Stationary moments, diffusion limits, and extinction times for logistic growthwith random catastrophes

Brandon H. Schlomann∗

Department of Physics and Institute of Molecular Biology,University of Oregon, Eugene, Oregon, 97405

(Dated: September 28, 2018)

A central problem in population ecology is understanding the consequences of stochastic fluctu-ations. Analytically tractable models with Gaussian driving noise have led to important, generalinsights, but they fail to capture rare, catastrophic events, which are increasingly observed at scalesranging from global fisheries to intestinal microbiota. Due to mathematical challenges, growthprocesses with random catastrophes are less well characterized and it remains unclear how theirconsequences differ from those of Gaussian processes. In the face of a changing climate and pre-dicted increases in ecological catastrophes, as well as increased interest in harnessing microbes fortherapeutics, these processes have never been more relevant. To better understand them, I revisithere a differential equation model of logistic growth coupled to density-independent catastrophesthat arrive as a Poisson process, and derive new analytic results that reveal its statistical structure.First, I derive exact expressions for the model’s stationary moments, revealing a single effectivecatastrophe parameter that largely controls low order statistics. Then, I use weak convergencetheorems to construct its Gaussian analog in a limit of frequent, small catastrophes, keeping thestationary population mean constant for normalization. Numerically computing statistics alongthis limit shows how they transform as the dynamics shifts from catastrophes to diffusions, enablingquantitative comparisons. For example, the mean time to extinction increases monotonically by or-ders of magnitude, demonstrating significantly higher extinction risk under catastrophes than underdiffusions. Together, these results provide insight into a wide range of stochastic dynamical systemsimportant for ecology and conservation.

I. INTRODUCTION

Stochastic fluctuations are important drivers ofecological and evolutionary processes [1–4]. Under-standing their consequences is essential for ecologicalmanagement, as well as for explaining observed pat-terns of biodiversity [2]. Given that data is oftenlimited, general principles of stochastic populationdynamics derived from the mathematical analysis ofminimal models can be immensely useful [2, 5]. Forexample, in classic work [6] Beddington and Mayderive for a stochastic logistic growth model howharvesting yields become less predictable as harvest-ing rates increase, a phenomenon that was suggestedby historical fisheries data at the time [7]. Exten-sions of this analysis have led to threshold harvestingstrategies that are proven optimal for a wide classof stochastic growth models that include extinction[8]. Beyond harvesting theory, analytically tractablemodels have led to diverse ecological and evolution-ary insights [1, 3, 9].

In these types of analyses, stochasticity is oftenmodeled by coupling growth to a Gaussian noise pro-cess, leading to stochastic differential equations thatare amenable to well established tools from diffusiontheory [2, 10]. However, large, abrupt catastrophes

∗ bschloma@uoregon.edu

are not captured by Gaussian models and are bet-ter modeled by discontinuous stochastic processes.These catastrophes are increasingly observed in a va-riety of ecological systems. On global scales, ecolog-ical catastrophes have already been observed as theresult of rapid warming and are expected to becomemore frequent as the climate continues to change[11]. At the opposite extreme, the intestinal micro-biomes of humans and other animals are observedto undergo abrupt compositional changes followingperturbations, such as antibiotic treatments [12–15].At all scales, efforts to understand and manipulateecological systems would greatly benefit from gen-eral, quantitative principles of how perturbationsand catastrophes shape population statistics.

I address this issue here by analytically and nu-merically studying a single-species model of logis-tic growth coupled to discontinuous, multiplicativejumps that arrive as a Poisson process, introducedin [16] and referred to here as the Logistic RandomCatastrophe (LRC) model (Figure 1A). Using themethod of moment equations [17], I derive exact ex-pressions for the stationary moments of the popu-lation distribution, neglecting the possibility of ex-tinction. These results provide a direct look into thestatistical structure of the LRC model, revealing asingle, effective catastrophe parameter that largelycontrols ensemble statistics. This effective param-eter was recently observed empirically in computersimulations and aided the analysis of experimental

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707.

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PE]

7 D

ec 2

017

2

rt

Xt/K

rt

Xt/K

FIG. 1. Sample paths of LRC and LES models. A: A sample path from the LRC model. Simulation parameters:r = 1, K = 104, λ = .07, f = 10−2, dt = .01. B: A sample path from the LES model. Simulation parameters: r = 1,K = 104, σ = .53, dt = .01.

data, but there was no theoretical basis for its exis-tence [15].

With this insight, I then turn to an old and funda-mental problem: which dynamics, intermittent ran-dom catastrophes or continuous stochasticity, posesa higher risk of extinction? For models of exponen-tial growth up to a hard wall carrying capacity inthe presence of either multiplicative Gaussian noise,called environmental stochasticity, or random, mul-tiplicative Poisson catastrophes, Lande [18] deriveshow the mean time to extinction scales as a powerlaw in the carrying capacity for positive long-rungrowth rate, with the exponent depending on thedetails of the particular model. This similarity inscaling behavior implies similar extinction risk in aqualitative sense, but it remains unclear how to con-struct a meaningful quantitative comparison, sincethe noise parameters of the two models describe dis-tinct processes.

To circumvent this issue, I propose a method thattreats the models not as distinct processes, but asextreme versions of the same process. Using func-tional generalizations of the Central Limit Theo-rems [19] and drawing inspiration from renormal-ization methods in theoretical physics [20, 21], I an-alytically construct the diffusion analog of the LRCmodel, referred to here as the Logistic Environmen-tal Stochasticity model (Figure 1B), in the limit ofinfinitely frequent, infinitesimal catastrophes, suchthat the stationary mean of the process remains con-stant. In this way, the problem of quantitativelycomparing two distinct models is traded for the morestraightforward problem of computing statistics ofone model as a function of parameters, specifically,

along a particular limit in parameter space. I applythis method to the comparison of extinction timesand find that the mean time to extinction increasesmonotonically along this limit by orders of magni-tude in a wide region of parameter space, implyingsignificantly higher risk of extinction under randomcatastrophes dynamics in general.

Taken together, these results highlight the powerof analytically tractable models of stochastic popula-tion dynamics. The expressions derived here aid theanalysis of experimental and observational data, in-form the design of computer simulations, and revealdeep connections between distinct stochastic pro-cesses relevant for a wide range of ecological systems.

II. BACKGROUND ON THE LOGISTICRANDOM CATASTROPHE MODEL

Hanson and Tuckwell [16] introduce an ideal min-imal model for the study of random catastrophesin isolation from additional complications: Single-species logistic growth coupled to constant fractioncatastrophes that arrive as a Poisson process, re-ferred to here as the Logistic Random Catastrophe(LRC) model. The LRC model can be written an-alytically as an Ito Stochastic Differential Equation(SDE):

dXt = rXt

(1− Xt

K

)dt− (1− f)Xt−dNt. (1)

The first term on the right hand side, of order dt,encodes deterministic logistic growth with growthrate r and carrying capacity K. The second term

3

rt

FIG. 2. Analytic results reveal statistical structure of LRC model. A: Analytic results for stationarycumulants agree with numerical simulations. Time evolution of the first 4 cumulants, Cn, of the LRC model,computed numerically (solid lines). Dashed lines indicate the asymptotic values predicted by the analytic results,with the cumulants computed from the moments given by equation (2). Parameters: r = 1, K = 104, λ = .1,f = .0012, dt = .01, Ntrials = 5 ·105. B: Range of validity of λ ln f as an effective catastrophe parameter. Parameterswere scaled according to λ′ = βλ and ln f ′ = β−1 ln f by dimensionless scale factor β. Dashed lines are analyticresults for first 4 stationary cumulants as a function of β. Parameters same as in A.

encodes random catastrophes with the use of a dif-ferential Poisson process, dNt, which is equal to oneif a catastrophe happens at time t and zero other-wise. Poisson catastrophes arrive with a constantprobability per unit time, λ, and have a size setby f , the fraction of the population remaining af-ter catastrophe. The notation Xt−dNt indicates theIto integration convention [22]. By including logisticgrowth, the LRC model captures realistic density-dependent regulation; by including catastrophes ofconstant fraction, it captures the realistic featurethat larger populations can experience larger losses,assuming that all individuals are equally suscepti-ble to the disturbance. Despite its simplicity, muchabout the statistical structure of the LRC model re-mains mysterious, due to the combined complica-tions of the discontinuous Poisson process and non-linear logistic growth.

III. RESULTS

A. Deriving exact expressions for LRCstationary moments

I present here exact results for the stationary mo-ments of the LRC model in absence of extinction,derived with the method of moment equations. Themethod of moment equations turns a stochastic dif-ferential equation into an deterministic differential

equation for the moment in question by averaging.For nonlinear SDEs this results in a hierarchy ofmoment equations, in which each moment is cou-pled to higher moments, that generally cannot besolved exactly. However, in the absence of extinc-tion, this hierarchy reduces in the steady state to analgebraic recursion relation, which in the case of theLRC model is a simple relation between E[Xn+1]and E[Xn] (Appendix A). This recursion relationcan be iterated to express each moment just as afunction of the mean. Computing the mean indepen-dently (Appendix A) therefore determines all sta-tionary moments:

E[Xn] = Kn

(1 +

λ

rln f

) n−1∏m=1

(1− λ(1− fm)

mr

),

(2)from which expressions for the stationary mean andvariance are readily obtained,

E[X]LRC = K

(1 +

λ

rln f

), (3)

Var[X]LRC = K2λ

r(− ln f − (1− f))

(1 +

λ

rln f

)(4)

(recall that f ∈ (0, 1), so ln f is negative for f < 1).These results agree well with simulations, as

shown in Figure 2A in the form of cumulants [23],

4

which generally provide more intuitive informationthan moments. The solid lines show the time evolu-tion of the first 4 cumulants, Cn, of the LRC model,computed via stochastic simulation of the Poissonprocess with no absorbing state representing extinc-tion (Materials and Methods). The dashed lines arethe analytic results, computed from the expressionsfor the moments in equation (2) [23]. Each cumulantasymptotes to the analytic value.

B. A single, effective catastrophe parameterlargely controls LRC moments

These analytic results suggest that the parametercombination λ ln f plays an important role in deter-mining population statistics. To investigate its role,I computed the response of the first four stationarycumulants of the LRC model to simultaneous, recip-rocal scaling of λ and ln f via a dimensionless scalefactor, β. In regions of parameter space where statis-tics depend only on the effective parameter λ ln f ,curves of cumulants as a function of β will be flat.The results are shown Figure 2B for β ranging from10−2 − 103, with catastrophe parameters scaled asλ′ = βλ and ln f ′ = β−1 ln f . Stationary momentswere computed for each value of β using the ana-lytic results derived above and converted to cumu-lants [23]. The stationary mean is invariant underthis scaling, as indicated by equation (3). Higherorder cumulants are approximately invariant for lowvalues of β, which correspond to rare, large catastro-phes, but decay to zero for large values of β, whichcorrespond to frequent, small catastrophes.

The large β limit is a dynamic analog to the lawof large numbers, in which the Poisson process thatdrives the LRC model tends to its average value.Recall that for a Poisson process, all cumulants areequal to the mean, λt, just as all cumulants of aPoisson distribution are equal to the mean. The nth

cumulant of the scaled process (1−f)Nt is therefore(1 − f)nλt. The limit β → ∞ corresponds to λ′ →∞ and f ′ → 1, such that λ′ ln f ′ is constant. Inthis limit, −(1 − f) is well approximated by ln f ,so, higher cumulants of the Poisson process decay asβ−(n−1), resulting in a deterministic model.

The effective parameter λ ln f has an intuitiveinterpretation: It is the correction to the long-rungrowth rate due to catastrophes, and has been pre-viously identified as an important quantity in a va-riety of related models [18, 22, 24]. Its existence hasimportant consequences for analyzing experimentaldata. As was done in [15], fitting ensemble statis-tics with the effective catastrophe parameter reducesthe number of parameters that needs to be estimatedfrom data. In fact, attempting instead to fit both the

rate (λ) and size (f) independently results in highlyunconstrained parameter estimates [15] and shouldbe avoided. The analytic results derived here putthe use of the effective parameter, λ ln f , on firmerground and explicitly delineate the range of its va-lidity.

C. The diffusion limit shows that randomcatastrophes pose higher extinction risk than

environmental stochasticity

I now consider the problem of quantitatively com-paring extinction risks in the LRC model and itsenvironmental stochasticity analog, referred to hereas the Logistic Environmental Stochasticity (LES)model. The LES model can be written as an SDE,

dXt = rXt

(1− Xt

K

)+ σXt−dBt, (5)

with Bt the standard Brownian motion process[10, 22], whose intervals are independent, Gaussiandistributed variables with E[Bt] = 0 and Var[Bt] =t, and σ setting the strength of the noise. Histor-ically, there has been no obvious way of quantita-tively comparing extinction risk between the twomodels across parameter space, since the noise pa-rameter σ and the catastrophe parameters, a rateλ and size f , describe distinct, model-specific pro-cesses [2, 18, 25].

To circumvent this issue, I propose an approach inwhich the LES model is viewed not as a distinct pro-cess, but as a special case of the LRC model. Thisnotion has been expressed qualitatively for decades[18, 25], but, to my knowledge, has never been madeexplicit. This can be done using a functional gener-alization of the Central Limit Theorem (CLT) [19],which says that fluctuations of the Poisson processabout its mean converge in distribution to Brow-nian motion in the limit of infinite jump rate andinfinitesimal jump size. In Appendix B, it is shownthat the relevant limits are

ln f(Nt − λt)λ→∞, f→1−−−−−−−→

√λ ln fBt, (6)

such that√λ ln f is constant. Consequently, the

mean drift of the scaled Poisson process diverges as√λ, the variance remains finite and all higher cu-

mulants go to zero. These are functional analogsto what happens when a Poisson distribution lim-its to a Gaussian in the classical CLT. In this case,the diverging drift - which is proportional to effec-tive catastrophe parameter λ ln f discussed above -is a manifestation of the fact that catastrophes areunidirectional, whereas noise in the LES model is

5

FIG. 3. The diffusion limit and extinction times. A: Smoothly transforming the stationary distribution ofLRC model to that of LES model in the diffusion limit. LRC model (no extinction) was simulated for Tmax = 300units of inverse growth rate for 5 values of the scale parameter α, rescaling parameters according to equation (7).Frames i-v depict the stationary distribution of logX (for visual clarity) for α = 1, 2.94, 8.66, 25.49, 75.00 respectively,estimated from 106 paths. Frame vi depicts the stationary distribution of logX for the target LES model. Parameters:r = .68, K = 6800, λ = .1, σ = .8, ln f = −σ/

√λ, dt = .01. B: Mean time to extinction increases as the LRC

model is morphed into the LES model (bottom), despite the stationary mean remaining constant throughout thetransformation (top). Parameters: Same as in A but α ranges logarithmically from 1 to 500 and σ = 1. C: Meantimes to extinction for the beginning (purple circles) and end (green squares) points of the diffusion limit as a functionof carrying capacity. Parameters: r = 1, λ = .1, f = .01, Ntrials = 5000, x0 = 10, x∗ = 1. For LES endpoints,σ2 = λ ln2 f . D: . The exponent of this power law, obtained by linear regression, as a function of effective noisestrengths. Parameters: r = 1, f = .01, Ntrials = 5000, λ is varied from .06 to .4. For each value of λ, τ vs K iscomputed for 10 values of K ranging from 100 to 10000. The last 5 values are used to compute ν. For the LRCmodel, the x-axis corresponds to σ2 = λ ln2 f . The exponent appears to asymptotically follow a power law in σ2,consistent with [18]

bidirectional. To obtain a non-trivial limiting pro-cess, this drift must be subtracted off manually be-fore taking limits. This subtraction can be absorbedinto a rescaling of the growth rate and carrying ca-pacity, similar to renormalization methods in theo-retical physics [20], such that the final transforma-

tion from the LRC model to the LES model involvesrescaling all four LRC model parameters.

The complete transformation from the LRC toLES model will be parameterized by a dimensionlessscale parameter, α. The prescription is as follows.Start from an LRC model together with a target

6

LES noise strength σ and fix λ ln2 f = σ2. Thentransform the LRC parameters according to

λ′ = αλ, ln f ′ = −σ/√λ′,

r′ = r

(1− λ′ ln f ′

r

), K ′ = K

(1− λ′ ln f ′

r

).

(7)It is shown analytically in Appendix B that in thelimit α → ∞, the LRC model LRC(r′,K ′, λ′, f ′)gets mapped to an LES model LES(rES ,KES , σ),with σ = λ ln2 f , rES = r(1 + (2r)−1σ2), andKES = K(1 + (2r)−1σ2). In this way, the station-ary mean of the process without extinction remainsconstant throughout this transformation, fixed atK. This is chosen as a convenient way to normal-ize the effects of noise. By adding constant off-sets, the transformation can be tuned to preserveother properties (Appendix B). This transformationis shown visually in Figure 3A, which depicts nu-merical results for the stationary distribution of theLRC model (in log variables for visual reasons) be-ing transformed with α increasing on the interval(1, 75) (Materials and Methods). The distributionapproaches that of the target LES model, shown ingreen in panel (vi) .

With this transformation, the question of rela-tive risks of extinction under the LRC model andthe LES model was revisited. The mean time to ex-tinction, τ , was computed via stochastic simulationof the LRC model for various values of α (Figure3B, bottom), rescaling LRC model parameters ac-cording to equation (7) for each α (Materials andMethods). The LRC model extinction time (purplecircles) increases with increasing α and asymptotesto the LES model extinction time (green square).Computed numerically, the stationary populationmean in the absence of extinction does indeed re-main constant throughout the transformation (Fig-ure 3B, top). The conclusion is again that thereexists a significantly higher risk of extinction underrandom catastrophe dynamics than under environ-mental stochasticity dynamics.

This conclusion is robust across parameter space.Plotting the beginning and end points of the curvein Figure 3B for various values of carrying capac-ity reproduces the asymptotic power law behaviordescribed by Lande for simpler models [18] (Figure3C), though the exponents obtained by linear fitting(Materials and Methods) are smaller for the LRCmodel across a wide range of effective noise strengths(Figure 3D). Note that because the growth rate isrescaled in this procedure, as long as the originalgrowth rate r is positive, the long-run growth rate[18] is positive for all values of σ. The conclusion isalso insensitive to the initial starting population, x0,

as the mean time to extinction becomes independentof x0 above a critical threshold (see [2, 16] and Sup-plementary Figure 1). In addition to this methodbased on the diffusion limit, an alternative approach,in which the stationary means of the LRC and LESmodels equated simply by mapping σ2 = −2λ ln f ,leads to the same conclusion (Supplementary Figure2, Appendix C).

IV. DISCUSSION

This work presented new results for the Logis-tic Random Catastrophe (LRC) model, a modelthat serves both as a foundation for understandingthe ecological consequences of random catastrophesand as an empirical model that describes real data[15, 16]. Exact analytic results for its stationarymoments were derived using the method of momentequations. These expressions revealed that ensemblestatistics are largely controlled by a single parame-ter that combines the average catastrophe rate andsize, which is both a fundamental insight into themodel’s statistical structure and a useful result forthe analysis of ecological data [15]. They also re-vealed the similarity in structure between the LRCmodel and its Gaussian noise counterpart, the Logis-tic Environmental Stochasticity (LES) model, whichwas exploited to construct the latter as a limit ofthe former. The mean time to extinction increasedmonotonically along this limit by orders of magni-tude in relevant regions of parameter space, indi-cating higher extinction risk under random catas-trophe dynamics in general. This has implicationsfor the prioritization of conservations efforts in theface of different types of stochasticity. In addition,given that large fluctuations appear to be intrinsicto intestinal microbiota [12, 13, 15], the enhancedextinction risk of reported here may be importantfor understanding the evolution of functional redun-dancy across symbiotic taxa and of host biochemicalnetworks that sense fluctuating microbial products.

The relationship between the LES model and theLRC model constructed here has multiple interpre-tations, which together provide useful intuition. Forone, the LES can be thought of as existing in asubset of the LRC parameter space, namely, as aparticular limit in the direction λ → ∞, r → ∞,K → ∞, f → 1. This is analogous to how expo-nential growth can be considered a special case oflogistic growth with infinite carrying capacity. Al-ternatively, if the Central Limit Theorem is inter-preted as dictating the fixed point of a coarse grain-ing procedure, in which case it becomes an exam-ple Renormalization Group methods from statisti-cal physics (albeit a simple one) [21], its applica-

7

tion to stochastic processes can be interpreted as astatement about iterative temporal coarse graining:any stochastic process with independent increments,zero mean, and finite variance resembles Brownianmotion when viewed on long time scales with appro-priate rescaling. As a result, the LRC model flowsto the LES model.

Computing statistics along this limit lead toquantitative insight into extinction risks. Thismethod is readily applied to the study of otherstatistics of the LRC model. It can also be easilyadapted to other Markov models, including multi-species models [19, 24]. The conditions for conver-gence are given in [19] and amount to reasonableboundedness conditions on the transition kernel ofthe Markov process. These generalizations allow formore computations, analogous to extinction times inFigure 3B, that could provide useful insight. For oneexample, it would be useful to revisit optimal con-trol problems relevant for ecological management inthe presence of random catastrophes, such as theharvesting strategies for fisheries considered in [26],and study how optimal policies evolve when dis-continuous jumps limit to continuous environmen-tal stochasticity. For another, evolutionary studiesof bet hedging in the presence of catastrophes [27]could be directly mapped to the analogous problemin the presence of continuous noise [3], connectingecological and evolutionary dynamics relevant for awide variety of systems.

V. MATERIALS AND METHODS

All code was written in MATLAB and is availableat https://github.com/bschloma/lrc.

A. LRC and LES model simulations

Sample paths of the differential Poisson processwere generated as Bernoulli trials [22]. These pathswere then used in the numerical integration of theLRC model. For all calculations except for thediffusion limit calculations in Figure 3, the logis-

tic growth equation was integrated with the Eulermethod between jump times, at which the popula-tion was reduced by a factor of f . The LES modelwas integrated with a straightforward application ofthe Milstein method [28].

B. The diffusion limit

In the diffusion limit, jump sizes approach the sizeof deterministic growth in one numerical timestep.So, the deterministic contribution of order ∆t mustbe retained, resulting in a more straightforwardEuler-type integration scheme. In this case, an adap-tive timestep is used, scaling ∆t′ = ∆t/

√α, identi-

cally to ln f , which sets the size of the jump. Thisscaling will lead to numerical artifacts when theprobability of catastrophe in one timestep, λ′∆t′,approaches unity. Since λ′ = αλ, this will occur atαc ∼ (λ∆t)−2, and so can be put off by starting witha sufficiently small time step.

C. Extinction times

Extinction times were computed by straightfor-ward stochastic simulation, following population tra-jectories from an initial population, x0 until theyreached the extinction threshold, x∗. To extract theexponent, ν, of the asymptotic relationship τ ∼ Kν ,a linear fit to log-transformed variables was done forthe larger half of the carrying capacity values, typi-cally 5 data points.

VI. ACKNOWLEDGEMENTS

I thank Raghuveer Parthasarthy, Pankaj Mehta,and David Levin for helpful feedback. Researchreported in this publication was supported by theNIH as follows: by the NIGMS under awardnumber P50GM098911 and by training grant T32GM007759.

Appendix A: Detailed calculation of stationary moments

In this section an expression for the nth stationary moment for the LRC model is derived. The approachis analogous for the LES model and since the results are already known [29], a detailed derivation isn’t given,though one remark is made on the application of this method to diffusion processes.

8

1. LRC model

Before beginning, the chain rule for jump processes [22] is stated without proof, for reference. Let Xt bya general process given by

dXt = f(Xt, t)dt+ h(Xt− , t−)dNt (A1)

with f and h deterministic functions, Nt a Poisson process with rate λ, and t− denoting the Ito conventionas in the main text. Further let Yt ≡ F (Xt, t) be a transformed process. Then Yt is governed by

dYt = (∂tF (Xt, t) + f(Xt, t)∂XtF (Xt, t)) dt+ ∆Y jumpt− dNt (A2)

with ∆Y jumpt− ≡ F (Xt− + h(Xt− , t−))− F (Xt− , t

−).Now recall the LRC model,

dXt = rXt

(1− Xt

K

)dt− (1− f)XtdNt. (A3)

The first step is to change variables to Xnt using the stochastic chain rule for jump SDEs. The result is

dXnt = nrXn

t

(1− Xt

K

)dt− (1− fn)Xn

t dNt. (A4)

Then, each term in this SDE is averaged. The expectation of Xnt−dNt can be factored: E[Xn

t−dNt] =E[Xn

t− ]E[dNt] = E[Xnt− ]λdt. Intuitively, this is because the two processes appear mutually independent. The

Poisson process has independent increments, and since the Ito convention was used, Xnt− is independent of

Nt, which occurs in the future. This is certainly true for a discrete time model, but care must be taken inthe continuous limit.

A more rigorous argument can be made using the Dominated Convergence Theorem. The case n = 1 isconsidered without loss of generality. Consider Xj , a discrete partition of the continuous time process Xt,such that Xj → Xt in probability. Then, sums of Xj converge in probability to integrals, in particular,∑

j

Xj−1∆Nj →∫T

Xt−dNt, (A5)

where ∆Nj is a partition of the Poisson process. The Dominated Convergence Theorem says that if Xt isdominated by an integrable function on the interval T ,

E

∑j

Xj−1∆Nm

→ E[∫

T

Xt−dNt

](A6)

in probability. Since populations in the LRC model are bounded by the carrying capacity for all time, thisis always valid. The expectation of the sum is straightforward, leading to the result,

E[∫

T

Xt−dNt

]=

∫T

E[Xt− ]λdt, (A7)

from which the infinitesimal version follows as a special case.Factoring the expectation results in an ODE for the nth moment. In the steady state, this becomes the

recursion relation

E[Xn+1] = K

(1− λ(1− fn)

nr

)E[Xn]. (A8)

Defining

cn ≡(

1− λ(1− fn)

nr

), (A9)

9

the nth moment can be expressed in terms of the mean as

E[Xn] = Kn−1

(n−1∏m=1

cm

)E[X]. (A10)

To complete the recursion relation, the mean must be computed independently. This is accomplished bychanging variables to lnXt using the chain rule for jump processes:

d lnXt = r

(1− Xt

K

)dt+ ln fdNt, (A11)

which in the steady state gives an expression for the stationary mean,

E[X] = K

(1 +

λ

rln f

). (A12)

Plugging this back into equation (A10) gives the final result

E[Xn] = Kn

(1 +

λ

rln f

) n−1∏m=1

(1− λ(1− fm)

mr

). (A13)

Evaluating this equation for n = 2 leads to the expression for the variance in the main text:

Var[X]LRC = K2λ

r(− ln f − (1− f))

(1 +

λ

rln f

). (A14)

2. LES model

The derivation is analogous for the LES model, except that Ito’s chain rule for diffusion processes is used.Since the results are already known [29], derived with traditional methods, a detailed computation will notbe given. However, one remark worth making concerns the expectation of Xt−dBt. The intuitive argumentoutlined for the LRC model - that since the Ito convention was employed the expectation of the product canbe factored - gives the correct answer in this case, but is in fact not generally valid. Essentially, for processesgoverned by equations of the form

dXt = f(Xt, t)dt+ g(Xt− , t−)dBt, (A15)

the integral∫g(Xt− , t

−)dBt can acquire non-zero expectation if the function g grows too quickly. A classicexample is the CEV model of quantitative finance [30], which is of the form f(Xt, t) = Xt and g(Xt, t) = Xγ

t

for γ > 1. However, one can use the fact that the exponential version of the LES model, i.e. K → ∞, isa well known SDE for which E

[∫Xt−dBt

]= 0. This model is known as Geometric Brownian Motion and

describes asset prices in the Black-Scholes model of quantitative finance [30]. Since paths of the exponentialmodel almost surely dominate paths of the LES model,

∫Xt−dBt for the LES model inherits the martingale

property from the exponential case, which implies zero expectation.Following the same procedure as for the LRC model, factoring expectations of Xn

t−dBt, results in

E[Xn]LES = Kn

(1− σ2

2r

) n−1∏m=1

(1 +

(m− 1)σ2

2r

). (A16)

Special cases of this include

E[X]LES = K

(1− σ2

2r

)(A17)

and

Var[X]LES =K2σ2

2r

(1− σ2

2r

). (A18)

10

Appendix B: The diffusion limit and the Central Limit Theorem

This section contains details of the construction of the LES model from the LRC model in the limit ofinfinitely frequent, infinitesimal catastrophes, referred to here as the diffusion limit. The complete trans-formation involves all four LRC model parameters and is specified as follows. Let α be a scale parameter,LRC(r,K, λ, f) an LRC model, and σ be the target noise-strength parameter of the limiting LES model.Fix λ ln2 f = σ2, and scale

λ′ = αλ, ln f ′ = −σ/√λ′

r′ = r

(1− λ′ ln f ′

r

)= r

(1 +

σ√λ

r

√α

),

K ′ = K

(1− λ′ ln f ′

r

)= K

(1 +

σ√λ

r

√α

). (B1)

The claim is that in taking the limit α → ∞, the LRC model LRC(r′,K ′, λ′, f ′) gets mapped to an LESmodel LES(rES ,KES , σ), with σ2 = λ ln2 f , rES = r(1 + (2r)−1σ2), and KES = K(1 + (2r)−1σ2), suchthat the stationary means of both models are equal. I first motivate the form of this transformation, whichinvolves all four LRC model parameters, by studying the behavior of the stationary moments. I then showhow the precise form of these limits, namely λ → ∞, f → 1, such that λ ln2 f → const., follows fromfunctional generalizations of the Central Limit Theorem (CLT), in which a scaled, compensated Poissonprocess limits to Brownian motion. Finally, I show analytically how the full transformation maps the LRCmodel into the LES model.

1. Motivation

As discussed in the main text, taking the limits λ→∞, f → 1, such that λ ln f → const. is analogous tothe law of large numbers, leading to a deterministic limit. The correct limits instead are λ→∞, f → 1, suchthat λ ln2 f → const., which I show below is analogous to the CLT. To motivate the final four parametertransformation, let us first consider the behavior of the LRC variance under these limits:

Var[X] = K2λ

r(− ln f − (1− f))

(1 +

λ

rln f

)limits−−−→ K2λ ln2 f

2r

(1 +

λ

rln f

)= K2 c

2

2r−K2 c

3

r2

√λ. (B2)

with c = const = −√λ ln f . In taking the limit f → 1, the relation (− ln f − (1− f))→ 2−1 ln2 f was used,

based on a 2nd order Taylor expansion.The variance diverges, but a part of it remains finite. The finite piece of the variance in this limit is exactly

the variance of an LES model with σ2 = λ ln2 f and increased growth parameters KES = K(1 + (2r)−1σ2),and rES = r(1 + (2r)−1σ2). Looking at the behavior of the mean in this limit leads to the same conclusion.This suggests that this limit does take the LRC model into an LES model, but one that is accompaniedby a noise-induced drift that diverges as

√λ. This is divergence should be expected, as it reflects the

unidirectionality of jumps in the LRC model, which is absent in the LES, analogous to the divergence of themean of a Poisson distribution when it limits to a Gaussian. To obtain a non-trivial limiting process, thisdrift needs to be subtracted off, for example, by adding a term −λ ln fdt to the LRC model SDE. This isequivalent to rescaling the growth rate and carrying capacity each by a factor of (1− r−1λ ln f), leading tothe full four parameter transformation.

11

2. Functional Central Limit Theorems

The form of the limits λ → ∞, f → 1, such that λ ln2 f → const., is a direct consequence of the CLT.The classical CLT says that given a set of n random variables, {ξj}, that are identically and independentlydistributed (i.i.d.) with mean µ and finite variance σ2, the sum of the deviations of these variables fromtheir mean, when rescaled by

√n, tends in distribution to a Gaussian variable as n→∞:

limn→∞

∑j ξj − nµ√

n= η ∼ N (0, σ2) (B3)

where N (µ, σ2) is a Gaussian distribution with mean µ and variance σ2. The condition that the variablesξj follow identical distributions can be relaxed, but we focus on this restricted case here.

A vast body of mathematical literature concerns the construction of generalization of the CLT to stochasticprocesses. One important generalization, which we will employ in the study of the Poisson process, is

Donsker’s theorem [19]. Donsker’s theorem dictates the limit of a sequence of stochastic process, X(n)t ,

constructed from sums of i.i.d. random variables ξj with zero mean and finite variance via

X(n)t ≡ 1√

n

[ns]∑j=1

ξj , ns ≡ t. (B4)

Here we have introduced time as multiples of a unit s, such that t = ns, and [...] denotes the integer part.

Donsker’s theorem says that as n → ∞ with s → 0 such that ns → t for arbitrary t, the processes X(n)t

converge in law to Brownian motion,

X(n)t → Bt. (B5)

Donsker’s theorem can be used to show the convergence of the compensated Poisson process to Brownianmotion in particular limits. The idea is to write the Poisson process as a sum of intervals which themselvesare i.i.d. random variables that meet the criteria for Donsker’s theorem, and then scale the jump size andrate in the ways that map onto the n → ∞ limit. This approach is based on a method known as finitedimensional convergence, which is only applicable to processes with independent increments [19].

Consider breaking a scaled, compensated Poisson process, εNt into a sum of finite intervals,

εNt = ε

[ns]∑j=1

∆Nj . (B6)

From inspection, we see that the appropriate mapping is λ→ nλ, ε = σλ−1/2, in which case

εNt = σ

[ns]∑j=1

∆Nj/√λ√

n. (B7)

Donsker’s theorem can then be applied with ξj ≡ ∆Nj/√λ, resulting in

εNtλ→∞, ε→0−−−−−−−→

√λεBt. (B8)

In the LRC model, collapse size is forced to zero by taking f → 1. This still leaves room for how exactlyλ and f should map on to σ. One choice would be to take ε = −(1− f), such that σ2 = λ(1− f)2. In thiscase, a quick calculation shows that the limiting process would be an LES model with unchanged growthparameters, (r,K), and consequently a reduced stationary mean of K(1 − (2r)−1σ2), whereas the originalLRC process, after removing the divergence of λ ln f , has a stationary mean of K. Alternatively, one couldtake ε = ln f , such that σ2 = λ ln2 f . This is the case examined above, which results in an LES model withan unchanged stationary mean, but altered growth parameters. Since our present goal is to normalize theeffect of noise to construct a fair comparison of extinction risk, the latter choice is more appropriate.

12

3. Convergence of LRC to LES

I now discuss the convergence of the LRC model to the LES model via the convergence of the Poissonprocess to Brownian motion discussed above. General conditions for the convergence of a pure jump Markovprocess to a diffusion are given in [19]. Rather than verify these general conditions here, I’ll take a more intu-itive approach that exploits the simplicity of the present models and uses the fact that both the LRC modeland the LES model possess unique, strong solutions, as follows from special cases of a general result derivedin [24]. This allows us to uniquely define a sequence of processes X ′t(α) ≡ F [Nt(α); r(α),K(α), λ(α), f(α)],where F is the solution to the LRC model depending on parameters r, K, λ, and f , and the limit of thesequence X∗t ≡ limα→∞X ′t(α). Existence and uniqueness of solutions to both models allows us in principleto take the limit and then invert the solution, recovering a diffusion SDE. In practice, we can take the limitdirectly in the context of the LRC model SDE. To begin, recall the LRC model,

dXt = rXt

(1− Xt

K

)dt− (1− f)Xt−dNt. (B9)

Let us expand (1− f) in powers of ln f to second order and write the Poisson process in terms of its meanand compensated process.

dXt = rXt

(1− Xt

K

)dt+

(ln f +

1

2ln2 f

)Xtλdt+

(ln f +

1

2ln2 f

)Xt−dNt +O(λ ln3 f) (B10)

Now let us absorb the mean drift of the Poisson process as scaling factors for the growth rate and carryingcapacity

dXt = r

(1 +

λ

rln f +

λ

2rln2 f

)Xt

(1 +

Xt

K(1− λ

r ln f + λ2r ln2 f

)) dt+(ln f +1

2ln2 f

)Xt−dNt+O(λ ln3 f).

(B11)Now we apply the transformation [19] with α finite and evaluate r′ in terms of r and K ′ in terms of K.

This has the effect of canceling all λ ln f terms, as intended.

dX ′t(α) = r

(1 +

λ′

2rln2 f ′

)X ′t

(1− X ′t

K(1 + λ′

2r ln2 f ′)) dt+(ln f ′ +

1

2ln2 f ′

)X ′t−dN

′t+O(λ ln3 f) (B12)

where primed variables depend on α. Before taking the α→∞ limit, we can identify λ′ ln2 f ′ as σ2, a finiteconstant independent of α,

dX ′t(α) = r

(1 +

σ2

2r

)X ′t

(1− X ′t

K(1 + σ2

2r

)) dt+

(ln f ′ +

1

2ln2 f ′

)X ′t−dN

′t +O(λ ln3 f). (B13)

We can now evaluate the α→∞ limit, knowing how Nt transforms: ln f ′dNt → σdBt in law, ln2 f ′dNt → 0,resulting in

limα→∞

dX ′t(α) = dX∗t = r

(1 +

σ2

2r

)X∗t

(1− X∗t

K(1 + σ2

2r

)) dt+ σX∗t−dBt. (B14)

The limiting process is an LES model with increased growth parameters rLES = r(1 + (2r)−1σ2) andKLES = K(1 + (2r)−1σ2). Comparing this model to the transformed LRC model of [31] using the analyticresults for the stationary mean equations (A12) and (A17) reveals that the two models do indeed have thesame stationary mean.

13

Appendix C: An alternative mapping that equates stationary means

The stationary means of the LRC and LES models can also be equated by using the same growth rate andcarrying capacities and mapping σ2 = −2λ ln f , as is clear from equations (A12) and (A17). This mappingprovides an alternative method of quantitatively comparing the two models, though one that is perhaps lessmeaningful than the diffusion limit approach. It can be understood intuitively by plotting the time evolutionof the mean population of both models in the presence and absence of extinction (Supplementary Figure2A). In the absence of extinction, both models asymptote to the same value. In the presence of extinction,the LRC model average decays to zero faster than the LES model average, indicating higher extinction risk.Computed directly, the mean times to extinction for the LRC model are significantly shorter than for theLES model (Supplementary Figure 2B), supporting the conclusions of the diffusion limit-based method.

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15

Supplementary Figure 1

100

101

102

0

2

4

6

Mean time to extinction is largely independent of initial starting population. Mean time toextinction, τ , plotted on a shifted log scale as a function of initial starting population, x0. Green squaresdenote the LES model, purple circles denote the LRC model. The mean extinction time, defined as the firsthitting time to x∗ = 1, starts from 0 but rapidly increases to a value independent of x0. Parameters: r = 1,K = 104, λ = .1, f = .01, Ntrials = 500.

16

Supplementary Figure 2

σ2/2r,rt

The LRC model has higher extinction risk than the LES model for equivalent stationarymeans. A: Illustration of the mapping. Numerical results for the average population plotted over timein the LRC (purple) and LES (green) models, showing both the cases of no extinction (dark solid lines)and extinction (light dashed lines) via an absorbing state at x∗ = 1. LES model has the same growth rateand carrying capacity as the LRC model and σ is determined by σ2 = −2λ ln f , such that the two modelshave equal stationary means (Appendix C). Parameters: r = 1, K = 104, λ = .1, f = .0012, σ = 1.16,dt = .01, Ntrials = 5 · 105. B: Mean time to extinction, τ , in units of inverse growth rate, for LRC (purplecircles) and LES (green squares) models as a function of noise strength, with σ2 = −2λ ln f . Parameters:r = 1, dt = .01, Ntrials = 5 · 103. For LRC model, f = .01 and λ was varied from .065 to .195. Inset: Meantime to extinction as a function of carrying capacity. Parameters: r = 1, λ = .13, f = .01, σ = 1.09, dt = .01,Ntrials = 5 · 103.