Long range dependence and the dynamics of exploited fish populations
aa r X i v : . [ q - b i o . P E ] J un Long range dependence and the dynamics ofexploited fish populations
Hugo C. Mendes ∗† , Alberto Murta ∗ and R. Vilela Mendes ‡§ Abstract
Long range dependence or long memory is a feature of many processesin the natural world, which provides important insights on the underlyingmechanisms that generate the observed data. The usual tools availableto characterize the phenomenon are mostly based on second order cor-relations. However, the long memory effects may not be evident at thelevel of second order correlations and may require a deeper analysis of thenature of the stochastic processes.After a short review of the notions and tools used to characterize longrange dependence, we analyse data related to the abundance of exploitedfish populations which provides an example of higher order long rangedependence.
Long range dependence or long memory is an important notion in many pro-cesses in the natural world. Studies involving this notion pervade fields frombiology to econometrics, linguistics, hydrology, climate, DNA sequencing, etc.Although, at times, it has been considered a nuisance in the study of theseprocesses, the existence of long memory is in fact a bonus, in the sense thatit provides further insight in the nature of the process. Whereas short or nomemory just points out to the essentially unstructured nature of the phenomena,long memory, by contrast, may provide a window on the underlying mechanismsthat generate the observed data.The most popular definitions of long range dependence are based on thesecond-order properties of the processes and relate to the asymptotic behaviorof covariances, spectral densities and variances of partial sums. However thereare other different points of view, some of which are not equivalent to the char-acterization of second-order properties. They include ergodic theory notions,limiting behavior, large deviations, fractional differentiation, etc. [1] [2] [3]. ∗ Instituto Portuguˆes do Mar e da Atmosfera, Avenida Bras´ılia, 1300-598 Lisboa † [email protected] ‡ CMAF and IPFN, Univ. Lisbon, Av. Prof. Gama Pinto 2, 1649-003 Lisboa § [email protected] When based on second order properties, long range dependence in a stationarytime series X ( t ) occurs when the covariances γ ( τ ) = E { X ( t ) X ( t + τ ) } (1)tend to zero so slowly that their sum ∞ X τ =0 γ ( τ ) (2)diverges. Alternative definitions of long-range dependence are based on thepower-law behavior of the covariances, namely P nτ = − n γ ( τ ) ∽ n α L ( n ) when n → ∞ ; 0 < α < γ ( τ ) ∽ τ − β L ( τ ) when τ → ∞ ; 0 < β < f ( ν ) ∽ | ν | − ξ L ( | ν | ) when ν → < ξ < f ( ν ) being the spectral density f ( ν ) = 12 π ∞ X τ = −∞ e − iντ γ ( τ ) (4) L , L being slowly varying functions at infinity and L slowly varying at zero.If γ ( τ ) is monotone as τ → ∞ , the definitions (3) are equivalent to thedivergence of the sum in (2) with α = 1 − β and ξ = 1 − β .In science, one of the main purposes when observing natural phenomena isthe construction of models. A useful approach in this endeavour is the com-parison of natural time series with the behavior of well-studied mathematicalstructures. In the context of long range dependence a central role is played bythe theory of self-similar stochastic processes X ( at ) d = a H X ( t ) (5)2ith stationary increments X ( t + h ) − X ( t ) d = X ( t ) − X (0) (6) d = meaning equality in distribution. These processes are denoted as H − sssi processes and H is called the Hurst exponent. Notice that there is a closerelation between self-similarity and stationarity. If X ( t ) is self-similar then Y ( t ) = e − tH X ( t ) is stationary and conversely if Y ( t ) is stationary X ( t ) = t H Y (ln t ) is self-similar.A finite σ variance H − sssi process has a covariance E { X ( s ) X ( t ) } = σ n | t | H + | s | H − | t − s | H o (7)Throughout this paper E {· · · } will denote the expected value which, for all thedata examples, one approximates by the empirical average h· · · i .There are non-Gaussian H − sssi processes as well as H − sssi processes withinfinite covariance [4]. However, the simplest example of a H − sssi process is aGaussian process uniquely defined by the covariance (7) and normalized to have σ = 1. It is called fractional Brownian motion (fBm) B H ( t ) and the incrementprocess Z ( t ) = B H ( t + 1) − B H ( t ) (8)is called fractional Gaussian noise (fGn). For H = , B / ( t ) is Brownianmotion. Fractional Gaussian noise has covariance γ ( τ ) = 12 n | τ + 1 | H − | τ | H + | τ − | H o (9)hence, if H = it has γ ( τ ) = 0 (no memory) and for H = and large τγ ( τ ) ∽ H (2 H − | τ | H − τ → ∞ (10)For < H < γ ( τ ) > H > and γ ( τ ) < H < , the process is called persistent in the first caseand anti-persistent in the second. For the spectral function at H = f ( ν ) ∽ ν − H ν → H > blows up at the origin.The Hurst exponent ( H ) as an index of long range dependence quantifies thetendency of a time series either to regress strongly to the mean or to persist ina deviation from the mean. An H value between 0 . . B H ( t ) = Z −∞ n ( t − u ) H − − ( − u ) H − o dB ( u )+ Z t ( t − u ) H − dB ( u ) (12)where B ( u ) = B ( u ). One practical implication is that to extract an eventualfractional behavior from the data it is not sufficient an observation of short timeintervals where the process may easily be confused with an uncorrelated process.Another way in which Brownian motion intervenes in modelling processeswhich are neither uncorrelated nor simple Brownian motion is through the fol-lowing fundamental result of stochastic analysis [5]: If X ( t ) is a random variablethat is square-integrable in the measure generated by Brownian motion, then dX ( t ) = µ ( t ) dt + σ ( t ) dB ( t ) (13)where µ ( t ) and σ ( t ) are well defined stochastic processes. Therefore althoughthe increments of X ( t ) have a representation in terms of the increments ofBrownian motion, the process may be very different, depending on the natureof the processes µ ( t ) and σ ( t ).Whenever long range dependence is modelled by fractional Gaussian noiseone benefits from the extensive theoretical and computational framework thatis available for this process. However, fractional Gaussian noise is quite rigid inthe sense that it specifies the correlations at all time lags, not only at τ → ∞ .It may therefore not be suitable for modeling long range dependent phenomenawhere the covariance at short time lags differs from fGn. This motivated thedevelopment of other models through Gaussian linear sequences X ( t ) = ∞ X j = −∞ c t − j ǫ j (14)where P ∞ j = −∞ c j < ∞ and { ǫ j } j ∈ Z are independent identically distributed(i.i.d.) normal random variables, called innovations . A Gaussian linear se-quence is stationary and for the convergence of the sum in (14) one requires- If ǫ j = N (cid:0) µ, σ (cid:1) with µ = 0, P j | c j | < ∞ - If ǫ j = N (cid:0) µ, σ (cid:1) with µ = 0, P j | c j | < ∞ An example is the FARIMA(p,d,q) process (fractional autoregressive inte-grated moving average) [6] [7] X ( t ) = Φ − p ( S ) Θ q ( S ) ∆ − d ǫ t t ∈ Z (15) { ǫ t } is an i.i.d. N (cid:0) , σ (cid:1) sequence and Φ p ( S ) , Θ q ( S ) are polynomials on theshift operator Sǫ i = ǫ i − (16)4 − d being ∆ − d = ( − S ) − d = ∞ X i =0 Γ ( i + d )Γ ( d ) Γ ( i + 1) S i (17)The fractional differencing ∆ − d for 0 < d < models long range dependence,whereas the auto regressive Φ p ( S ) and the moving average Θ q ( S ) polynomialsprovide flexibility in modeling the short-range dependence.Finally, as mentioned on the introduction, there are other ways to deal withlong range dependence for which the behavior of covariances does not play themain role. A potentially promising way is based on ergodic theory becausethe notion of memory is related to the connection between a process and itsshifts. Then, a possible definition of long range dependent process would be onethat is ergodic but non-mixing. However the mixing property is probably notsufficiently strong to imply that a mixing stationary process has short memory.Stronger requirements may be needed. These notions will not be used here andwe refer to [1] [8] for a discussion. Long range dependence has been rarely documented in marine ecology, presum-ably because of the scarcity of long time series. This lack of extended timeseries has limited research on long memory in fish stock sizes, whose fluctua-tions are more often attributed to human exploitation, because most studiesfocus on highly exploited populations (such as the North Atlantic stocks) andover relatively short time periods. However, for a few fish populations, studieson long-term fluctuations have found long ranging trends related to human ac-tivity, mostly through overexploitation and pollution of spawning and nurseryareas, environmental changes that affect the recruitment period inducing natu-ral fluctuations in stock size and biotic processes, such as predation, cannibalismand competition [9] [10] [11].A few years ago Niwa [12] studying the time series of 27 commercial fishstocks in the North Atlantic concluded that the variability in the populationgrowth (the annual changes in the logarithm of population abundance S ( t )) r ( t ) = ln (cid:18) S ( t + 1) S ( t ) (cid:19) (18)is described by a Gaussian distribution. That is, the population variabilityprocess would be a geometric random walk r ( t ) = dS ( t ) S ( t ) = σ r dB ( t ) (19)for some constant σ r depending on the species. The independence of the in-crements of Brownian motion would then imply that r ( t ) is a purely randomprocess. 5f completely accurate this would be a sobering conclusion. Natural processesthat look purely random, are processes that depend on some many uncontrol-lable variables that any attempt to handle them is outside our reach. Thiswould be a serious blow to, for example, the implementation of sustentabilitymeasures.In this paper we reanalyze some of the same type of data to confirm orsharpen the conclusions in [12]. To explore the variability in fish populationgrowth we extracted information on the Spawning-stock biomass (SSB) data oncommercial fish stocks in the North Atlantic. The available SSB time-series dataare derived from age-based analytical assessments estimated by the 2013 work-ing groups of the International Council for the Exploration of the Sea (ICES),based on the compilation of relevant data from sampling of fisheries (e.g. com-mercial catch-at-age) and from scientific research surveys. From the collectionof available assessment data we selected three North Atlantic stocks for whichthe annual time-series of SSB covers at least 60 years, namely Northeast Arc-tic cod (Gadus morhua), Arctic haddock (Melanogrammus aeglefinus) and theNorth Sea autumn-spawning herring (Clupea harengus). At present, ICES clas-sifies these stocks as having above average biomass levels with full reproductivecapacity and being harvested sustainably under active management plans. Thestock assessment detailed information is available at the ICES webpage [13].For these three species we analyze the autocorrelation functions for r ( t ) and | r ( t ) | C ( r, τ ) = E { r ( t ) r ( t + τ ) } σ (20) C ( | r | , τ ) = E {| r ( t ) | | r ( t + τ ) |} σ (21)The results are shown in the Fig.1One sees that, already for time lags of one year, autocorrelations are atnoise level, suggestive of uncorrelated processes. However, if S ( t ) is indeed ageometrical Brownian motion, to rely on correlations or fitting of probabilitydistribution functions is not sufficient. The scaling properties of r ( t ) should bechecked. As Niwa [12] rightly points out, defining r ∆ ( t ) = ln (cid:26) S ( t + ∆) S ( t ) (cid:27) = ∆ X i =1 r ( t + i ) (22)the geometrical Brownian motion hypothesis would imply (cid:0) E (cid:8) r (cid:9)(cid:1) / ∽ ∆ / (23)Normalizing E (cid:8) r (cid:9) by the covariance σ r for each species and taking the averageover all species, Niwa has obtained a behavior roughly consistent with (23).However, when we analyzed each species separately, the hypothesis does nothold. In Fig.2 we have plotted in loglog scale the computed (cid:0) E (cid:8) r (cid:9)(cid:1) / as afunction of ∆ for the three species analyzed in this paper.6
10 15 20−0.500.51 τ Cod5 10 15 200.60.70.80.91 τ τ τ τ τ Figure 1: Autocorrelation functions for r ( t ) (upper plot) and | r ( t ) | (lower plot) −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 ∆ (years) < r ∆ > /
203 5 CodHaddock Herring
Figure 2: (cid:0) E (cid:8) r (cid:9)(cid:1) / as a function of ∆7ne sees that at the species level the geometrical Brownian motion is not agood hypothesis. Even for Herring, where the data seems to follow a scaling law,the slope at large ∆ is closer to 0 . .
5. The conclusion is that whatever isactually determining the stochastic process for each species is somehow washedout when averaging over all the 27 species as Niwa did.Actually this is no surprise. Recall the stochastic analysis result (13). Tofind a process r ( t ) that has features close to Brownian motion, but is not exactlyBrownian motion only means that the process is square-integrable with respectto the (Wiener) measure generated by Brownian motion. All the interestingfeatures actually lie on the dynamics of the process σ ( t ). That is, on thedynamics of the amplitude of the fluctuations. In fact this makes sense inbiological terms because it is known [14] [15] that fishing magnifies fluctuationsin exploited species.To reconstruct the dynamics of σ ( t ) from the data we use a standard tech-nique. First using a small time window we compute the local value of σ ( t )by the standard deviation of r ( t ). For the numerical results presented here awindow of 6 years has been used. Then form the cumulative processes t X i =1 σ ( i ) = β t + R ( t ) t X i =1 ln σ ( i ) = β t + R ( t ) (24) β and β being the average values of σ and ln σ and R and R the cumulativeprocesses of the fluctuations about the average. To obtain a model for the fluc-tuations one looks for the scaling properties of R and R , namely the behaviorof E | R ( t ) − R ( t + ∆) | and E | R ( t ) − R ( t + ∆) | as a function of the timelag ∆.The conclusion is that although not perfect, which would not be expectedwith data covering at most 68 points, we may assume that R and R obey anapproximate scaling law with exponents H in the range 0 . − .
9. Therefore R and R may be modelled by fractional Brownian motion implying that thefluctuations of σ and ln σ , away from an average value, are modeled by Gaussianfractional noise. We have looked for scaling both for the cumulative σ and ln σ todecide which one would provide a simpler model for the amplitude fluctuations.However, with the data available, there is no clear decision. Therefore twoalternative models are proposed for the population fluctuations dS ( t ) = σ ( t ) S ( t ) dB t σ ( t ) = β + α ( B H ( t ) − B H ( t − dS ( t ) = σ ( t ) S ( t ) dB t ln σ ( t ) = β + α ( B H ( t ) − B H ( t − −1 ∆ (years) E | R ( t ) − R ( t + ∆ ) | HerringHaddock Cod 203 5
Figure 3: The scaling behavior of R
53 20CodHaddockHerring ∆ (years) E | R ( t ) − R ( t + ∆ ) | Figure 4: The scaling behavior of R H and H H H Cod 0 .
86 0 . .
89 0 . .
93 0 . H − values one sees that the dynamics of the fluctuations is a longrange memory process. In addition we have found out that the processes seemto be species-dependent. For illustration we plot the cumulative amplitudefluctuations R R in the Figs.5 and 6.Processes with such high H values are almost deterministic processes. Thisis an important outcome because they may provide clues on the causes of thatparticular dynamics.In conclusion, this methodology of separation of the pure random featuresfrom the non-trivial dynamics of amplitude fluctuations may be useful for theanalysis of other natural processes. In fact it has a solid mathematical basis as aconsequence of the stochastic analysis result mentioned in (13). The dynamicsof exploited fish populations provides a good example of a phenomenon wherethe long range dependence features appear not at the level of second ordercorrelations but only through the analysis of higher order effects. H -values associatedto the fluctuations, one sees that the long range memory process seems to bespecies dependent. This is the reason leading to Niwa’s random walk conclusionbecause, when averaging over species, the σ ( t ) process in Eqs. (25) and (26)would be simply replaced by a fixed number σ ( t ) (the average of σ ( t )).2 - A full discussion of whatever is actually determining the stochastic pro-cess for each species is beyond the scope of this study but the question why fishpopulations fluctuate has generated much attention from fishery scientists andmarine ecologists over the past century. Three general hypotheses have beenproposed to answer this question: (i) species interactions generate fluctuat-ing and cyclic population dynamics; (ii) nonlinearity in single-species dynamicsgenerates deterministic fluctuations; and (iii) changes in the environment de-termines variation in vital rates and recruitment, which in turn drive variationin abundance [17] [15] [20]. These are not mutually exclusive hypotheses as all10
950 1960 1970 1980 1990 2000 2010 2020−2−1012 R ( t ) Cod1950 1960 1970 1980 1990 2000 2010 2020−2−101 R ( t ) Haddock1950 1960 1970 1980 1990 2000 2010 2020−1−0.500.51 R ( t ) Herring
Figure 5: Dynamics of the cumulative fluctuations of σ ( t ) R ( t ) R ( t ) Haddock1950 1960 1970 1980 1990 2000 2010 2020−505 R ( t ) Cod
Figure 6: Dynamics of the cumulative fluctuations of ln σ ( t )11hree could act together to increase variability. For exploited species, fishing canalso vary from year to year and translate directly into population variability orcould interact with the other drivers to enhance fluctuations in fish abundance[15] [18].Long range trends are frequently related to external forcing on the popula-tions and are usually derived from human exploitation or environmental change.The dependence in the population growth observed for the haddock, herring andcod stocks could be derived from the different and varying exploitation regimesand/or from large-scale environmental changes as the North Atlantic Oscilla-tion, that can induce low (or high) productivity regimes in fish recruitment (e.g.[16] [22] [23]).Bjørnstad et al. [10] have shown that short-term variability in recruitmentcaused by environmental change, combined with intercohort interactions canbe echoed through the population age structure inducing persistent cycles andlong-term fluctuations. It is also known [21] that noise in recruitment combinedwith a large number of classes of spawners could lead to long-term variations inspawning stock biomass and yields, as well as to regular cycles, depending on thelifespan of the species. Furthermore, the stocks analyzed here showed speciesspecific stochastic processes that should be analyzed considering the contrastinglife history traits. The distinct growth rates, age at maturity, spawning durationand lifespan are characteristics that make some fish stocks more or less vulner-able to exploitation and environmental conditions [15] [19]. The small bodiedand younger herring population should be less able to smooth out environmen-tal fluctuations and more prone to exhibit unstable dynamics due to changingdemographic parameters.3 - There is no doubt that fisheries management would profit from a clearerunderstanding of the mechanisms determining the dynamics of the amplitude ofthe fluctuations in exploited fish stocks. The failure of many fish stocks, despitethe implemented management measures, to recover rapidly to former levels ofabundance, might arguably be related to the long range memory observed inthis study. However, acknowledging the uncertainty arising from the reducednumber of species analyzed stresses the importance of retaining more time seriesof population abundance over long periods, if the aim is to detect and describethe underlying mechanisms that drive population variability. References [1] G. Samorodnitsky;
Long Range Dependence , Foundations and Trends inStochastic Systems 1 (2006) 163–257.[2] G. Dominique;
How can we define the concept of long memory? An econo-metric survey , Econometric reviews 24 (2005) 113–149.[3] P. Doukhan, G. Oppenheim, and M. S. Taqqu;
Theory and Applications ofLong-Range Dependence,
Springer, Berlin 2003.124] G. Samorodnitsky and M. S. Taqqu;
Stable non-Gaussian processes:Stochastic models with infinite variance , Chapman and Hall, New York1994.[5] D. Nualart;
The Malliavin calculus and related topics , Springer, Berlin2006.[6] C. Granger and R. Joyeux;
An introduction to long-memory time seriesand fractional differencing , J. of Time Series Analysis 1 (1980) 15-30.[7] J. Hosking;
Fractional differencing , Biometrika 68 (1981) 165-176.[8] R. Bradley;
Basic properties of strong mixing conditions. A survey andsome open questions , Probability Surveys 2 (2005) 107-144.[9] R. Dickson and K. Brander;
Effects of a changing windfield on cod stocksof the North Atlantic , Fisheries Oceanography 2 (1993)124-153.[10] O. Bjørnstad, J. M. Fromentin, N. C. Stenseth and J. Gjøsæter;
Cycles andtrends in cod populations , Proceedings of the National Academy of SciencesUSA, 96 (1999) 5066–5071.[11] J. M. Fromentin, R. M. Myers, O. Bjørnstad, N. C. Stenseth, J. Gjøsæterand H. Christie;
Effects of density-dependent and stochastic processes onthe stabilization of cod populations , Ecology 82 (2001) 567–579.[12] H.-S. Niwa;
Random-walk dynamics of exploited fish populations
Fishing elevates variability in the abundance of exploited species ,Nature 443 (2006) 859-862.[15] C. N. K. Anderson, C.-h. Hsieh, S. A. Sandin, R. Hewitt, A. Hollowed, J.Beddington, R. S. May and G. Sugihara;
Why fishing magnifies fluctuationsin fish abundance , Nature 452 (2008) 835-839.[16] C. S. Leif, G. Ottersen, K. Brander, K. Chan and N. C. Stenseth;
Codand climate: effect of the North Atlantic Oscillation on recruitment in theNorth Atlantic , Marine Ecology Progress Series 325 (2006) 227–241.[17] A. O. Shelton and M. Mangel;
Fluctuations of fish populations and the mag-nifying effects of fishing . Proceedings of the National Academy of SciencesUSA, 108 (2011)7075–7080.[18] P. Turchin and A. D. Taylor;
Complex dynamics in ecological time series .Ecology 73 (1992) 289–305. 1319] J. Reynolds, S. Jennings and N. K. Dulvy;
Life histories of fishes andpopulation responses to exploitation , In : Conservation of Exploited Speciespp. 147-168, J. D. Reynolds, G. M. Mace, K. H. Redford and J. G. Robinson(eds.), Cambridge University Press, Cambridge 2001.[20] J. E. Overland, J. Alheit, A. Bakun, J. W. Hurrell, D. L. Mackas and A. J.Miller;
Climate controls on marine ecosystems and fish populations , Journalof Marine Systems 79 (2010) 305–315.[21] J. Fromentin and A. Fonteneau;
Fishing effects and life history traits: acase study comparing tropical versus temperate tunas . Fisheries Research53 (2001) 133-150.[22] M. F. Borges, H. C. Mendes and A. M. P. Santos;
Sardine (Sardinapilchardus) recruitment is strongly affected by climate even at high spawningbiomass in West Iberia/Canary upwelling system in Science and Manage-ment of Small Pelagics, S. Garcia, M. Tandstad and A. M. Caramelo (eds.),FAO Fisheries and Aquaculture Proceedings 18 (2011) 237-244.[23] K. Brander;