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Climate modelling with delay differential equations

Andrew Keane∗, Bernd Krauskopf and Claire M. PostlethwaiteDepartment of Mathematics, The University of Auckland,

Private Bag 92019, Auckland 1142, New Zealand

May 2017

Abstract

A fundamental challenge in mathematical modelling isto find a model that embodies the essential underly-ing physics of a system, while at the same time beingsimple enough to allow for mathematical analysis. De-lay differential equations (DDEs) can often assist in thisgoal because, in some cases, only the delayed effects ofcomplex processes need to be described and not the pro-cesses themselves. This is true for some climate systems,whose dynamics are driven in part by delayed feedbackloops associated with transport times of mass or en-ergy from one location of the globe to another. Theinfinite-dimensional nature of DDEs allows them to besufficiently complex to reproduce realistic dynamics ac-curately with a small number of variables and parame-ters.

In this paper we review how DDEs have been usedto model climate systems at a conceptual level. Moststudies of DDE climate models have focused on gaininginsights into either the global energy balance or into thefundamental workings of the El Niño Southern Oscilla-tion (ENSO) system. For example, studies of DDEs haveled to proposed mechanisms for the interannual oscilla-tions in sea-surface temperature that is characteristic ofENSO, for the irregular behaviour that makes ENSO dif-ficult to forecast and for the tendency of El Niño eventsto occur near Christmas. We also discuss the tools usedto analyse such DDE models. In particular, recent de-velopments of continuation software for DDEs make itpossible to explore large regions of parameter space inan efficient manner in order to provide a “global picture”of the possible dynamics. We also point out some direc-tions for future climate modelling that we believe couldimprove the descriptive power of DDE models.

1 Introduction

A delay in a physical system is the time required forsubsystems to communicate, process information and re-act. In a way, delays are always present, but in manycases of mathematical modelling, they are shown to be

∗corresponding author: [email protected]

“harmless” with regard to the dynamics of the model,as shown in [21]. On the other hand, there are manysituations where delays have a significant impact on thebehaviour of the model; generally, when the delays aresufficiently large relative to the intrinsic time scale ofthe system being considered. The effect of delays havebeen investigated in a wide range of physical systems;for example, in ecology [46], control theory [63], mod-els of genetic regulatory systems [18, 59], neural systems[9, 72], epidemics [52], coupled chemical oscillators [7]and laser systems [53, 47].

Large delays occur naturally in climate systems ona variety of time scales, mainly as a result of mass orenergy transport across the globe and/or throughoutthe atmosphere. Consider, for example, energy-balancemodels (EBMs), which focus on the global balance be-tween incoming and outgoing radiation on the Earth. Inthe paleoclimate context of investigating transitions be-tween glacial and interglacial states, the time scales ofinterest are 103–105 years [5]. There exists a fundamen-tal feedback loop in EBMs because the global tempera-ture of the Earth depends on the albedo of the Earth,which measures how much incoming solar radiation isreflected back into space. In turn, the albedo is relatedto the level of cloud and snow/ice cover, which dependson the global temperature. Yet, it can take a long timefor clouds or ice of a significant size on the global scaleto form or dissipate. Therefore, albedo values do notdepend on present temperatures alone, but on tempera-tures in the past; this can be modelled by a delay on theorder of 103–104 years.

Another important example where delays play a rolein climate variability, is the El Niño Southern Oscilla-tion (ENSO) system, which is a coupled climate sys-tem with an oceanic component (El Niño) and an at-mospheric component (Southern Oscillation). Figure 1provides evidence for this coupling. The blue curve inpanel (a) displays the average anomaly in sea-surfacetemperature (SST) in the eastern Pacific Ocean (NINO3index) and the red curve is the normalised surface airpressure difference between Tahiti and Darwin, Australia(Southern Oscillation Index; SOI), throughout the years1964–2014. Generally, the blue and red time series are inanti-phase synchronisation, so that high NINO3 indices

1

Climate modelling with delay differential equations 2

1970 1980 1990 2000 2010

-4

-2

0

2

4

time [years]

(a)

J A S O N D J F M A M J0

2

4

6

calendar month

freq

uen

cy

(b)

Figure 1: Panel (a) shows the monthly NINO3 index andSOI deviations from 1964–2014 as blue and red curves,respectively. Calendar month locations of warm events(peaks above 1°C) for the NINO3 index data of panel (a)are displayed in panel (b). The NINO3 data is fromNOAA and the SOI data is from the Climatic ResearchUnit, University of East Anglia.

coincide with low SOI and vice-versa. There are manyextrema in both time series of Fig. 1(a) on a small, intra-seasonal time-scale. However, it is only the larger peaksin the NINO3 index that represent El Niño events, thewarm phase of ENSO, while large drops represent thecool phase known as La Niña. These events tend to oc-cur every four to seven years with significant variabilityin strength. Interdecadal variability can also be seen inpanel (a). What is not so clear in the time series, yet iscommonly know, is that El Niño events generally occurat the same time of the year near Christmas. This hasbeen attributed to seasonal locking of the time series, asillustrated by the histogram in panel (b), which showsthe monthly positions of unusually warm events (above1°C) from the NINO3 data shown in panel (a). Clearly,there is a tendency for El Niño events to occur in bo-real winter. The annual cycle therefore represents theintrinsic time scale of the ENSO system.

According to the so-called delayed action oscillator de-scription of ENSO, which we discuss in detail in sec-tion 3, certain aspects of the observed SST fluctuationscan be attributed to the movement of oceanic wavesacross the Pacific Ocean. These waves form feedbackloops that provide delayed effects to the SST in the east-ern Pacific Ocean on the order of months.

Mathematically, delayed effects are described by delaydifferential equations (DDEs). The vast majority of pub-lished studies into DDE climate models assume that thedelays involved remain constant over time. Therefore,in our review of DDEs in climate modelling, we focus onDDEs with constant delays, which take the form:

ẏ(t) = f(y(t), y(t− τ1), y(t− τ2), . . . , y(t− τN ), t). (1)

Here y ∈ Rn is a state vector and there are N delay terms

— one for each feedback loop in the climate system andwith a delay τi associated with its underlying physicalprocesses; for example, the time for ice sheets to melt.

There is a well established theory of DDEs with a finitenumber of constant delays [20, 33, 78, 77]. In contrast toordinary differential equations (ODEs) with n variablesand the phase space Rn, the phase space of the DDEis C([−max(τi), 0];Rn)×R, where C([−max(τi), 0];Rn)is the infinite-dimensional space of continuous functionsover the delay interval with values in Rn and t ∈ Rrepresents time. Therefore, an initial condition for theDDE consists of a whole function segment over the timeinterval [−max(τi)+t0, t0], often referred to as an initialhistory.

The bifurcation theory for constant DDEs is analo-gous to that for ODEs, because the solutions to con-stant DDEs depend smoothly on the parameters andinitial history and the linearisation of equilibria and pe-riodic solutions have at most a finite number of unstableeigendirections [22, 33]. Centre manifold and normalform reductions can be applied to a constant DDE toyield an ODE that describes the dynamics locally neara bifurcation. Therefore, one encounters the same typesof bifurcations in DDEs as for ODEs. For example, itis known that in the case of negative delayed feedback,ẏ(t) = −g(x(t − τ)) for a large family of odd functionsg, the trivial zero solution undergoes a Hopf bifurcationfor a critical value of τ , creating a family of periodicsolutions of period 4τ [12, 14, 58].

In most cases of past literature on conceptual climatemodels in the form of DDEs numerical simulation isemployed to investigate their time-dependent solutions.Generally, existing methods for simulating ODEs can beadapted for DDEs; for example, popular software rou-tines include Matlab’s dde23 [74], dde solver [83], radar5[32] and the solver of xppaut [24].

Figure 2 shows example solutions of model (4) below;they were computed with the improved Euler methodand a constant initial history h(t) ≡ 0 to demonstratetypical types of dynamics discussed later. The modelcontains two delays, τp and τn, and is subject to peri-odic forcing, representing seasonal effects, with time tmeasured in years. The solutions are displayed (aftertransients have died down) as a time series, a projec-tion in the (h(t), h(t − τp), h(t − τn))-space, a strobo-scopic trace in the (h(t), h(t − τp), h(t − τn))-space anda logarithmic power spectrum obtained as the Fouriertransform of the time series over 300 years. The strobo-scopic traces are constructed according to [8, 48]: aftereach forcing period the first point of the solution, whichis a function segment, is plotted in projection onto the(h(t), h(t − τp), h(t − τn))-space. The forcing period inthis model is one year, so effectively, we plot h(t) when-ever t ∈ N.

Row (a) shows a periodic solution of period one. Thetime series shows periodic motion, which corresponds toa closed curve in the phase space projection. For peri-

Climate modelling with delay differential equations 3h(t)

log[a.u.]

(a)

h(t)

log[a.u.]

(b)

h(t)

log[a.u.]

(c)

h(t)

th(t− τn) h(t− τp

) h(t− τn) h(t− τp)

log[a.u.]

freq

(d)

Figure 2: Stable solutions found by numerical integration of model (4) for κ = 0.9 (a), κ = 1.2 (b), κ = 1.5 (c)and κ = 2 (d) represented as a time series (far left column), as a projection in the (h(t), h(t− τp), h(t− τn))-space(middle-left column), as a stroboscopic trace in the (h(t), h(t− τp), h(t− τn))-space (middle-right column) and as apower spectrum on a logarithmic scale in arbitrary units [a.u.] (far-right column). Other parameters are a = 2.02,b = 3.03, c = 2.6377, du = 2.0, dl = −0.4, τp = 0.0958 and τn = 0.4792.

odic solutions the number of points in its stroboscopictrace corresponds to the period of the solution, so thissolution is of period one. Finally, the power spectrumhas a single dominant peak at one year. Note that allpower spectra have a peak at one year due to the seasonalforcing. The solution shown in row (b) is quasiperiodic,which is an unlocked solution on a torus. A slight modu-lation of local peak heights is not so obvious in the timeseries; nevertheless, it is clear that the projection of thesolution in the (h(t), h(t − τp), h(t − τn))-space is not aclosed loop but rather a torus, which is filled densely bytrajectories. The closed loop formed by the points ofthe stroboscopic trace is further evidence of a quasiperi-odic solution. As such, the distinct peaks in the power

spectrum are incommensurate with the frequency of theseasonal forcing. The solution shown in row (c) is peri-odic, now with a period of four. The solution projectionin (h(t), h(t− τp), h(t− τn))-space is a closed curve andthere are four points in the stroboscopic trace. In row (d)the solution is chaotic. This is evident by the irregulartime series and the broad power spectrum, which con-tains contributions from all frequencies as is typical fora chaotic solution.

In the context of conceptual climate modelling, it isgenerally difficult to estimate or even physically inter-pret certain parameters, which may realistically be sub-ject to considerable fluctuations over time. Therefore, itis necessary to gain insight into the possible dynamics


of a model across a large range of its parameter space.However, it can be impractical to explore large param-eter ranges by numerical integration because one has todeal with many possible initial conditions, transients andmultistabilities. These factors can make it difficult tointerpret the model dynamics correctly, understand howtypical certain model behaviour is and how certain dy-namical features relate to different model components.

One way to overcome these challenges is to employcontinuation software, which numerically continues (ortracks) equilibria or periodic solutions while a parameteris varied. Such software can also calculate the stabilityof the solutions in order to identify codimension-one bi-furcations. These bifurcations can in turn be continuednumerically as curves in a two-dimensional parameterplane by fixing constraints on the stability of the solu-tions. Conducting a bifurcation analysis by such meansallows one to organise the parameter space into regionsof different solution types, providing a comprehensiveoverview of the dynamical capabilities of the model. Twoready-to-use continuation software packages for DDEsare DDE-Biftool [23, 75] and Knut (formerly, PDDE-Cont) [81]; the numerical methods implemented in bothpackages are reviewed in [67].

Modelling climate systems with DDEs has two impor-tant advantages. Firstly, it can improve the accuracy ofthe model. Accuracy of models is often an issue; see,for example, the review [51] on DDEs in the contextof engineering applications. An example from climatescience concerns the albedo discussed above. Earlier lit-erature assumed an instantaneous relationship betweenalbedo strength and global temperatures. Delays werepurposely introduced into existing models in order tomake them more realistic.

Secondly, a representation of the system by a DDEhas the potential to describe the behaviour of interestwith a much smaller number of variables and parame-ters, thus, offering a means of model reduction. Indeed,a system involving transportation phenomena can be de-scribed by partial differential equations (PDEs); differ-ences in intrinsic time scales can also be approximatedby ordinary differential equations (ODEs) in the form offast-slow systems [50]. The advantage of DDEs is thatthe explicit details of the transportation process itself isno longer required as part of the model. Consider, forexample, the ENSO system already mentioned above.In order to describe the time evolution of SST anomaliesin the eastern equatorial Pacific Ocean, it is sufficientto consider only the effects of the delayed feedbacks interms of their strengths and the length of the associateddelays. This constitutes a model reduction comparedto the full PDE for zonal wind surface anomaly, surfacepressure anomaly, momentum and thermal damping co-efficients, etc.

The purpose of this paper is to present a review of howDDEs have been used for conceptual modelling of cli-mate systems. We review the inclusion of delayed effect

in EBMs, palaeoclimatology models and ENSO modelsand discuss some results from relevant studies. As tobe expected from the conceptual level of modelling, theresults are of a phenomenological and fundamental na-ture. Most results in the existing literature are achievedby linear stability analysis of steady-state solutions, sim-ulation of time-dependent solutions and often their spec-tral analysis. More recent work has utilised state-of-the-art continuation software to develop a more detailed andcomprehensive picture of possible model behaviour in de-pendence on parameters, and we provide some examplesof such results. For future work, we suggest that con-tinuation methods provide an efficient means to analyseDDE climate models across appropriately large param-eter ranges. We also observe that a common modellingassumption is that the delays associated with feedbackeffects in climate models are generally constant, and sug-gest that non-constant delays (in particular, distributedor state-dependent delays) be considered more in DDEclimate models.

2 Energy-balance models andpalaeoclimatology

The incorporation of delay terms into mathematicalmodels of climate systems first appeared in an EBM [27].The simplest type of EBM approximates the Earth as asingle point in space, and a variable T represents a glob-ally averaged surface temperature. Although mathemat-ically this is a one-dimensional model because it has onevariable, in climate science such a model is convention-ally referred to as a zero-dimensional EBM because ithas no spacial dimensions. Similarly, a one-dimensionalEBM has one spatial dimension in the latitudinal (merid-ional) direction around the Earth and a two-dimensionalEBM is one where temperature varies across the wholesurface of the Earth.

In EBMs the albedo of the Earth, α, parametrises howmuch solar radiation is reflected back into space by theEarth’s atmosphere and surface; thus, given a solar inso-lation S, power per unit area, Sα is reflected and S(1−α)is absorbed by the Earth. As described in section 1, de-layed effects can be incorporated in EBMs by assumingthat the albedo of the Earth depends on temperaturesin the past.

Some analytical results have been achieved for de-lay EBMs. For example, hysteresis of fixed point so-lutions was found in a one-dimensional EBM [19] and ina two-dimensional EBM [37, 34, 35, 36]. Nonetheless,DDE models relevant to palaeoclimatology have primar-ily been investigated by means of simulation. Variousmethods of numerical integration have been used; forexample, the Euler method with constant initial histo-ries [2] or the Crank-Nicolson finite difference methodwith initial histories taken from simulated trajectoriesfor zero delay [5]. Often, however, the precise method of


integration has not been stated in past literature.In past literature, simulations were almost always run

with the initial histories chosen to be a constant value,which, however, constitutes only a one-dimensional sub-space of the space of continuous functions C. Note thatthe authors of [1] did compare the use of both constantinitial histories and those taken from associated zero-delay simulations, but found that the different initialhistories had no effect on their results. As we will seelater, there are indeed cases where the initial historiesused does become important.

Generally, the inclusion of delay effects into EBMs wasfound to create surprisingly complicated dynamics. In[1] the authors studied a zero-dimensional EBM of theform:

C0Ṫ (t) = Q0[1− α∗(T (t− τ))]− σg(T )T 4, (2)

where T is temperature, C0 is the averaged global heatcapacity, Q0 is mean solar radiative input and σ is theStefan-Boltzmann constant. The term T 4 appears ac-cording to the Stefan-Boltzmann law for black body radi-ation. Function g(T ) is the parametrised effective emis-sivity coefficient and α∗(T −τ) describes how the overallalbedo of the Earth depends of temperature with a con-stant delay. They found that for a large enough delayone of the steady-state solutions loses stability, leadingto self-sustaining oscillations. As the delay is increased,the dynamics becomes chaotic via a cascade of period-doubling bifurcations.

A similar result was observed in [2] for a zero-dimensional EBM with three albedo variables, as illus-trated schematically in figure 3, with αc and αs repre-senting the cloud and surface albedos for incoming ra-diation, respectively. Infra-red radiation emitted fromthe surface, L, has longer wavelengths, so the albedo ofclouds for this radiation is different and denoted αcp. Itis assumed that each albedo is affected by the same con-stant delay. Further to the cascade of period-doublingbifurcations observed in [1], the authors of [2] also foundevidence of an alternative route that may be related tothe intermittency route to chaos.

In a one-dimensional EBM studied in [27] the delayis due to albedo dependence on continental ice-sheet ex-tension across the globe. These authors demonstratedthat varying the insulation parameter leads to the ap-pearance of finite amplitude self-sustaining oscillationswithout the presence of external forcing and they sug-gested that this mechanism could play a role in glaciationcycles.

In the context of palaeoclimatology, specific Earthsubsystems have also been described by DDEs. Thethermohaline circulation in the Atlantic Ocean and itsinfluence on glacial cycles were studied in [28, 96] inthe framework of discrete space variables as a so-calledBoolean delay equation. The variable describing the vol-ume of ice sheets in the Northern hemisphere, for exam-ple, takes on one of two states: low or high ice volume.

S

Sαc

S(1− αc)S(1− αc)αs

S(1− αc)(1− αs)

LLαcp

L(1− αcp)

Figure 3: Schematic representation of the role of albe-dos, based on the model studied in [2]. S represents solarinsolation and L is radiation emitted by the Earth sur-face. The parameters αc, αcp and αs are the albedos ofthe clouds in relation to S, the clouds in relation to Land the surface, respectively.

This is similar to the discrete variable used in delayedswitching [76]. Delays were included in the model of[28, 96] due to the gradual expansion of ice sheets andthe overturning time of the Atlantic Ocean circulation.It was found by numerical simulation that the length ofthe glacial episodes depends continuously on the delayparameters and that, depending on the initial historyused, the system demonstrates multistability.

In an effort to compare modelling results with real-world data, the authors of [66] used a DDE to investigateaspects of the Vostok ice core data, which provides proxytemperature data for the last 430,000 years. Their modelis a logistic growth DDE that describes the competi-tion between a positive ice-albedo feedback and a neg-ative delayed precipitation-temperature feedback with aparametrically forced delay time. A spectral analysis oftime series generated with the model for different forcingstrengths of the delay time showed similarities to the icecore data. A similar version of the logistic growth DDEwas coupled to a simple energy-balance equation in [65].Time series and power spectra from this model capturedsignificant features from deep-sea sediment and ice coredata at Milankovitch and millennial scales, including thecharacteristic saw-tooth shape of the data time series,the mid-Pleistocene climate switch and the Dansgaard-Oeschger oscillations. Generally speaking, the resultssuggested that the Earth’s climate may be only weaklydriven by astronomical forcing with most “intriguing”dynamical features being the result of internal nonlinearprocesses and feedbacks.


3 The delayed action oscillator —Models of ENSO

Conceptual ENSO models have been developed that fo-cus on the interactions of key mechanisms in order tobetter understand some basic dynamical features, suchas those described in section 1. Most of these models arebased on the so-called delayed action oscillator (DAO)paradigm. In this paradigm, ENSO dynamics are drivenby feedback loops that form through the coupling of at-mospheric and oceanic processes above and in the equa-torial Pacific Ocean, as illustrated in Fig. 4. The variableh represents deviations of the thermocline depth from itslong-term mean at the eastern boundary of the equato-rial Pacific Ocean. The thermocline is a relatively thinlayer of the ocean that separates the deeper cold watersfrom the warmer well-mixed waters. Its depth can beconsidered a proxy for sea-surface temperature (SST), soa large h corresponds to the warming of the eastern equa-torial Pacific Ocean observed during El Niño events. Apositive perturbation in h slows down the easterly tradewinds. This induces transport of warm surface water to-wards the equator in the central part of the ocean, wherethe ocean-atmosphere coupling is strongest, as indicatedby the blue arrows in Fig. 4. The SST rises, creating apositive signal that is carried back to the eastern bound-ary as a so-called Kelvin wave to form a positive feedbackloop. Simultaneously, a negative signal is created off theequator, where there is a deficit of warm surface water.This signal travels westward as a so-called Rossby wave,which is then reflected at the western edge of the Pacificbasin before travelling back to the eastern boundary asa Kelvin wave, thus closing the negative feedback looprepresented by the bar in Fig. 4. In this system the de-lays are due to the finite speeds of the Kelvin and Rossbywaves, which typically take 1–6 months to carry a signalacross the Pacific Ocean.

In the DAO models that we discuss below, as men-tioned above, the variable h describes anomalies of thethermocline depth, which is considered a proxy for SST,in the eastern equatorial Pacific Ocean. A modelling as-sumption made in all the DAO models is that the delaytimes associated with the travelling oceanic waves areconstant. The models were generally studied by simu-lations; for example, with a variable-order, variable-stepAdams method [87, 26] or a forward-stepping algorithm[69, Appendix 1]. In many cases the precise integrationmethod was not stated. Nonetheless, the use of constantinitial histories seems to be common practise when run-ning simulations (except for [44, 45, 43]).

3.1 Basic DAO with delay

The story of investigating ENSO by means of DDE mod-els begins with a DAO model introduced in [80] and given

h

+

Atmosphere

Rossby wave

Kelvin wave

Equator

Figure 4: The variable h represents deviations from themean thermocline depth at the eastern boundary of theequatorial Pacific Ocean. The coupling between oceanand atmosphere allows for the creation of positive andnegative delayed feedbacks. The green arrows and barsrepresent processes of positive and negative reinforce-ment, respectively (see text for details).

by:

ḣ(t) = h(t)− h(t)3 − αh(t− τ). (3)

The first term reflects a “local” (instantaneous) positivefeedback, whereby a SST perturbation heats the atmo-sphere, whose wind response drives ocean currents toreinforce the original perturbation. The growth of theinstability due to the positive feedback is limited by ef-fects such as advective processes in the ocean and moistprocesses in the atmosphere, which are represented bythe second term in Eq. (3). The third term describes theeffect of delayed oceanic waves. A linear stability anal-ysis of this simple model was conducted in [80] to showthat the steady-state solution loses stability for certainparameter values of α and τ . The resulting periodic so-lutions have periods of at least twice the length of delay,demonstrating that this simple feedback can provide amechanism for ENSO’s oscillatory behaviour on an ap-propriate interannual timescale.

In [4] it is shown that model (3) can be derived from anintermediate coupled ocean-atmosphere model [3] that isvery similar to the Zebiak-Cane model [98] 1. Throughtheir rigorous derivation of the DDEs, the authors of[4] were able to relate the parameters to backgroundstates of the ocean and atmosphere and the geometryof the Pacific basin. By comparing alternative back-ground states and geometries, the results helped to ex-plain why ENSO-like variability is not observed in thetropical Indian or Atlantic Ocean. Supporting evidenceof the DAO paradigm was provided by empirically de-rived time-delay models, which were shown to produceoscillations similar to observable data in [30] and to asophisticated oceanic general circulation model in [71].

1The Zebiak-Cane model is significant because, despite beingdeterministic, it was shown to successfully forecast El Niño events[11].


3.2 Multiple feedback loops and seasonalforcing

The simple negative delayed feedback in model (3) pro-vides a mechanism for SST oscillations. There have sincebeen numerous extensions to this model, in an effort tomake the models more realistic by including additionaltime-scales. For example, the inclusion of only one de-lay term is a modelling assumption that only one modeof Rossby wave is important. In fact, many modes ofRossby waves form at different latitudes with differentphase speeds. The authors of [94] showed, by simulationand spectral analysis of a DAO model, that delay timesassociated with sets of Rossby waves forming at approx-imately 7◦N, 12◦N and 18◦N result in biennial, inter-annual and decadal signals similar to those observed indata [84].

The positive delayed feedback formed by Kelvin waves,as described above, and seasonal forcing were includedin [87] in the DDE model:

ḣ(t) = aA(κ, h(t− τp))− bA(κ, h(t− τn)) + c cos (2πt), (4a)

with A(κ, h) =

{du tanh(

κduh) if h ≥ 0,

dl tanh(κdlh) if h < 0.

(4b)

The positive and negative feedbacks are associated withfixed delay times, τp and τn, respectively. The seasonalforcing is represented by additive forcing with a periodof one year. The function A(h) is a special case of theocean-atmosphere coupling function justified in [55] withcoupling strength κ and horizontal asymptotes dh > 0and dl < 0.

The model (4) is significant because it demonstratedthat the irregularity characteristic of ENSO could be re-produced as chaotic behaviour of this simple DDE. Thisis in contrast to the alternative view that the irregular-ity is driven by noise, in particular, by small-scale, high-frequency stochastic forcing; for example, see [60, 82].Specifically, the authors of [87] observed a transition tochaos upon increasing the parameter κ and presented thesame solutions shown here in Fig. 2. Based on these so-lutions, the authors suggested that the chaotic behaviourat κ = 2.0 was due to the coexistence of mode-locked so-lutions or, in other words, overlapping resonances. Fur-thermore, they suggested that the coexistence of differ-ent mode-locked solutions depends on the strength ofnonlinearity in the model, so that κ must be sufficientlylarge to observe irregular behaviour. The view thatENSO irregularity is primarily due to low-order chaoticprocesses gained further support in the study of a differ-ent DAO model with two fixed delay times and a seasonalcycle in [69].

Ghil et al. considered a simplification of model (4),referred to here as the GZT model, with a = 0, du = 1and dl = −1 that focusses on the interaction between the

c

τ n

max[h(t)]

Figure 5: Maximum map of the GZT model display-ing the maximum value of h(t) according to the colourscheme in the (c, τn)-plane. Parameters are b = 1and κ = 11 and the initial history is h(t) ≡ 1; t ∈[−τn, 0]. Reproduced from M. Ghil, I. Zaliapin andS. Thompson, A delay differential model of ENSO vari-ability: Parametric instability and the distribution of ex-tremes, Nonlinear Processes in Geophysics, 15 (2008),pp. 417–433 under CC-BY licence.

negative delayed feedback and the seasonal forcing. De-spite its simplicity, the GZT model demonstrated com-plicated dynamics; specifically, it reproduced importantENSO features such as intraseasonal oscillations, inter-decadal variability, frequency locking for varying param-eters (observed as a Devil’s terrace or two-dimensionalDevil’s staircase), as well as phase locking with the sea-sonal forcing. A primary tool utilised by Ghil et al. wasthe computation of so-called maximum maps, which plotthe maxima of simulated solutions as a function of twoparameters. In order to avoid transient effects, a tra-jectory is given sufficient time (thousands of years) toapproach and reach a stable attractor. To ensure accu-racy, the length of time from which the maximum wasobtained needed to be sufficiently long, because the so-lution may not necessarily be periodic.

Figure 5 shows an example of a maximum map from[29] with parameters c and τn and a single fixed ini-tial history of h(t) ≡ 1. One can easily identify tworegimes — one in the upper-left and one in the lower-right side of the parameter plane — divided by a sharpinterface, which represents where there is a rapid transi-tion in max[h(t)]. Further sharp interfaces form elon-gated shapes in the upper-left side of the parameterplane. Given that generic stable solutions to the modeldepend continuously on the parameters and initial his-tory, as proven in [29], the observed sharp interfaces im-ply the existence of stability loss and families of unstablesolutions.

The existence of multistabilities in the GZT model was


uncovered in a follow-up paper to [29] by studying the ef-fect of different constant initial histories [97]. It was alsodemonstrated in [97] that the phase locking observed in[29], which is an important feature of the model becauseit agrees with the tendency of El Niño events to occurprimarily towards the end of the calendar year, is a ro-bust feature of the model, except when the forcing isweak.

Tziperman et al. studied the phase locking mechanismwith a more complicated model:

ḣ(t) = aA(κ(t− τp), h(t− τp))− bA(κ(t− τn), h(t− τn))− ch(t),

κ(t) = k0 + dk sin (π

6t), (5)

withA(κ, h) given by Eq. (4b). The last term of the DDEreflects attenuation due to dissipation. Motivated bystudies of more complex models, the effect of the seasonswas not included as an additive term, but as parametricforcing of the ocean-atmosphere coupling strength. Theparameters k0 and dk are the mean coupling strengthand annual variation, respectively. Time t is measuredin months. The authors analysed a particular solu-tion of model (5) that demonstrated non-periodic be-haviour with large extrema of varying size occurring ev-ery three years. They showed that for this solution theEl Niño events can only reach their peak when the cou-pling strength is at its minimum strength, which is atthe end of the calendar year, and that this mechanismis very robust to parameter changes. This phase-lockingmechanism was later shown in in [26] to be essential ina more sophisticated DDE model derived from a sim-plified version of the Zebiak-Cane model by followingsimilar physical assumptions as in [38, 39]. This modelretains the use of constant delays, but incorporates notonly oceanic wave propagation times but also SST re-sponse times to changes in thermocline depth.

3.3 Continuation results for model (5)

In order to gain a more comprehensive view of the possi-ble dynamics of model (5) and confirm the observationsmade in [86], the authors of [49] employed the continua-tion software DDE-Biftool to conduct a bifurcation anal-ysis in the (dk, k0)-plane. Figure 6 illustrates that thereare qualitative changes of solutions, which occur as pa-rameters pass through curves of torus bifurcations (TR),period-doubling bifurcations (PD) and saddle-node bi-furcations of periodic orbits. In the autonomous case ofdk = 0, the trivial zero-solution loses stability at a Hopfbifurcation and creates a family of periodic orbits. Oncethe forcing is introduced with dk > 0, the zero-solutionbecomes a trivial periodic solution with an amplitudeof zero and a period of one, which remains stable forsmall k0 and dk. While increasing dk, the trivial periodicsolution loses stability in a period-doubling bifurcation,

creating period-2 solutions. On the other hand, increas-ing parameter k0 leads to stability loss through torusbifurcations. Therefore, above the curve TR there existinvariant tori and associated p :q resonance tongues, aslabelled in Fig. 6 for all q ≤ 11. The resonance tongues,which are bounded by curves of saddle-node bifurcationsof periodic orbits, contain families of stable and saddlep :q periodic orbits that are frequency locked. Therefore,within each resonance tongue the invariant torus has afixed rational rotation number p/q. In between the res-onance tongues, quasiperiodic solutions exist.

It should be noted that theory predicts the existenceof an infinite number of resonance tongues, one for ev-ery rational rotation number of the torus, which becomeincreasingly narrow with increasing q. Figure 6 demon-strates that one can obtain an effective overview of allpossible dynamics of model (5) for a relevant range of pa-rameters. The insert in the lower-right corner of Fig. 6is an enlargement of the parameter plane near the rootpoint of the 1:3 resonance tongue. The cross indicatesthe parameters used in [86] and reveals why the solutionto model (5) demonstrated aperiodic motion, yet closelyresembled a period-three periodic orbit. The authors of[49] also plotted all stable 1 :3 and 1:4 solutions (notshown) from the parameter range considered in Fig. 6in order to show that the phase locking to the seasonalcycle is indeed a robust feature, in agreement with theclaim made in [86].

3.4 Continuation results for theGZT model

In [44] we used DDE-Biftool to conduct a bifurcationanalysis of the GZT model studied in [29, 97]. Figure 7shows maximum maps overlaid with a bifurcation set inthe (c, τn)-plane; white, grey and black curves representsaddle-node bifurcations of periodic orbits, torus bifurca-tions and period-doubling bifurcations, respectively. Incontrast to the maximum map shown in Fig. 5, whichwas calculated for a single fixed initial history, the max-imum maps in Fig. 7 are calculated such that for eachrow of fixed delay τn the parameter c is scanned up anddown (using previous solutions as initial histories), asindicated by the arrows. This convenient and system-atic approach means that the simulated trajectory stayson the same branch of solutions while c is slowly varieduntil stability is lost. Comparing panels (a) and (b) ofFig. 7, the maximum maps reveal regions of bistability,where the maxima of observed solutions depend on thedirection in which the parameter c is varied.

The bifurcation set in Fig. 7 divides the (c, τn)-planeinto regions of different solution types. Generally, thedynamics is driven by two independent mechanisms thatcreate self-sustaining oscillations: the negative delayedfeedback and the seasonal forcing. Along the line c = 0in Fig. 7, the periodic orbits depend only on the negativedelayed feedback term, creating periodic solutions for


Figure 6: Bifurcation set of model (5) in the (dk, k0)-plane. Curves of torus (TR) and period-doubling (PD)bifurcations are shown. Saddle-node bifurcations of peri-odic orbits are the boundaries of p :q resonance tongues.The inset is an enlargement to highlights the loca-tion of the parameters used in [86]. Other parame-ters are a = 0.2535, b = 0.1901, c = 0.1601, du = 3,dl = −1, τp = 1.15 and τn = 5.75. Reproduced fromB. Krauskopf and J. Sieber, Bifurcation analysis ofdelay-induced resonances of the El-Niño Southern Oscil-lation, Proceedings of the Royal Society A, 470 (2014),348.

τn > π/(2κ) [12, 14, 58]. On the other hand, solutionswith large c are dominated by the seasonal forcing, sothat they all have a period of one and appear sinusoidal-like. As the parameter c is decreased in panel (b), thosesolutions lose stability in torus bifurcations at curve TR.Therefore, between line c = 0 and curve TR is a regionwhere both oscillations interact and give rise to dynamicson an invariant torus. The locked solutions on the torusare organised into resonance tongues, which appear aselongated shapes in Fig. 5 as best observed in panel (a).

The sharp interfaces of the maximum maps in Fig. 7that could not be explained by bifurcations of periodicorbits are shown in [44] to be more complicated bifur-cations involving folding tori. These folding tori wereanalysed in detail in [43] and were shown to possessa complicated bifurcation structure that occurs as toriapproach each other and break-up. As briefly demon-strated in [43], this phenomenon can be considered as amechanism for climate tipping that, in contrast to simplefold bifurcations of equilibria or periodic orbits, has theadditional bifurcation structure associated with foldingtori that could offer precursor information about an ap-proaching tipping point. This result is a good exampleof how new phenomena, which are easily overlooked incomplex models, can be found and studied in conceptualmodels.

0 2 4 6 8

0

0.5

1

1.5

2

0.5 1 1.5 2

τ n

max[h(t)]

-

(a)

4:3�1:1

1:2 3:7PPi 3:8PPi

1:3

1:4

1:5

1:6

1:7

0 2 4 6 8

0

0.5

1

1.5

2

τ n

c

�

(b)

4:3�1:1

1:2 3:7PPi 3:8PPi

1:3

1:4

1:5

1:6

1:7

Figure 7: Maximum maps of the GZT model with curvesof saddle-node bifurcations of periodic orbits, period-doubling and torus bifurcations; drawn in white, greyand black, respectively. Several frequency ratios of reso-nance tongues are indicated. Other parameters are b = 1and κ = 11.

3.5 Continuation results for model (4)

The combination of continuation software with bifurca-tion theory was applied to model (4) in [45]. Although itwas a crucial result that this simple deterministic modelcould produce irregular behaviour reminiscent of real-world data, there remained important open questions:Is this behaviour characteristic of the model? In otherwords, how robust is this behaviour to changes in param-eters? By what mechanism does the solutions becomechaotic?

Figure 8 addresses these questions; it shows a bifur-cation set in the (c, κ)-plane with curves of saddle-nodebifurcations of periodic orbits (blue), torus bifurcations(red), and period-doubling bifurcations (black). Themaximal Lyapunov exponent of each solution, in cases


0 1 2 3

0.9

1.2

1.5

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3

maximal Lyapunov exponent

κ

c

a

b

c

d

0:1

1:4

1:5

1:6

1:7

Figure 8: Bifurcation set of model (4) in the (c, κ)-planewith saddle-node bifurcations of periodic orbits (blue),torus bifurcations (red), and period-doubling bifurca-tions (black). Shown p :q resonances are labelled, as arethe points (a)–(d) corresponding to solutions consideredin [87] and shown in Fig. 2. The colour scheme indi-cates the positive maximal Lyapunov exponent. Otherparameters are a = 2.02, b = 3.03, du = 2.0, dl = 0.4,τp = 0.0958, and τn = 0.4792.

where it is positive indicating chaotic behaviour, is dis-played by a colour scheme. The maximal Lyapunov ex-ponents are calculated according to the algorithm forDDEs described in [25]. Also shown in Fig. 8 are theparameter points of solutions (a)–(d) referred to in [87]and displayed in Fig. 2.

In Fig. 8 we see a curve of torus bifurcations for smallvalues of κ and resonance tongues that are rooted atthe zero-forcing line c = 0. The resonance tongues con-tain cascades of period-doubling bifurcations that ap-pear where different resonance tongues overlap, lead-ing to chaotic solutions with positive maximal Lya-punov exponent. Some positive maximal Lyapunov ex-ponents appear in regions where the displayed resonancetongues are not overlapping. This is simply becausein-between those shown exist smaller, higher-order reso-nance tongues that will overlap.

The parameter point of solution (a) in Fig. 2 is lo-cated in the parameter region of Fig. 8 that is domi-nated by the seasonal forcing; hence, it is a solution ofperiod 1. Upon increasing κ for fixed c = 2.6377 thesolution loses its stability at a torus bifurcation, so thatquasiperiodic can also be observed. The parameter pointof solution (b) is not inside any resonance tongue and istherefore observed in [87] to be quasiperiodic. On theother hand, the parameter point corresponding to solu-

0 10 20 30 40 50 60 70 80 90 100

-1

0

1

h(t)

t

Figure 9: Time series found by numerical integration ofmodel (4) for a = 2.02, b = 3.03, c = 2.6377, du = 2.0,dl = 0.4, τp = 0.0958, τn = 0.4792 and κ = 2.0.

tion (c) lies within the 1:4 resonance tongue, in agree-ment with the period-4 solution seen in [87]. The chaoticsolution (d) in Fig. 2 is situated in a parameter region ofoverlapping resonance tongues in Fig. 8. Furthermore,according to the bifurcation set, the same changes of so-lution type will be observed while increasing κ for anyfixed 1.5 . c ≤ 3.2 and possibly for larger c, albeitfor different values of κ. Therefore, the transition tochaos observed in [87] and illustrated in Fig. 2 is indeeda prominent feature of model (4) and leads to chaoticbehaviour across a substantial range of parameters.

Figure 9 shows the time series of solution (d) over alonger time window. It is a good example to demon-strates how simple feedback mechanisms can account forboth the spatial and temporal sense of irregularity thatis observed in the magnitude and frequency of El Niñoevents.

Particular routes to chaos that account for the irreg-ular behaviour in ENSO models have been discussed atlength; for example, in the literature reviews [64, 68, 89].In various ENSO models routes to chaos have beenidentified as either the quasiperiodic route [42, 85, 86],period-doubling route [10, 13, 55] or the intermittencyroute [91]. Interestingly, it was shown in [45] that differ-ent routes to chaos can coexist, even to the same chaoticsolution, depending only on the chosen path through pa-rameter space. While increasing κ in model (4) leads tochaos via the period-doubling route, an alternate routeto the same chaotic attractor is found when decreasingκ for fixed c. In this case, chaos appears just belowthe lower boundary of the 1:6 resonance tongue seenin Fig. 8. After exiting the 1:6 resonance tongue, as κis decreased, episodes of periodic behaviour become in-creasingly shorter until they apparently disappear. Thisis evidence of the so-called intermittent transition [62],which is characterised by the sudden appearance of chaosat a saddle-node bifurcation.

Notice that many aspects of the bifurcation set inFig. 8 are similar to that of Fig. 6. One difference be-tween the two is that, while the resonance tongues inFig. 8 overlap each other, the resonance tongues in Fig. 6appear to only approach each other without overlapping.Whether this reflects a difference between additive and


multiplicative forcing, or whether the resonance tonguesin Fig. 6 do overlap for larger parameter values thanthose considered is not clear and warrants further anal-ysis.

3.6 Alternative ENSO paradigms

Apart from the DAO paradigm, other ENSO paradigmshave been considered and modelled as DDEs. The so-called western Pacific oscillator paradigm focusses on thecompetition between central equatorial and western off-equatorial Pacific thermocline and demonstrates the po-tential importance of western Pacific variability in re-lation to ENSO. It is introduced in [92] and modelledby a set of linear DDEs with four variables describinganomalies in: equatorial thermocline depth in the east-ern Pacific Ocean, off-equatorial thermocline depth inthe western Pacific Ocean and the zonal wind stressesabove the west-central and western Pacific Ocean. Thethermocline variables depend on delayed wind stresses,modelled with two fixed delays. It was shown that thedelays are not necessary to produce oscillations. Al-though the oscillations are periodic, the authors arguethat irregularities could be introduced into the dynam-ics by, for example, adding nonlinearities or stochasticforcing to the model. This study inspired further in-vestigation of the role of western Pacific variability in amore complex model related to the Zebiak-Cane modelin [90], which produced co-oscillating anomaly patternsin the western and eastern Pacific Ocean that were con-sistent with observations.

In [88] a DDE model is derived that encapsulates fourdifferent ENSO paradigms: the DAO, the western Pa-cific oscillator, the recharge-discharge oscillator[38, 39]and advective-reflective oscillator [61]. The model con-sists of four variables describing anomalies in: the av-erage SST in the eastern equatorial Pacific Ocean, theoff-equatorial thermocline depth in the western PacificOcean, the zonal wind stresses above the central andwestern equatorial Pacific Ocean. The author of [88] sug-gests that naturally varying parameters could alter therelative role of each paradigm in different El Niño andLa Niña events. For example, the western Pacific oscil-lator might play a larger role in strong El Niño events,since strong equatorial wind anomalies are known to oc-cur in the western Pacific Ocean at times of strong ElNiño events.

4 Discussion and outlook

In this paper we reviewed how DDEs have been used inclimate modelling. In order to reflect the balance of ex-isting literature, we chose to focus on the application ofDDEs to EBMs, palaeoclimatology and ENSO models.Upon including delayed effects into the respective phe-nomenological models, the dynamics becomes consider-

ably more complicated and allows for the reproductionand investigation of certain dynamical features that areobserved in nature. For example, a possible phase lock-ing mechanism is identified as an explanation why ElNiño events tend to occur around Christmas. Most ofthe results have been achieved by running simulations.We, therefore, highlighted some more recent results thatwere obtained by continuation methods that are ableto deal effectively with multistabilities, transients andthe difficulty of choosing appropriate initial conditions.These reviewed results include the identification of fold-ing tori, which are particularly interesting in the contextof climate tipping, as well as clarification of the precisemechanism by which chaotic behaviour was observed insimulations of an ENSO model.

Beyond the focus of this brief review, DDEs have beenused to describe other climate systems; these includean interdecadal cycle in the Arctic and Greenland Sea[16], an interdecadal cycle in the subpolar North Atlantic[95], vegetation interaction with global climate fluctu-ations [17], heat transport between the Pacific extra-tropics and tropics [31], rainfall with land-atmospherecoupling through soil moisture, vegetation and surfacealbedo [6] and boundary reflections of oceanic waves inthe Indian Ocean [93]. Furthermore, we have consid-ered here deterministic models, yet stochastic behaviouris undoubtedly present in climate systems and couldhave highly relevant effects, even if a system is primarilydriven by deterministic processes. For examples of DDEclimate models with noise, see [6, 31, 70, 79, 93].

4.1 Feedback loops with nonconstant de-lays

We now briefly discuss nonconstant delays in DDE cli-mate models — a subject we believe has considerable po-tential for rendering phenomenological models more real-istic and relevant from a climate modelling point of view.Moreover, DDEs with nonconstant delays are presentlya research area of considerable interest from a dynam-ical systems point of view, and climate DDEs arise asnatural test-bed models for new theory and numericalapproaches.

A first class of nonconstant delays, referred to asdistributed or weighted delays, describe the situationthat a variable might depend on a range of past times.Then the delay term y(t− τi) becomes an integral term∫ τmax0

w(τ)y(t − τ)dτ , where values of y are consideredover a delay range of [0, τmax] and w(τ) is the associ-ated kernel or delay distribution that says how the dif-ferent past times contribute to the overall feedback. Inthe literature on DDE climate models, distributed de-lay was considered in a very early publication [5]. De-spite this, the trend towards the use of a constant delayterm quickly set in, because it simplifies the analysis ofthe DDE. In fact, the study of the constant-delay DDEserves as a starting point for any investigations of the


role of nonconstant delays, but we believe it promis-ing and necessary that future work reconsider the useof distributed delays in climate models. For example,in the above mentioned DDE ENSO models the delaysare always assumed to be constant, while oceanic wavevelocities are distributed around mean velocities [?]. Itwould be interesting to include into the model the asso-ciated distribution of delays, which is influenced in nosmall part by the geometry of the Pacific basin near theequator.

A second type of nonconstant delays arises when a de-lay depends on the state of the system itself, which leadsto a delay term of the form τi = τi(t, y(t)); one speaks ofstate-dependent delays. For example, the delays in theDAO paradigm of ENSO are determined in part by theposition in the Pacific Ocean where the oceanic wavesform, which is influenced by the position of the westernPacific warm-pool. Yet, the position of the warm-poolitself is influenced by changes in the thermocline depth.An implicit expression for a state-dependent delay in thiscontext has already been suggested in [15].

State-dependency also arises when taking into accountsubsurface ocean adjustment dynamics. In a series of pa-pers [56, 41, 57, 40], three regimes of ENSO dynamicswere studies: the fast-SST regime, where the SST adjust-ment to changes in thermocline depth is instantaneous;the fast-wave regime, where the speeds of oceanic wavesare infinite; and the mixed-mode regime, where bothSST adjustment and oceanic wave propagation times areessential. These authors showed that the mixed-moderegime is the most realistic. One could incorporate theSST adjustment time into the model introduced in [29]by adding an additional delay time that will depend onthe position of the thermocline itself. As a quite sim-ple example we consider linear state-dependent delays,of the positive and negative feedback loops, of the form:

τp/n(t) = τp/n + η h(t). (6)

It represents the respective overall delay as the delay ofthe oceanic wave dynamics τp/n plus the time η h(t) re-quired for the upwelling process to carry the signal fromthe thermocline to the sea-surface. The parameter ηrepresents the speed of the upwelling process, as well asthe strength of the state-dependency of the delay. As-suming h is scaled such that the distance between themean thermocline depth and the sea-surface in the east-ern equatorial Pacific Ocean is approximately 1, η = 0.04corresponds to a SST response time of about two weeks,as estimated in [99].

DDEs with distributed or state-dependent delays canbe studied with the continuation software DDE-Biftool;for example, see [54, 8]. Figure 10 shows how the 1:2resonance tongue of Fig. 7 changes under the influence ofstate-dependency as given by Eq. (6). As the value of ηis increased from η = 0 as in panel (a), there is a changein the parameter region where period-2 solutions exist.

0.46

0.5

0.54

τ n

(a)

T

SN

0.46

0.5

0.54

τ n

(b)

T

SN

PD

0 1 2

0.46

0.5

0.54

τ n

c

(c)

T

SN

PD

Figure 10: Bifurcation set of the GZT model in the(c, τn)-plane with a state-dependent delay given byEq. (6) with η = 0 (a), η = 0.01 (b) and η = 0.04 (c).Curves of torus (T) and period-doubling (PD) bifurca-tions are shown. Saddle-node bifurcations of periodicorbits (SN) form boundaries of a 1:2 resonance tongue.Other parameters are b = 1 and κ = 11.

More specifically, there is a symmetry-breaking withinthe family of period-2 solutions that results in curves SNof saddle-node bifurcations of (originally symmetricallyrelated) periodic solution to no longer lie on top of eachother; see panels (b) and (c) of Fig. 10. At the sametime, the root point of the resonance tongue on curveT grows into a region of period-doubling. Notice alsothat the overall region with locked dynamics increasesconsiderably with η.

Figure 11 shows a time series of model (4) with Eq. (6)for η = 0.04 to further demonstrate the effect of state-dependency; it was computed with the Matlab solverddesd [73]. The time series shows irregular large tem-perature events and appears to be chaotic. This typeof behaviour persists for about 2000 years but then the


0 10 20 30 40 50 60 70 80 90 100

-1

-0.5

0

0.5

h(t)

(a)

2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2500

-1

-0.5

0

0.5

h(t)

t

(b)

Figure 11: Time series of model (4) with Eq. (6) for η =0.04, calculated with Matlab’s ddesd. Other parametersare a = 2.02, b = 3.03, c = 2.6377, du = 2.0, dl = 0.4,τp = 0.0958, τn = 0.4792 and κ = 1.5.

trajectory finally settles into a complicated periodic so-lution with a very long period.

Equation (6) constitutes the simplest possible, yet re-alistic way in which to introduce state-dependence intoan ENSO model. Despite this, and even though η issmall relative to τn in Figs. 10(c) and 11, our resultsshow that state-dependency has the potential to play asignificant role for generating realistic observable modelbehaviour. We believe that state-dependence in DDEclimate models will emerge as an interesting directionfor future research. Indeed, there are already a num-ber of interesting questions: Is Eq. (6) an appropri-ate representation of state-dependent delay in a DAOmodel? Are there other important state-dependent rela-tionships? Would a distributed delay be more realisticand accurate? Should more focus be put on the result-ing transient behaviour rather than eventual asymptoticbehaviour?

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