Alternate bearing and possible long-range communication of Olea europaea
aa r X i v : . [ phy s i c s . b i o - ph ] J a n Alternate bearing and possibile long-rangecommunication of
Olea europaea
Sergei Esipov
Quant Isle Ltd., New York, USAand
Clara Salue˜na
Department of Mechanical Engineering,Av. Pa¨ısos Catalans 26, 43007 Tarragona, Spain
July 28, 2018
Abstract
Spatio-temporal analysis typically performed in horticulture and in statistical physicsreveals persistent correlations of olive yields which range depend on the size of the av-eraging regions. Mapping spatially correlated regions unveils areas which resemblehistorical spread of
Olea europaea . These yield patterns are remarkable given the in-tensive nature of modern agriculture, and cannot be attributed to weather or pollenindices due to inability of these variables to properly predict yields, and their differentcorrelation patterns. Long-range correlations between yields of olive trees may indicatelong-range communications among trees. Introduction
Another problem of evaluation, useful when discussing alternation,seems to have not been quantitatively approached: synchronizationof different plants within a single orchard or of different orchardswithin a single region. Such an attempt would provide a basis toevaluate to what extent external factors (common to a grove, anarea, etc.) are dominant as against internal factors of trees or factorscommon to a restricted area, such as microclimate,soil-rootstock-cultivar relationships, etc. [1], p. 131
About four decades have passed since the above request for a quantitative model ofalternate bearing was written, these decades saw an intensifying research into the originand control of alternate bearing - a phenomenon where crops alternate, year after year, inan approximate ’year on - year off’ but otherwise seemingly random fashion. Olive trees,with their praised fruits, and relatively detailed historical records, are similar to many otherplants in this regard. A year of abundance is usually followed by a year of low crops, and viceversa, but not necessarily, and not everywhere. Alterations extend over at least 5-6 ordersof magnitude in space, ranging from a single olive tree (leaving aside the branch-to-branchalterations) to scales of hundreds if not thousands of kilometers, at times - across a bodyof water. This phenomenon is known to resist human control which routinely includes suchdrastic measures as cultivation of young trees, aggressive annual pruning, planting trees ina optimized dense grid [2], and a wide spectrum of biochemical applications.On one side, truly biannual alterations could only be in one of two phases (on or off for agiven tree in a given year), which would make quickly decaying random spatial patterns afterspatial averaging were it not for synchronized regions which may extend over hundreds ofkilometers, and include geographically separated trees. On the other side, there is no knownclear predictor of yield at large scales (except, obviously, the yield itself), while there is noshortage of factors considered, [3]. Among external factors, attempts to use pollen [4], [5],and weather-related variables for crop predictions are repeatedly revisited [6], [7], [8], [9], [10].Below, starting with data analysis, we briefly present a comparison of different correlationmeasures which either have been of could be of interest regarding alternate bearing, andargue that these measures cannot be explained endogenously, suggesting a potential long-range communication between olive trees. By age anywhere from one to three dozen years, depending on local agricultural practice, the trees areconsidered to lose their ’vigor’, and are replaced by young trees. Spatio-temporal correlations in olive plantations
In this section we will address the correlations of the surface density of the annual olive crops.This variable is termed ’yield’ in agriculture. The time scale t , for yield, Y t ( x ) is discrete(integer indices t , t + 1, t + 2, ... refer to years), while x are spatial scales. Historical data areseldom available at the sub-tree level, but tree-level data are sometimes available at researchinstitutions, and public data can be found at county, province and country level and can beobtained from government or non-for-profit sources and from The Statistics Division of TheUnited Nations [11].We first perform a study of the type usually done in statistical physics [12]. Since plantsare inherently Malthusian systems, we consider correlations of normalized logarithmic yields, y t ( x ) = 1 σ ( x ) [log Y t ( x ) − h log Y ( x ) i ] , (1)where h ... i is an average over multiple years (hence, no time index), and σ ( x ) is the standarddeviation of the logarithm (again, no time index). The average ρ ( t ′ − t ) = h y t ( x ) y t ′ ( x ) i , (2)is then the correlation coefficient over the lag of t ′ − t years, called the autocorrelation below.Here t ′ − t = 1 , , ... . A certain degree of alternation is observed in the time series of yield,and one can see in Table 1 that on average one-year autocorrelations are negative, ρ (1) < ρ ( x ′ − x ) = h y t ( x ) y t ( x ′ ) i (3)is a spatial correlation coefficient over the distance x ′ − x . An important property of thiscorrelation is its decay, which we will model by means of a single correlation length, ρ ( x ) ∝ exp (cid:0) − x /L (cid:1) . (4)Remarkably, the fitted correlation length, L , is found to depend on the dataset scale, seeFig1. For the province of Tarragona it is 67 km, for Andalusia - 189 km, and for the Eu-rope/Africa/Middle East dataset it is 603 km, see also Table 1. The largest of thesecorrelations lengths is comparable to the latitudinal span of The Mediterranean Sea. Scale- This allows one to focus more on the spatio-temporal relationships rather than on the details of the yieldmagnitude. Similar phenomena exist in turbulence, where fluid velocity correlations depend on the sampling scale,because progressively bigger eddies begin to contribute [12]. ρ (1) L , km B I B S I S Montsi`a 2008-2013 − . ± .
23 n/a 1 0.28 n/a n/aM´alaga 1999-2013 − . ± .
18 n/a 0.77 0.17 n/a n/aTarragona 2006-2013 − . ± .
35 67 0.83 0.15 0.5 to 1 0.37 to 0.45Lleida 2006-2013 − . ± .
35 248 0.5 0.18 0.5 to 1 0.31 to 0.36Andalucia 1999-2013 − . ± .
23 189 0.61 0.18 0.5 to 1 0.20 to 0.32Spain 1999-2013 − . ± .
16 n/a 0.5 0.11 n/a n/aWorld 1999-2013 − . ± .
24 603 0.52 0.03 0.52 to 0.55 0.40 to 0.57Table 1: Different measures of alterations. ρ is the yield one-year autocorrelation, (2), L is the correlation length (3), B - biennuality (5), I B - intensity (6), S - synchronization (7), I S - relative amplitude of spatial fluctuations (8). At the locations, where we did not havesubdivision data available, spatial indices are given as ’n/a’. Synchronicity, S , depends ontime, and the minimum and maximum values are given. M´alaga is chosen over Jan becausethe latter determines the entire Andalusia, and its values are close to Andalusian ones.dependent spatial correlations of olive yields indicate a presence of hierarchical spatial struc-tures. One should then expect that there are subsets of locations where correlations decaymuch slower with distance, as compared to what (3) prescribes, and subsets which decaymuch faster. And indeed, further investigation shows that correlations decay slower withdistance in Lleida province of Catalonia than in Tarragona, and that C´ordoba, Granada,Ja´en, and M´alaga act as a single entity in Andalusia, and the same can be said, for example,about olive yields in Turkey, Syria, Lebanon, Cyprus, et cetera. It is then of interest toobtain a correlation map, which gives a visual representation of these connections, see Fig2. A large number of factors which are exogenous to olive trees have been considered as yieldpredictors. These factors include weather (namely, temperature, humidity and precipitation)and solar irradiation. As these factors are themselves time-dependent, various filtered sumshave been suggested as yield predictors [6], [7], [8], [9], [10]. In addition the significance ofolive pollen has been recognized, [4], [5]. A belief that there might exist a very detailedweather-related predictor, along the lines of climate definitions and cluster analysis is persis-tent due to complexity of weather records. The following paragraph is taken from [10], “Themeteorological variables considered in the present study were: monthly average of maximum4
50 100−1−0.8−0.6−0.4−0.200.20.40.60.81 L = 74km S pa t i a l C o rr e l a t i on County to county in Barcelona 0 200 400−1−0.8−0.6−0.4−0.200.20.40.60.81 L = 245kmProvince to province in Andalucia 0 500 1000 1500−1−0.8−0.6−0.4−0.200.20.40.60.81 L = 636kmCountry to country Figure 1: Decay of spatial correlations ρ at different scales. x -axis is distance in kilometersin all subplots. Vertical lines show error bars. Red lines are fits using Eq (3) with correlationlength L given below each subplots.and minimum temperatures ( ◦ C) and relative humidity (%), monthly accumulated precip-itation (mm) and evapotranspiration (mm day1), summer period (21st of June to 21st ofSeptember) accumulated mean temperature ( ◦ C) and accumulated precipitation (mm), andaccumulated precipitation since the pre-flowering start date (end of March) until the peakpollination day (mm).” 5igure 2: Green and pink countries submitted official reports to FAOSTAT for at leasta decade. The countries, which are pairwise correlated with ρ > /
2, are colored greenand connected by straight blue lines. The entire countries are colored (even if their oliveplantations are far from being uniform within country’s borders). The correlated countriesform two clusters. Italy, Albania and Greece form one, and seven countries in the Middle Eastform another cluster. In Spain we show a province-level cluster in Catalonia and Andalusia.We do not have data for other provinces in Spain.Indeed, the scales of the largest clusters in Fig2 are comparable to correlation lengths formonthly averages of weather variables used in climate definitions. Nevertheless, the maps ofspatial correlations of pollen, which reflect wind patterns, and other weather-based variableslook quite different from Fig 2 (see below). Since the yield time series which are regressed on these external factors are relativelyshort (they usually cover a couple of decades), it is always possible to find a meteorologicalcandidate for yield regression, especially using filtered variables (such as a variable exceedingan optimized threshold). While the original findings of Hartmann and Porlingis [6] have beenverified on numerous occasions, it is important to have sufficient out-of-sample analysis to See monthly maps of wind, temperature, precipitation, etc available from the International ResearchInstitute for Climate and Society, Earth Institute, University of Columbia, https://iri.columbia.edu.
Horticulturists do not consider temporal or spatial correlations introduced above. Instead,they rely on frequency (known as B , for ’biennuality’) and severity (known as I , for ’inten-sity’) of alterations, defined as [1] B = 1 n − n − X t =2 sign ( Y t +1 − Y t ) sign ( Y t − Y t − ) , (5) I B = 1 n − n X t =2 | Y t − Y t − | Y t + Y t − (6)Bienniality indicates how frequently a trend changes, while intensity evaluates relative am-plitudes of ’swings’. Both quantities supersede correlator (2) which origin is in gaussian (i.e.normal) statistics. On the other side, bienniality specifically focuses on two-year periods,while many olive plantations display more complex patterns (e.g. Ja´en in 1999-2006 hada triennial alternation, rather than biennial). Both quantities have their analogs in termsof spatial correlations. Their spatial frequency is accessed through the ’synchronization’parameter S = 12 + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A X x sign [ Y t ( x ) − Y t − ( x )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (7)which measures relative excess of trend increase or decrease in a given area. The value S = 0 . S = 1 is full synchronization. We have notseen a measure of the amplitude of spatial fluctuations (an analog of the intensity above)to be considered in horticultural literature, as it is not an exclusive measure of alternative7earing. An index of relative spatial fluctuations can be defined as I S = 1 A X x,x ′ | Y t ( x ) − Y t ( x ′ ) | Y t ( x ) + Y t ( x ′ ) . (8)Examples of numerical evaluation of these parameters can be found in Table 1. One can seethat coarse-graining non only leads to decreasing of absolute values of one-year autocorrela-tion, ρ (1), but also (i) the biennuality, B , is similarly suppressed, while remaining positive,(ii) the intensity of alternations, I B , consistently diminishes with a characteristic distance of10 km, (iii) the spatial synchronization, S , of fluctuations becomes small only at the globallevel (Lleida or Andalusia, represented by counties, could still have S = 1), and (iv) theindex of relative spatial fluctuations, I S , remains large at any level.Thus, the alternate bearing can be averaged over, but it takes an effort: several decadesin time and global spatial scales. In other words, for the data in Table 1 the alternate bearingcan be found on all spatial scales. Data for an experimental grove, where different rootstocks are used, and trees are cared forin the same fashion as in agriculture, are particularly valuable, since they might containdetailed information about alternate bearing. These data require a considerable dedication:one has to maintain experimental groves for decades (IRTA).The trees in the grove were arranged in a rectangle on a square grid: 10 blocks, 11 treesper block.The overall behavior of the grove yield is shown in Fig 3. During the first 4 years ofbearing (1989-1992, where 1989 is not shown), young trees grew exponentially in size andgave increasing yields, while their spatial synchronization, I S , reached maximum in 1990,temporal synchronization, S reached maximum in 1991, and yield, Y t reached maximumin 1992, accompanied by a decrease in both S and I S . Then, in 1993-1997 we have aconsolidation phase, in response to increasing pruning, where synchronization in time isfully recovered as soon as pruning eased in 1996, while synchronization is space plunged toits minimum, allowing trees a semi-individualistic freerun. The yields reached a maximumin 1997 and were again pruned (c.f. the red line). In response to pruning, alternate bearingwas fully developed by year 1999 with large negative autocorrelation ρ (1) = − .
8, andthe grove entered an alternating regime where synchronization in time remained high, andsynchronization is space was alternating out-of-phase with the yield (high Y t correspondedto lower I S ). In year 2007 trees were severely pruned, and it was accompanied by a phaseshift of alternate bearing: year 2008 is the first even calendar year with a high yield. Overall8here is a clear connection between alternate bearing and pruning: pruned olive trees entera consolidation stage, regroup and re-synchronize in space to produce high yield in the year t + 1 following excessive pruning in year t . Figure 3: Time series of normalized net yield (rescaled to maximal yield ever observed) -green, synchronization parameter, S , - blue and spatial synchronization, I B , - magenta, andnormalized tree volume - red.To understand whether yield synchronization is a collective phenomenon we compara-9ively time traced the behavior of the individual trees with maximal yields. For every twoconsecutive years, beginning with year 1990, a tree with the highest yield was selected, andits yield time series were plotted jointly with that of a tree which had the highest yieldin the subsequent year (beginning with 1991, respectively). In Fig 4 we show subsequentsynchronization of such consecutive leaders. As one can see, in many cases it took about twoyears for ex-leaders to synchronize in terms of alternate bearing. Some pairs (i.e. 1998/1999)did not subsequently synchronize, however in such infrequent cases the yield of one of theleaders subsequently declined. This may indicate that synchronization represents a healthybehavior. Some pairs (i.e. 2004/2005) were already synchronized even while being leadersin subsequent years. Interestingly, a minority subset of trees can always be identified whichexperiences alternate bearing out-of-phase with the majority. The lifetime of this out-of-phase alternate bearing does not exceed a couple of years, when the participating minoritytrees rejoin the quorum, while few new majority trees may join the out-of-phase minority.The out-of-phase alternate bearing may be used by the tree community for exploration of’predators’ appearance during the majority off-years, and whether it is beneficial to switchto the opposite on-off phase instead.Fig 4 shows that synchronization is reestablished in majority of cases, and it is not,therefore, reducible to external factors. Indeed, if a particular winter chilling were to lead toan increased yield in a given year (say, year 1997 when the alternating pattern started), thiscannot explain why leaders of the consecutive subsequent years had any incentive first toappear and then to re-synchronize with the majority. While external factors, such as winterchilling, are significant, they cannot substitute for the drivers of self-regulation.Since Fig 4 only accounts for leaders, it is indicative of synchronization affecting thecases of maximal severity. It is even more instructive to see what happens to frequency ofparticipation in the majority and minority of trees experiencing alternate bearing. With thisin mind in Fig 5 we plotted the number of trees in both groups, along with the number oftrees in transition between majority and minority and vice versa.As one can see, the grove first maintains a comparable number of trees in both majorityand minority groups (in terms of ’on/off’ years), and most trees do not participate in eithergroup. Beginning with year 1997, when volume reduction due to pruning was (sufficiently?)large, the majority group for the first time reaches half of the grove, then the minority isgiven a chance to increase, but none of these adjustments helped to reduce pruning, and themajority is given a chance to approach 85% of the trees. When even this does not help,in 2007, following intensive pruning, a switch to minority ’on/off’ pattern is performed bythe entire grove skipping one ’on’ year, accumulating resources, and reaching the highest netyield ever in 2008. This section is a ’local’ discussion, where communication takes place within the grove. In view of the
990 2000 2010050100 1990 2000 2010050100 1990 2000 2010050100 1990 2000 20100501001990 2000 2010050100 1990 2000 2010050100 1990 2000 2010050100150 1990 2000 20100501001501990 2000 2010050100150 1990 2000 2010050100150 1990 2000 2010050100150 1990 2000 201001002003001990 2000 20100100200300 1990 2000 2010050100150 1990 2000 2010050100 1990 2000 2010050100
Figure 4: Synchronization of leaders in two subsequent years. The blue circle is the maximalyield in a given year t , and blue line connects yields of that ’blue’ tree in all other years. Themagenta circle is the maximal yield in the following year, t + 1, and magenta line connectsyields of that ’magenta’ tree in all other years. The x -axis is calendar years, and y -axis isyield in kilograms.In terms of spatial synchronization we did not observe any persistent spatial patterns.Synchronized trees seemed to occupy random slots, and so did out-of-phase trees. In view correlation studies in the previous sections we expect that the grove may participate in external communi-cations.
990 1995 2000 2005020406080100120 Number of participating trees minoritymajoritymaj2minmin2majtotal1990 1995 2000 200500.20.40.60.81 Fraction of trees in transition maj2min fractionmin2maj fraction
Figure 5: Top subplot. Number of trees participating in the majority group having either ’on’ year ( Y t > Y t − and Y t > Y t +1 ) or ’off’ year ( Y t < Y t − and Y t < Y t +1 ) is given by thegreen line, minority group - having the opposite year - is given by red line, trees which weremembers of the majority group and ended up as minority 2 years later are shown in cyan,and minority group trees synchronized with the majority are shown in magenta. Bottomsubplot. The fractions obtained by dividing trees in transition to the total number of trees inthe group they started from. In year t = 1993 all trees had a yield increase, Y t − < Y t < Y t +1 ,so that this year cannot be classified as either ’on’ or ’off’ year for any tree.12f high synchronization, this implies that the communication between the trees is long-ranged, since, for the randomness to be produced, every tree had to have information atleast about the average state of the entire grove if not about every other tree in the grove.The mechanism of this long-ranged communication(s) remains to be identified. Note thatthis mechanism is not reducible to interaction via pollen (which is known to be correlatedwith yield, see above), since fruit bearing requires considerable resources to be accumulatedin advance. Moreover, in view of the complex multistage phenological process unfoldingduring flowering and fruit bearing, any synchronization mechanism has to operate at leaston multiple occasions if not continuously.We conclude that alternate bearing is a collective phenomenon, accompanied by syn-chro-ni-za-tion in time, and alternate synchronization in space. Collective strategies, followed bythe grove, are complex and show a feedback to pruning. Spatial synchronization of alternative bearing is usually attributed to exogenous factors,such as pollen or weather indices. We find that the yield correlations do not follow thecorrelations of the ’explanatory’ pollen or weather variables. The observed synchronizationin crop yields, which takes many months for trees to bear, is much stronger than what canbe explained by pollen or weather. Given that trees are adaptive at all stages of their cycle,any persistent long-term, long-range synchronization requires similar and very frequent ifnot continuous communication. Given that mature olives still exist in the region [14], wesuggest to consider whether humans interact with a plant organism of the size comparableto the Mediterranean scales, exceeding the so-called ’Pando’ organism of 43.6 ha of aspentrees [15]. Note than inside the Pando organism one also finds more than one genetic variateeven among aspens.
The authors are most grateful to colleagues at IRTA (Institute of Agrifood Research andTechnology, Generalitat de Catalunya, Spain) for access to experimental data on olive groves.13 eferences [1] S.P. Monselise and E.E. Goldschmidt. Alternate bearing in fruit trees.
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Western NorthAmerican Naturalist , 68(4):493–498, 2008.15 L = 27kmGirona, delay = 0 0 100 200−1−0.500.51 L = 230kmCatalonia, delay = 0 0 100−1−0.500.51 L = 176kmValencia, delay = 0 0 200 400−1−0.500.51 L = 222kmAndalusia, delay = 00 50−1−0.500.51 L = ∞ Girona, delay = 1 0 100 200−1−0.500.51 L = 143kmCatalonia, delay = 1 0 100−1−0.500.51 L = 124kmValencia, delay = 1 0 200 400−1−0.500.51 L = 220kmAndalusia, delay = 10 50−1−0.500.51 L = 34kmGirona, delay = 2 0 100 200−1−0.500.51 L = 219kmCatalonia, delay = 2 0 100−1−0.500.51 L = 77kmValencia, delay = 2 0 200 400−1−0.500.51 L = 135kmAndalusia, delay = 2940 1960 1980 2000012345 Chortoicetes terminifera in AustraliaYear S c a l e o f ou t b r ea k k , yrs ρ ( k ) Olea europaea in ItalyYear
Log o f no r m a li z ed y i e l d k , yrs ρ ( kk