Spatial Phase Synchronisation of Pistachio Alternate Bearing

15 Nonlinear physics and agroecosystems can be of great relevance in the synchronisations 16 of chaotic oscillators. The endogenous dynamics of the seed production of perennial 17 plant species which include alternate bearing and masting, portray typical 18 synchronisation patterns in nature and can be modelled using a tent map known as a 19 resource budget model (RBM). This study investigates the collective rhythm in 9,562 20 pistachio trees caused by their endogenous network dynamics and exogenous forces 21 (common noise). Common noise and a local coupling of RBMs are the two primary 22 factors emerging from the bearing phase synchronisation in this orchard. The in- 23 phase/out-of-phase analysis technique quantifying the strength of the phase 24 synchronisation in trees (population /individual) allows us to study the observed spatial 25 synchrony in detail. We demonstrate how three essential factors, i.e. (a) common noise, 26 (b) local direct coupling, and (c) the gradient of the cropping coefficient, explain the 27 spatial synchrony of the orchard. Here, we also show that the methodology employing 28 nonlinear physics to study agroecological systems can be useful for resolving practical 29 problems in agriculture including yield variability and spatial synchrony which often 30 compromise efficient resource management.

synchronisation in trees (population /individual) allows us to study the observed spatial 25 synchrony in detail. We demonstrate how three essential factors, i.e. (a) common noise, 26 (b) local direct coupling, and (c) the gradient of the cropping coefficient, explain the 27 spatial synchrony of the orchard. Here, we also show that the methodology employing 28 nonlinear physics to study agroecological systems can be useful for resolving practical 29 problems in agriculture including yield variability and spatial synchrony which often 30 compromise efficient resource management. 31

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Alternate bearing (biennial bearing) is a common synchronisation in several tree crops 34 in which a year of heavy yield (on-year state) is followed by a light yield (off-year 35 state). Citrus (e.g. oranges, lemons, and mandarins) and nuts (e.g. pistachio, pecan, and 36 walnuts) are typical alternate bearing crops 1-9 that generally show dominant period-two 37 cycle synchronisation. Masting is also a prevalent synchronisation among tree species in 38 which there are multiple and mixed cycles (i.e. period-two, period-three, period-four, or population levels for Citrus and Zerkova seratta 16 . The yield data that were analysed 50 here are remarkable since they included measurements pertaining to 9,562 trees 51 obtained over five years 17 . In this study, by introducing the in-phase/out-of-phase

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Spatial phase synchronisation identified in the orchard 93 Figure 1 shows the spatial phase synchronisation in the orchard. To explain the spatial synchrony of the orchard ,the indices for phase 101 synchronisations were defined as described in the Method-Statistics section. Figure 1 were calculated for each 105 tree i within the block to which tree i belonged to. was high enough to reach 1.0 in 106 the west blocks and was as low as 0.7 in the east blocks. Figure 1(c) shows (K = 1, 107 2, …, 30), which is the average for every five columns. (L = 1, 2, …, 9) is the 108 south to north directional average of for every nine rows. drops from 0.92 to 109 0.7 from west to east ( Fig. 2(b)). Conversely, only a small gradient of was 110 observed along the south-north direction (Fig. 2(c)). These results indicate that the 111 strength of the phase synchronisation significantly decreases from west to east but is In this section, we identify the states of the "on-year" and "off-year" of a population by year" state to "off-year" states in west to east. The coexistence of the "on-year" and 184 "off-year" states within pistachio orchards has only been known as general qualitative  those trees whose cycles are three years ,four years or five-years. For example, in the 223 eastern area, the fraction of period-two (FR(2)) is smaller than that in the west; further, 224 the fraction of period-three (FR(3)) in the east is larger than that in the west. For another 225 example, from the 9,562 trees, we can identify 57 trees whose period are five-years and 226 investigate why they behave in this unique manner.

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The results from the novel approach applied to the yield data of the orchard uncover 228 unique and important features of the phase synchronisations in the orchards:

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(1) The gradient of the strength of in-phase synchronisation from west to east.

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(2) The occurrence of perfect synchronisation in the whole orchard. fluctuates widely over the same range. The map of in i (Fig.5(b)) shows good 283 agreement with that of the real data shown in Fig.1-(a), and the actual and model plots 284 of plots (Fig.5(c)) also shows good agreement. In the absence of coupling (second 285 row), the short range spatial correlation (Fig. 5(d)) is much smaller than that of the 286 presence of coupling case (Fig.5(a)), while the map of in i (Fig.5(e)) and the actual consecutive perfect phase synchronisations in other words, this is the mechanism of the 361 three years consecutive perfect phase synchronisations (i.e. three years mode locking)).

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If CE (14) is at least greater than 6, t=14 will be a weak "on-year" state with partial 363 desynchronization and this mode lock will terminate.

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In the opposite case, t=6 was an "on-year" with a weak synchronisation mode (see The gradient of the strength of phase synchrony is a unique feature of the orchard.

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The results led us to one conclusion: the cropping coefficient (m) increases from 1 to 1.  In the pistachio orchard, male trees are evenly located; thus the pollen density is 404 uniform and the soil type is also uniform with a small geographical slope. It is also hard to assume that Z changes from 1 to 1.6 in the west to east of the orchard. Therefore, the Observing the trends of on-off cycle in circle maps expressed in Fig.3 and , it is  This is challenging in terms of both nonlinear physics and farming operations. In 451 orchards and forests, the uniformity of individual trees is ensured, and accurate spatial 452 arrangement information can be easily obtained. Therefore, the proposed approach can 453 be applied to tree crops other than pistachio and forest trees. In other words, nonlinear 454 physics is useful in fruit production, tree production, and ecosystem management and is 455 expected to be widely applied to these fields. (14) 544 We assume m varies from tree to tree in the orchard,      Spatio-temporal behaviour of phase synchronisations caused by endogenous dynamics with exogenous force. The numerical experiments were conducted with the best t external force (common noise) eC(t) and initial values of Si(1) (i = 1, 2, …, 9,562) determined in Fig.5-(b) and calculated 24 () and FIN(t).