3, yields the deterministic linear system of ODEs for the d2-dimensionalvector E(Y (t))

(2.3) dE(Y (t)) = S E(Y (t)) dt ,

where S is given by

(2.4) S = Id ⊗ F + F ⊗ Id +

m∑r=1

Gr ⊗Gr .

S is the stability matrix of the linear ODE system (2.3) and, consequently, may bereferred to as the mean-square stability matrix of the linear SDE system (2.1). Thefollowing result is standard.

Lemma 2.5. The equilibrium solution of (2.1) is globally mean-square asymptot-ically stable if and only if α(S) < 0.

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We will use S to investigate the asymptotic and transient properties of systemsof SDEs arising in the applications considered in this paper, taking advantage of thespecial properties of Kronecker sums and products summarized in section 2.2.

3. Application: A stochastically perturbed predator-prey model.

3.1. Background. Recent publications in ecological modeling, e.g., [14, 31, 23,24, 28], emphasize the importance of investigating both the transient and asymptoticdynamics of populations subject to initial perturbations. Traditionally, the analysisof models describing population dynamics focused on asymptotic properties of themodels: stability of equilibria. However, when dealing with longer-lived organisms—most plants, fish, and mammals—experimental observations may be a manifestationof transient rather than asymptotic behavior.

The linearization of a nonlinear model around an equilibrium yields the linearODE system

(3.1)dx(t)

dt= Ax(t) ,

where the community matrix A has constant entries that depend on the model pa-rameters. To provide a quantitative measure of the short- and long-term responses ofan ecological system to initial value perturbations in the vicinity of an equilibrium,Neubert and Caswell [23] proposed the following measures.

1. Resilience measures the asymptotic response of (3.1) to initial value per-turbations, describing how rapidly a system returns to a stable equilibrium afterperturbation. It is defined as

ν∞(A) = −R(λmax(A)),

where λmax(A) is the eigenvalue with largest real part; i.e., R(λmax(A)) is the spectralabscissa α(A) of A.

2. Reactivity measures the transient response of (3.1) to initial value perturba-tions and is defined as

ν0 = max‖x(0)‖2

(1

‖x(t)‖2d‖x(t)‖2

dt

)∣∣∣∣t=0

or, alternatively, as λmax

(12 (A+AT )

).

3. The amplification envelope ρ(t) is the maximum possible amplification thatany initial-value perturbation could induce in a solution at time t, given by

ρ(t) := maxx0 =0

‖x(t)‖2‖x0‖2 ,

which corresponds to ρ(t) = ‖eAt‖2.4. The maximum possible amplification ρmax is given by

ρmax := maxt≥0

ρ(t),

and the time at which it occurs is denoted tmax.Remark 3.1. Notice the following:1. Resilience corresponds to the magnitude of the maximal Lyapunov exponent

of the linearized system (3.1) when the zero equilibrium is exponentially as-ymptotically stable.

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2. Reactivity corresponds to the logarithmic matrix norm μ2[A] of A with re-spect to the Euclidean norm, defined by

μ2[A] = limδ→0+

(‖I + δA‖2 − 1)/δ.

See, for example, [27] for further details on logarithmic matrix norms.3. The amplification envelope may be understood as follows: solutions to (3.1)

are given by x(t) = etAx(0), and therefore ρ(t) = ‖etA‖2, considered asa function of t, describes the growth and decay of solutions over time. Infact, the asymptotic properties of the amplification envelope are connectedto resilience in that

limt→∞

1

tlog ‖ρ(t)‖ = α(A).

However, it is also useful to consider finite-time behavior of the amplifica-tion envelope when A is nonnormal. In particular, ρmax gives a bound onany finite-time transient behavior that may be present. See [29] for furtherdiscussion.

The above measures capture the response of a linearized deterministic system toa single isolated perturbation and as such require further development. Neubert andCaswell [23] (for example) point out that biological populations are generally subjectto continual stochastic disturbance. The mathematical forms of such disturbancesare potentially many and varied; some are catalogued in Spagnolo, Valenti, and Fias-conaro [34]. Moreover, analysis in Higham and Mao [15] and Buckwar and Kelly [8]highlights the fact that the manner in which noise manifests itself in a nonnormalsystem can have a profound effect on the dynamics of the resulting stochastic system.Our goal in this section is to show how the notions of reactivity, resilience, and am-plification envelope can be generalized in order to capture the short- and long-termdynamics of ecological systems subject to persistent, equilibrium-preserving, stochas-tic perturbation.

3.2. Generalization to stochastic systems. Suppose that the model systemis subject to m equilibrium-preserving stochastic perturbations, yielding, after lin-earization, the linear stochastic system

(3.2) dX(t) = AX(t) dt+

m∑i=1

BiX(t) dWi(t).

We can use (2.4) to construct the mean-square stability matrix of (3.2) as

S = Id ⊗A+A⊗ Id +m∑i=1

Bi ⊗Bi

and redefine the notions of resilience, reactivity, and amplification envelope in termsof the second moment of solutions of (3.2) as follows.

Definition 3.2.

1. The root-mean-square resilience of (3.2) is given by

−1

2α(S).

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2. The root-mean-square reactivity of (3.2) is given by

1

2λmax

(S + ST

2

).

3. The root-mean-square amplification envelope of (3.2) is given by

√‖eSt‖2

with ρmax and tmax defined as before.Lemma 3.3. Suppose that (3.2) is such that Bi = 0 for each i = 1, . . . ,m. Then

parts 1–3 of Definition 3.2 correspond to the definitions of resilience, reactivity, andamplification envelope for the deterministic model system (3.1).

Proof. Note that if Bi = 0 for each i = 1, . . . ,m, then

S = Id ⊗A+A⊗ Id = A⊕A.

By part 1 of Lemma 2.4, the eigenvalues of a matrix formed by this Kronecker sumare the pairwise sums of the eigenvalues of the summed matrix terms. Therefore, part1 of the statement of the lemma follows, since

−�(λmax(S)) = −�(λmax(A⊕A)) = −2�(λmax(A)).

By part 1 of Lemma 2.3 and part 3 of Lemma 2.4, part 2 of the statement of thelemma follows because

S + ST

2=

(A+AT

2

)⊕(A+AT

2

),

and therefore

λmax

(S + ST

2

)= 2λmax

(A+AT

2

).

Finally, part 3 of the statement of the lemma follows by part 2 of Lemma 2.4 since,for any induced matrix norm ‖ · ‖,

‖eSt‖ = ‖e(A⊕A)t‖ = ‖eAt ⊗ eAt‖ = ‖eAt‖2.Remark 3.4. While the idea of considering a linear ODE system that governs the

mean-square dynamics of a linear SDE system is not new (see [17], for example), theparticular form of S given by (2.4) allows us to take advantage of the known propertiesof Kronecker sums and products summarized in section 2.2, hence contributing to thesimplicity of the proof of Lemma 3.3.

3.3. Example: A perturbed predator-prey model.

3.3.1. The deterministic model. The authors in [23, 24] consider the simplenonlinear predator-prey model

dN

dt= rN

(1− N

K

)− aNP

N + b,(3.3a)

dP

dt=

caNP

N + b− dP .(3.3b)

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Here, the prey and predator populations at time t are denoted N and P (the t de-pendence is generally suppressed). The right-hand side of (3.3a) incorporates logisticgrowth of the prey population in the absence of the predator with intrinsic growthrate r and carrying capacity K, and a Holling type II saturating functional responsewith saturation level a and half-saturation constant b. The additional parameters onthe right-hand side of (3.3b) correspond to the predator’s yield coefficient c and itsmortality rate d.

System (3.3) can be nondimensionalized with the change of parameters

y1 = N/b , y2 = aP/(rb) , τ = rt ,κ = K/b , α = (ac)/r , β = d/(ac) ,

yielding a system of the form

dy1dt

= y1

(1− y1

κ

)− y1 y2

y1 + 1,(3.4a)

dy2dt

= α

(y1 y2y1 + 1

− βy2

).(3.4b)

Of three equilibrium solutions of (3.4), the authors in [23, 24] consider the asymptoticand transient properties of the one at which the predator and prey populations coexist:

(3.5) y∗1 =β

1− β, y∗2 = (1 + y∗1)

(1− y∗1

κ

).

Linearizing and centering around the equilibrium given by (3.5) yields a linear systemof the form (3.1) with community matrix

(3.6) A =

⎛⎜⎝

(1− 2y∗1

κ − y∗2(1 + y∗1)

2

)−β

αy∗2

(1 + y∗1)20

⎞⎟⎠ .

3.3.2. A stochastic sensitivity analysis. It is our goal to investigate theeffect of persistent stochastic perturbations on the equilibrium of system (3.4), andour approach can be viewed as a sensitivity analysis of the deterministic equilibriumdynamics (as motivated by [18, 26]) rather than a stochastic model derived from firstprinciples; see, for example, [1, 21].

We will introduce a stochastic sensitivity function to the right-hand side of (3.4),an approach that is consistent with that of [33, 22] and (with a different setup) [6], asfollows: let W1 and W2 be independent Wiener processes, and consider the stochasticsystem

dY1 =

[Y1

(1− Y1

κ

)− Y1Y2

Y1 + 1

]dt+ σ1(Y1 − y∗1) dW1(t),(3.7a)

dY2 =

[α

(Y1Y2

Y1 + 1− βY2

)]dt+ σ2(Y2 − y∗2) dW2(t),(3.7b)

where y∗1 , y∗2 are as given in (3.5). Linearize and center the drift and diffusion coeffi-

cients of (3.7) around (y∗1 , y∗2), using the change of variables (Y1, Y2) = (Y1−y∗1 , Y2−y∗2)

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to get the system of SDEs

(3.8) d

(Y1

Y2

)=

⎛⎜⎝1− 2y∗1

κ − y∗2(1 + y∗1)

2 −β

αdy∗2

(1 + y∗1)2 0

⎞⎟⎠

(Y1

Y2

)dt

+

(σ1 00 0

)(Y1

Y2

)dW1(t) +

(0 00 σ2

)(Y1

Y2

)dW2(t).

Remark 3.5. The information gained from the analysis of a stochastic systemarising as part of a sensitivity analysis does not have the same explanatory power asinformation from stochastic models constructed from first principles and should notbe interpreted quantitatively. However, a qualitative analysis can indicate which partsof the system have significant influence on the transient and asymptotic dynamics ofthe deterministic equilibrium and are particularly sensitive to perturbation.

Setting d =y∗2

(1+y∗1 )

2 , we can apply (2.4) to compute the mean-square stability

matrix

S =

⎛⎜⎜⎜⎜⎝

2− 4y∗1κ − 2 d+ σ2

1 −β −β 0

αd 1− 2y∗1κ − d 0 −β

αd 0 1− 2y∗1κ − d −β

0 αd αd σ22

⎞⎟⎟⎟⎟⎠ .

While it is not possible to explicitly write the eigenvalues of S in closed form, we canuse S to develop graphical representations of root-mean-square resilience, reactivity,and amplification envelope over relevant parameter regimes.

Remark 3.6. Note that the eigenvalues of A, as defined in (3.6), have negative realparts when κ−1

2 < y∗1 < κ, guaranteeing that the equilibrium is asymptotically stablefor this range of parameters. The introduction of a stochastic perturbation, though itpreserves the equilibrium itself, may result in mean-square instability for a choice ofparameters where the equilibrium of the unperturbed system is asymptotically stable,and this must be kept in mind in any comparative analysis.

3.3.3. Numerical results: Reactivity and resilience. Figure 1 shows theeffect of changes in α (maximum predator growth rate) and β (predator mortalityrate) on the root-mean-square resilience and reactivity of the equilibrium of (3.8) inthe absence of perturbation (σ1 = σ2 = 0). Notice that, as explained by Lemma 3.3,these plots are identical to those for the resilience and reactivity of the equilibrium ofthe unperturbed model (3.4) given in Figure 7 (a) and (b) of [23].

The effect of changes to α and β on root-mean-square resilience and reactivity inthe presence of perturbations of the predator and prey species is illustrated in Figures2 and 3. We make the following observations for Figure 2.

1. Increasing the intensity of either perturbation has the effect of reducing therange of values of β over which the equilibrium is mean-square asymptotically stable(visible in each figure as the range of values over which the root-mean-square resilienceremains positive).

2. Similarly, increasing the intensity of either perturbation has the effect of re-ducing the root-mean-square resilience and increasing the root-mean-square reactivityover the range of parameter values for which the equilibrium remains mean-squareasymptotically stable.

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(a) (b)

Fig. 1. Root-mean-square resilience (solid line) and root-mean-square reactivity (dashed line)of the equilibrium solution of (3.7) with κ = 1 and σ1 = σ2 = 0. In (a), α = 0.05, and in (b),β = 0.4.

3. Both of these effects are visibly more pronounced for perturbations of thepredator species. Moreover, predator perturbations have a destabilizing effect at thetop and bottom of the range of values of β. Prey perturbations have a destabilizingeffect only at the bottom of the range.

We make the following observations for Figure 3.1. Perturbations of either species have a smoothing effect on root-mean-square

resilience in the vicinity of α = 1. It was pointed out in [23] that when α > 1, theequilibrium of the unperturbed system is a focus and the real parts of the eigenvaluesof A as defined in (3.6) are independent of α. We can see that this is not true of theequilibrium of the perturbed system, regardless of the species under perturbation.

2. Perturbations of the prey species have no visible effect on root-mean-squarereactivity; nor do they have a destabilizing effect over the range of values of α tested.However, perturbations of the predator species do have a destabilizing effect for smallvalues of α and increase root-mean-square reactivity for values of α where the equi-librium of (3.7) remains mean-square asymptotically stable. In particular, note thatroot-mean-square reactivity is nonzero when α = 2.

3.3.4. Numerical results: Amplification envelope. Figure 4 shows the ef-fect of perturbations of the predator and prey species on the root-mean-square am-plification envelope of the equilibrium solution of (3.7) with κ = 1, α = 0.05, andβ = 0.2. It can be seen from Figure 2 that, for this choice of parameters, the equi-librium solution will remain mean-square asymptotically stable for perturbations ofintensity up to σi = 2.6. We see that increasing the intensity of perturbation increasesthe amplification envelope regardless of the species subject to perturbation, but themost dramatic change is visible when the predator species is subject to perturbation.The effect of each perturbation on ρmax and tmax is shown in the table under eachgraph, where ρ(t) has been maximized over a discretized time axis with step size 0.01.

Taken together, these observations indicate the following: the mean-square as-ymptotic stability of the species coexistence equilibrium is more vulnerable to persis-tent perturbations of the predator species than of the prey species. Moreover, when

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Prey species perturbation Predator species perturbation

(a) (c)

(b) (d)

Fig. 2. Effect of varying β on root-mean-square resilience (solid line) and root-mean-squarereactivity (dashed line) of the equilibrium solution of (3.7) with κ = 1, α = 0.05. In the first column,σ2 = 0 and (a) σ1 = 0.13, (b) σ1 = 0.26. In the second column, σ1 = 0 and (c) σ2 = 0.13, (d)σ2 = 0.26.

the equilibrium is mean-square asymptotically stable, the transient dynamics of thesystem, as measured by reactivity and amplification envelope, are more sensitive toinitial value perturbations when the predator species (rather than the prey species)is subject to persistent stochastic perturbations.

Remark 3.7. In [23], Neubert and Caswell chose to define reactivity and ampli-fication envelope in terms of the l2-vector-norm and corresponding induced matrixnorm. It was pointed out in [31] that, although the l2-norm is computationally con-venient, the l1- and l∞-norms may have more appropriate biological interpretations.Since we seek to extend the framework proposed in [23] to stochastic systems, andto compare our results with theirs, we have continued to work with the l2-norm inour definitions of root-mean-square reactivity and amplification envelope. However,since these properties are now defined in terms of the mean-square stability matrix S

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(a) (c)

(b) (d)

Fig. 3. Effect of varying α on root-mean-square resilience (solid line) and root-mean-squarereactivity (dashed line) of the equilibrium solution of (3.7) with κ = 1 and β = 0.4. In the firstcolumn, σ2 = 0 and (a) σ1 = 0.13, (b) σ1 = 0.26. In the second column, σ1 = 0 and (c) σ2 = 0.13,(d) σ2 = 0.26.

rather than the community matrix A, it is certainly possible to further extend them toinclude other (potentially more realistic) norms. We have computed the root-mean-square amplification envelope under induced l1- and l∞-matrix norms and found thatthe greater sensitivity of the predator species to persistent stochastic perturbationsreported in this section is observed under these norms as well.

4. Application: Transition-to-turbulence in parallel shear flows.

4.1. Background. Trefethen et al. [30] proposed the 2-dimensional model

(4.1)

(u′

v′

)=

(−R−1 10 −2R−1

)(uv

)+ ‖u‖

(0 −11 0

)(uv

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σ1 ρmax tmax

0 1.406709 12.150.13 1.425792 12.480.26 1.505148 13.81

σ2 ρmax tmax

0 1.406709 12.150.13 1.502841 13.550.26 2.059184 23.57

(a) (b)

Fig. 4. Effect of perturbation on the root-mean-square amplification envelope of the equilibriumsolution of (3.7) with κ = 1, α = 0.05, and β = 0.2. In (a), σ2 = 0, and in (b), σ1 = 0.

as a means of understanding the role of linear nonnormality in the onset of turbulencein plane Couette flow. A discussion of the motivation behind (4.1), including a detailedinterpretation of u and v, may be found in Trefethen and Embree [29, Chapters 20and 21]. We summarize their presentation as follows.

The component v may be viewed as the amplitude of “mother” flow structures(streamwise vortices) that can sustain “daughter” flow structures (streamwise streaks)with amplitude u, and indeed this relationship is expressed in the nonnormality ofthe linear coefficient: notice that the linear dynamics of u receive positive input fromv, but not vice-versa.

The parameter R may be viewed as a stand-in for the Reynolds number. Rtakes only positive values, and therefore the eigenvalues of the linear coefficient arealways negative; this is consistent with a linear stability analysis of theoretical planeCouette flow showing that the laminar state is stable at all Reynolds numbers (seeRomanov [32]). In practice, however, such flows make the transition to turbulenceincreasingly readily as the Reynolds number grows large (see [29] again for a reviewof the long history of this observation). Trefethen et al. [30] proposed that the non-normality of the linear dynamics leaves the laminar state vulnerable to perturbation,which in (4.1) takes the form of the energy-conserving nonlinear mixing term.

Baggett and Trefethen [5] compare several low-dimensional models, including avariation on (4.1) of the form

(4.2)

(u′

v′

)=

(−R−1 10 −R−1

)(uv

)+ ‖u‖

(0 −11 0

)(uv

),

seeking to answer the following question: if ε is the minimum amplitude of all dis-turbances that may induce a transition-to-turbulence, and the relationship between ε

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and R has the form ε = O(Rα) as R grows large, then what is the value of α (referredto as the threshold exponent)?

Asymptotic analysis of the Navier–Stokes equations in Chapman [10] yields avalue of α = −1 for initial-value perturbations of plane Couette flow. A numericalinvestigation of model (4.1) in Baggett and Trefethen [5] yields a value of α = −3 fora certain class of perturbations chosen to induce large transient growth in the lineardynamics. Other low-dimensional models are also considered in [5] and also satisfyα < −1.

Remark 4.1. The use of such low-dimensional models is open to criticism: werefer readers again to Trefethen and Embree [29, Chapters 20 and 21] for a discussionof the issues. We acknowledge here that such models are too simplistic to providea detailed mathematical description of plane Couette flow and in particular do notattempt to describe such flows once they have made the transition-to-turbulence. Thepurpose of such models is to investigate aspects of the transition-to-turbulence itselfand in particular any potential role played by nonnormality in the dynamics of themodel after linearization around the laminar flow equilibrium. Low-dimensional mod-els allow us to isolate interactions between a nonnormal linearization and a stochasticperturbation that may contribute to such a transition, though we do not claim thatnoise is the only factor, or that the transition itself is a purely linear phenomenon.

4.2. A model with stochastic mixing. We replace nonlinear deterministicmixing in (4.2) with linear stochastic mixing, resulting in the SDE system

(4.3) d

(uv

)=

(−R−1 10 −R−1

)(uv

)dt+ σ

(0 −11 0

)(uv

)dW (t).

This linear system has a single equilibrium and, considering that it may be morerealistic to investigate the response of the flow to ongoing rather than isolated pertur-bation, we examine the effect of σ2 on global mean-square asymptotic stability, ratherthan the tendency of initial-value perturbations of magnitude ε to move solutions outof the basin of attraction of the laminar state.

The relationship between R and σ2 governing mean-square asymptotic stabilityof the zero equilibrium of (4.3) may be stated precisely as follows.

Theorem 4.2. The zero equilibrium of (4.3) is mean-square asymptotically stableif and only if

(4.4) − 2

R− 1

3σ2 +

1

3

3

√8σ6 + 27σ2 + 3

√3σ4(16σ4 + 27)

+4

3

σ4

3

√8σ6 + 27σ2 + 3

√3σ4(16σ4 + 27)

< 0.

Proof. The mean-square stability matrix that corresponds to (4.3) is given by

S =

⎛⎜⎜⎝−2/R 1 1 σ2

0 −2/R −σ2 10 −σ2 −2/R 1σ2 0 0 −2/R

⎞⎟⎟⎠ .

The result follows by direct inspection of the eigenvalues of S.Corollary 4.3. The transition threshold satisfying the asymptotic relation σ2 =

O(Rα) for the model defined in (4.3) is α = −3.

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Diagonal perturbation Orthogonal perturbation

(a) (c)

(b) (d)

Fig. 5. Diagonal perturbation: Eigenvalues for the mean-square stability matrix of (4.5) withR = 250 and (a) σ = 0 and (b) σ = 0.0006. Orthogonal perturbation: Eigenvalues for the mean-square stability matrix of (4.3) with (c) σ = 0 and (d) σ = 0.0006.

Proof. For σ2 < 1, we can expand the left-hand side of (4.4) in terms of σ2 aroundzero, giving

− 2

R+

3√2(σ2

)1/3+ O(σ2),

and the statement of the corollary follows.

4.3. The role of perturbation geometry in the transition-to-turbulence.Consider Figure 5. The first column represents the eigenvalues of the mean-squarestability matrix of the system with diagonal noise given by

(4.5) d

(uv

)=

(−R−1 10 −R−1

)(uv

)dt+ σ

(1 00 1

)(uv

)dW (t)

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with mean-square stability matrix given by

⎛⎜⎜⎝−2/R+ σ2 1 1 0

0 −2/R+ σ2 0 10 0 −2/R+ σ2 10 0 0 −2/R+ σ2

⎞⎟⎟⎠

for σ = 0 in part (a) and σ = 0.0006 in part (b). This matrix has repeated eigenvalues−2/R+σ2, and the perturbation, acting along the deterministic flow, is far too smallto have any effect on stability. We see in parts (c) and (d) that the effect on thespectrum of the stability matrix is far more dramatic when a perturbation of thesame intensity acts orthogonally, as in (4.2). The interaction with drift nonnormalityis sufficient to move the dominant eigenvalue into the right half of the plane.

4.4. Alternative low-dimensional models with stochastic mixing. In-cluding a linear stochastic mixing term in the original 2-dimensional model systemproposed in [30] gives the linear SDE system

(4.6) d

(uv

)=

(−R−1 10 −2R−1

)(uv

)dt+ σ

(0 −11 0

)(uv

)dW (t).

Three-dimensional models are additionally proposed in Baggett and Trefethen [5].We take one example and introduce stochastic mixing, resulting in the linear SDEsystem

(4.7) d

⎛⎝uvw

⎞⎠ =

⎛⎝−1/R 1 0

0 −1/R 00 0 −1/R

⎞⎠

⎛⎝uvw

⎞⎠ dt+

⎛⎝0 −σ −σσ 0 −σσ σ 0

⎞⎠

⎛⎝uvw

⎞⎠ dW (t).

We now ask whether or not there is a power-law relationship between R and σ2

corresponding to a transition threshold of α = −3 for the alternative models given by(4.6) and (4.7). We investigate this question computationally, since the eigenvaluesof the mean-square stability matrices for (4.6) and (4.7) are not in a form that allowsus to derive a sharp analytic stability condition as given by Theorem 4.2.

A visual representation of a finite part of the stability region for each model maybe produced by computing the eigenvalues of the mean-square stability matrix over arange of values in the (R, σ)-plane. We present these for all three models of interest inFigure 6 for values of R over the interval [500, 2000]. The computed stability region for(4.2) is presented in part (a) and reproduced on a log-log scale in part (b). Similarly,the computed stability regions for (4.6) and (4.7) are presented in parts (c) and (e),again using log-log scales in parts (d) and (f), respectively.

All three log-log plots show stability regions with boundaries that may be closelyapproximated by straight lines—an indication of a power-law relationship between σ2

and R. The reciprocal of the slope of each line, computed as

α =log(σ2

2)− log(σ21)

log(R2)− log(R1)

for points (σ21 , R1) and (σ2

2 , R2) chosen close to the start and endpoint of each linesegment, will correspond approximately to the transition threshold. We compute thisapproximation (denoted α) for each model in Table 1.

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(a) (b)

(c) (d)

(e) (f)

Fig. 6. Mean-square stability regions in the (σ2, R)-plane for the equilibria of (a) the 2-dimensional model (4.3); (c) the 2-dimensional model (4.6); and (e) the 3-dimensional model (4.7).Each plot is reproduced on a log-log scale in (b), (d), and (f), respectively.D

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Table 1

Estimated transition thresholds for (4.3), (4.6), and (4.7), using values from Figure 6.

Figure 6 α

(b) (σ21 , R1) = (2.7969 × 10−8, 501.868671806645)

(σ22 , R2) = (4.98× 10−10, 1994.966001150962) −2.914717927

(d) (σ21 , R1) = (1.1397 × 10−8, 501.874917391011)

(σ22 , R2) = (1.81× 10−10, 1994.775555880278) −3.002029594

(f) (σ21 , R1) = (2.7969 × 10−8, 501.868671806645)

(σ22 , R2) = (4.98× 10−10, 1994.966001150962) −2.914717927

These computations indicate that the transition thresholds for (4.6) and (4.7) maybe the same in each case as for (4.3), and hence we make the following conjecture.

Conjecture 4.4. The transition thresholds of the models defined in (4.6) and(4.7) are α = −3 in each case.

We finish by recommending that stochastic mixing should be further investigatedas a possible mechanism by which a loss of stability may be induced in low-dimensionalmodels of the transition to turbulence in plane Couette flow.

5. Application: Noise assisted high-gain stabilization in mean-square.

5.1. Background. Crauel, Matsikis, and Townley [11] demonstrated that theequilibrium of a linear feedback control system may be stabilized via a high-gainparameter, even if the deterministic dynamics are unstable, as long as the system issubject to skew symmetric stochastic perturbation of sufficient intensity, where theintegral is interpreted in the Stratonovich sense.

Specifically, they investigated the dependence of the mean-square asymptotic sta-bility of the equilibrium of the Stratonovich SDE

dX(t) =

(a− k bc d

)X(t) dt+

(0 −σσ 0

)X(t) ◦ dW (t)

on the high-gain parameter k, finding that, in the mean-square sense, noise can en-hance stabilization by proportional feedback. This was achieved by developing anasymptotic formula for exponential growth rates, valid for large values of k.

Their strategy was an application of Lyapunov theory to solutions of the equiva-lent Ito SDE

(5.1) dX(t) =

(a− k − 1

2σ2 b

c d− 12σ

2

)X(t) dt+ σ

(0 −11 0

)X(t) dW (t).

The relevant result may be stated as follows.Proposition 5.1. Let (X(t,X0))t≥0 be the strong solution of (5.1). Then

gk,σ(2) := limt→∞

1

tlogE sup

‖X0‖=1

‖X(t;X0)‖2 = 2d− σ2 +O(k−1),

and therefore limk→∞ gk,σ(2) = 2d− σ2.Notice that the action of the noise intensity σ in (5.1) is twofold: a stabilizing effect

through the drift coefficient and a destabilizing effect through the diffusion coefficient,which (as we saw in section 4.3) is potentially amplified by any nonnormality in thedrift. The statement of Proposition 5.1 indicates that the stabilizing effect through

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the drift will dominate for large k regardless of the nonnormality of the unperturbedsystem but does not speak to what happens when k is small. In the remainder of thesection, we seek to fill this gap by investigating how the competition of the two effectsinfluence mean-square asymptotic stability for small values of k and the role playedtherein by the nonnormality of the drift.

5.2. Mean-square stability analysis for small finite k. To separate theaction of the perturbation through the drift from that through the diffusion of (5.1)when the unperturbed system is nonnormal, we rewrite with c = 0 as

(5.2) dX(t) =

(a− k − 1

2ξ2 b

0 d− 12ξ

2

)X(t) dt+ σ

(0 −11 0

)X(t) dW (t),

which has stability matrix given by

(5.3)

⎛⎜⎜⎝2(a− k)− ξ2 b b σ2

0 a− k + d− ξ2 −σ2 b0 −σ2 a− k + d− ξ2 bσ2 0 0 2d− ξ2

⎞⎟⎟⎠ .

The eigenvalues of (5.3) are long, but computable, and may be represented as

a− k + d− ξ2 + σ2,

a− k + d− ξ2 − 1

3σ2 +

1

3

3

√f1(σ2) + 3

√f2(σ2)− 3

f3(σ2)

3

√f1(σ2) + 3

√f2(σ2)

,

a− k + d− ξ2 − 1

3σ2 − 1

6

3

√f1(σ2) + 3

√f2(σ2) +

3

2

f3(σ2)

3

√f1(σ2) + 3

√f2(σ2)

+ i

√3

2

⎛⎝1

3

3

√f1(σ2) + 3

√f2(σ2) + 3

f3(σ2)

3

√f1(σ2) + 3

√f2(σ2)

⎞⎠ ,

where f1, f2, and f3 are polynomials in σ2 with leading order terms 8σ6, 48b2σ8, and− 4

9σ4, respectively; all other coefficients depend on one or more of a, b, d, and k, but

not ξ2.This gives us some insight into the dual role of stochastic perturbation in stabiliza-

tion. The action through the drift (represented by ξ) is stabilizing, since −ξ2 appearsin the real part of each eigenvalue. The action through the diffusion (representedby σ) is more complex, though we can immediately see that any term that includesthe expression bcσd in the polynomial expansion of the real part of each eigenvaluemust have d < 2, and therefore sufficiently intense noise will have stabilizing effectsthrough the diagonal part of the drift that dominate any potentially destabilizingeffects arising from interaction of the diffusion with the nonnormal part of the drift.

We illustrate this in Figures 7 and 8, which show regions of mean-square asymp-totic stability for increasing values of k in the (d, σ2)-plane for special cases of (5.1)with a = 10 and b = 20 and 35, respectively. Working with the 2-dimensional system(5.1) allows us to plot precise stability regions. Consider each choice of k in turn:k = 9. Here, a−k = 1 > 0, and we see that, while the unperturbed system must have

an unstable equilibrium, stabilization in mean-square is possible for d < 0 aslong as the intensity of stochastic input σ2 is sufficiently large (a value thatincreases with b).

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k = 9 k = 15

k = 25 k = 100

Fig. 7. Mean-square stability regions in the (d, σ2)-plane for the equilibrium solution of (5.1)with a = 10, b = 20, and increasing k.

k = 15. Here, a − k = −5 < 0. When d < 0 we see that, when there is little or nostochastic input, or when d is large and negative, the system is asymptoticallystable. However, there is a range of values of d < 0 where increasing σ2

beyond a threshold induces instability, as the interaction of the stochasticperturbation with the nonnormality of the drift acts to destabilize. However,if the intensity of the stochastic perturbation increases further, then its actionthrough the drift reasserts dominance, and the equilibrium reverts to mean-square asymptotic stability. Increasing b emphasizes the phenomenon.

k ≥ 25. As a − k grows increasingly negative, the range of values of d for which thecompeting effects of the stochastic perturbation are visible becomes narrower,and the boundary between the regions of stability and instability rotatesclockwise around the origin and straightens out. By k = 100 the boundaryis approximately linear and clearly approaching the line 2d − σ2 = 0, aspredicted by Proposition 5.1. Increasing b, and hence the nonnormality of

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k = 9 k = 15

k = 25 k = 100

Fig. 8. Mean-square stability regions in the (d, σ2)-plane for the equilibrium solution of (5.1)with a = 10, b = 35, and increasing k.

the unperturbed system, appears to slow this process down; when d > 0, kand/or σ2 need to be larger before stabilization is possible.

6. Final remarks. In this article, we have considered three specific applicationswhere a nonnormal drift coefficient interacts with the geometry of a state-dependentstochastic perturbation. Our analysis highlights the potential importance of suchinteractions in stochastic modeling and demonstrates the utility of the mean-squarestability matrix technique for investigating stability and transience.

In future work, we hope to further investigate the effect of drift-diffusion interac-tion on transient response. We particularly refer to the presentation of Kreiss matrixbounds by Townley et al. [31] as alternatives to reactivity and amplification envelopeas measures of transient response in ecological models. It may also be productive toexplore potential links to the theory of pseudospectra described in [29].

We focus exclusively here on mean-square (rather than a.s.) asymptotic and tran-sient equilibrium dynamics. There is a large literature on stochastic stabilization and

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destabilization in the a.s. sense: a comprehensive review is contained in the introduc-tion to the paper by Appleby, Mao, and Rodkina [2], and we refer the reader to [3]for a more recent article where a stochastic control is used to stabilize the laminarflow equilibrium in the linearized Navier–Stokes equations. However, the emphasis ison analysis of top Lyapunov exponents, describing asymptotic rather than transientdynamics.

Moreover, we are not aware of any investigation that examines the effect of non-normal drift-diffusion interaction on a.s. equilibrium dynamics for model systems suchas those considered in this article. For example, the paper by Crauel et al. [11] (fromwhich we obtain the linear feedback control system considered in section 5) confirmsthat stabilization can be induced both in the mean-square and a.s. senses. However,just as in mean-square, the a.s. result is provided in the limit as the high-gain feed-back parameter k tends to infinity, and therefore the a.s. results in that paper do notspeak to the situation where k is small and the drift is nonnormal. We highlight thisgap for future research.

Acknowledgments. The authors would like to thank Professors Nick Trefethenand Stuart Townley for their comments on early drafts of this article. We addition-ally wish to thank the editor and anonymous referees for their careful reading andconstructive feedback, which has considerably improved the finished article.

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asymptotic and transient mean-square properties of stochastic systems arising in ecology, fluid...

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