Density Dependent Resource Budget Model for Alternate Bearing

Alternate bearing, seen in many types of plants, is the variable yield with a strongly biennial pattern. In this paper, we introduce a new model for alternate bearing behavior. Similar to the well-known Resource Budget Model, our model is based on the balance between photosynthesis (carbon accumulation) and reproduction processes. We consider two novel features with our model, 1) the existence of a finite capacity in the tree’s resource reservoir and 2) the possibility of having low (but non-zero) yield when the tree’s energy level is low. We achieve the former using a density dependent resource accumulation function, and the latter by removing the concept of the well-defined threshold used in the Resource Budget Model. At the level of an individual tree, our model has a stable two-cycle solution, which is suitable to model plants in which the alternate bearing behavior is pronounced. We incorporate environmental stochasticity by adding two uncorrelated noise terms to the parameters of the model associated with the carbon accumulation and reproduction processes. Furthermore, we examine the model’s behavior on a system of two coupled trees with direct coupling. Unlike the coupled Resource Budget Model, for which the only stable solution is the out-of-phase solution, our model with diffusive coupling has stable in-phase period-2 solutions. This suggests that our model might serve to explain spatial synchrony on a larger scale.


Introduction
"Alternate bearing" is the variability of fruit or nut pro-  (Monselise and Goldschmidt, 1982;Lavee, 2007). 28 Isagi pioneered a simple model to explain the mechanism 29 of variable acorn yield observed at the level of an individ-30 ual tree (Isagi et al., 1997). His model, called the Resource 31 Budget Model (RBM), is based on the dynamics of the tree's 32 energy resource (carbohydrate) which is accumulated as the 33 result of photosynthesis and consumed during the flowering 34 and nut production processes. 35 The Resource Budget Model, as originally proposed by 36 Isagi et al. (1997) and expanded in Iwasa and Satake (2004), 37 assumes the existence of a well-defined threshold for the 38 tree's resource levels below which the plant will not repro-39 duce. This means that during an OFF-year, the tree has no 40 yield. This assumption is appropriate for the plants like olive 41 and citrus, for which there is zero or near-zero yield dur-42 ing an OFF-year, but represents other species with low but 43 positive OFF-year yields less well. Once the resource level 44 of the tree exceeds the threshold, flowering and nut produc-45 tion happens. Both flowering and nut production processes 46 are costly and result in the depletion of the tree's resource 47 reservoir. The cost of flowering and nut production is as-48 sumed to be proportional to the amount of resources above 49 the threshold with the depletion coefficient (the parameter of 50 the model). The Resource Budget Model belongs to the cat-51 egory of tent maps for which there is no stable period-2 so-52 lution at the level of individual tree (except at the bifurcation 53 point). Systems of two trees do have an in-phase period-2 so-54 lution, but it is only stable if the trees are coupled via indirect 55 (mean-field) coupling, as with pollination, and not for trees 56 coupled directly (diffusively) through local interactions like 57 root grafting (Prasad et al., 2017). Together, these features 58 make the Resource Budget Model a simple and successful 59 like pistachio, during OFF years. 99 Inspired by an existing data set collected from a pista-  This is achieved by replacing the concept of a sharp cut-off 105 for reproduction (represented by a threshold function) with a 106 continuous function that accommodates the non-zero yield 107 when the tree's current energy level is low. Also, we take 108 into account the fact that there is a maximum capacity for the 109 plant to store photosynthate and therefore, its energy storage 110 cannot grow indefinitely.

111
In Sec. 2 we briefly review the rules and the character- with low yield during OFF-years. We analyze the model by 115 performing a bifurcation analysis. Furthermore, we discuss 116 Figure 1: The orbit diagram of the Resource Budget Model shows that the dynamics of the system goes from a stable fixed point for the depletion coefficient, < 1, to period-four oscillation for a very small range of and then quickly leading to chaos. and apply some necessary constraints on the model to make 117 it biologically meaningful and applicable. In Sec. 4, we add 118 stochasticity to the model to account for environmental vari-119 ation. As the preliminary step to understand the collective 120 behavior of the trees in an orchard or a natural forest, in Sec. 121 5, we study the dynamics of a two-tree system.

123
First we describe the Resource Budget Model: every year, 124 the resource level of an individual tree increases by a con-125 stant amount called . If the resource level exceeds a thresh-126 old, , the plant will flower and bear fruits/nuts which de-127 pletes the energy reservoir of the tree. The cost of flowering 128 is assumed to be proportional to the excess amount of re-129 sources above the threshold with a positive constant . The 130 cost of fruit/nut production is also considered to be propor-131 tional to the cost of flowering. The Resource Budget Model 132 is formulated as, where is the ratio of the cost of fruit/nut production to the 134 cost of flowering. The model can be written in terms of the 135 dimensionless variable = + − as, where = ( + 1) − 1 is called the depletion coefficient. 137 As it is shown in the model's orbit diagram ( Figure 1) 138 and discussed in Appendix A, the model has a stable fixed 139 point for < 1. At exactly = 1 (which is the bifurcation 140 point) the system shows two-cycle behavior. For > 1 the 141 system demonstrates a chaotic period-four oscillation for a 142 very small range of the parameter, followed by a single band 143 chaos (Prasad and Sakai, 2015).  Reproduction, we take into account the following consider- where is the current carbon level, is the tree's max-188 imum capacity to store carbon, and 1 is the efficiency of 189 the Carbon Accumulation process. Figure 2a shows the be-   Finally, we can write the nondimensionalized model as, 216 While equation 6 models the dynamics of a tree's carbon 217 level, the amount of nut production is the observable that is 218 actually measured for each tree. Since nut production is the 219 main sink of the tree's carbon resources during reproduction, 220 it can be taken to be proportional to the cost of reproduction. 221 We use = 1 1+ (− 2 + ) − 1 1+ to denote the nondimension-222 alized yield of a tree at time to also study the dynamics of 223 the observable of the system.

225
In this section we study the behavior of the model for 226 different values of efficiency rates and a fixed value of . 227 For simplicity, we choose the accumulation efficiency and 228 reproductive investment rates ( 1 and 2 ) to always be equal. 229 Therefore, we will have 1 = 2 = . This will simplify the 230 model to a one-dimensional, single parameter map, many ex-231 amples of which have been extensively explored (Strogatz, 232 1994;Devaney, 2003;Feigenbaum, 1980). 233 Figure 3a and 3b show the orbit diagram of the model 234 and the tree's yield ( ) respectively, when = 7. Quali-235 tatively, the orbit diagrams are similar to orbit diagrams of 236 one dimensional unimodal maps with one parameter, like 237 the quadratic map. The model has a stable fixed point for 238 ≲ 6.8. At ≈ 6.8 the first period-doubling bifurcation 239 happens. For 6.85 ≲ ≲ 8.6 the model shows a 2-cycle be-240 havior (the range of the parameter where the alternate bear-241 ing behavior can be modeled). At ≈ 8.6 a second period-242 doubling bifurcation happens and the system switches to a 4-243 cycle oscillation. Next period-doubling bifurcation happens 244

276
The Carbon Accumulation process should always result 277 in the increase of current carbon levels. This means that the 278 result of equation 4 should always be greater than zero when 279 the current carbon levels are below the maximum capacity 280 (i.e. < or < 1). As → ( → 1), 281 the Carbon Accumulation function should approach zero. In 282 other words, = ( = 1) should be the stable fixed 283 point of the model when the reproduction is turned off. This 284 means that the solution of equation 7 (Carbon Accumulation 285 = 0) should be * = 1.
In Figure 2a, * is where each curve crosses the horizontal 287 axis. As we can see, for any finite values of 1 , the solution 288 to the equation 7 is less than 1. This means for * < ≤ 1 289 the result of the Carbon Accumulation function is negative. 290 As it can be seen in Figure 2a, as 1 becomes larger, * gets 291 closer to 1 and the result of the Carbon Accumulation func-292 tion remains positive for a larger range of . Our goal is to 293 find a lower bound for 1 (let's call it ) so that for values 294 of 1 greater than the corresponding * is sufficiently 295 As presented in Figures 2b and 2c, for some combina-314 tions of 2 and , the above condition is not met. To find the 315 acceptable ( 2 , ) pairs, for which the condition is satisfied, 316 we solved equation 9 numerically. The shaded area in Figure 317 5 shows the acceptable pairs of ( 2 , ) for which the cost of 318 reproduction does not exceed the current carbon levels. The stochastic effect of environmental variation plays an 321 important role both in photosynthesis (the carbon accumu-322 lation process) and reproduction. Factors like the amount 323 of CO 2 , the intensity of radiant energy, and the tempera-324 ture affect the process of photosynthesis (Marshall and Bis-325 coe, 1980). On the other hand, the amount of precipitation 326 and the temperature during the reproduction season affect 327 flowering or nut production. We incorporate environmental 328 variability into the model by adding two noise terms to the 329 carbon accumulation efficiency and reproductive investment 330  ters are unlikely to go beyond the acceptable range. Figure   342 6a shows a perfect period-two behavior of the model without 343 noise. Figure 6b-d show the effects of noise with different 344 strengths (variances, ) on the amplitude and the phase of 345 the oscillation.

347
One of the mechanisms behind spatial synchrony, ob-348 served in the masting phenomenon, is the local interaction 349 between trees. The trees planted in proximity to one another 350 interact in complex ways including exchanging their carbon 351 through root grafts (Klein et al., 2016). Grafting is known as 352 direct interaction or diffusive coupling (Prasad et al., 2017). 353 Trees also interact through pollination via external agents 354 (e.g. birds, insects, and wind). This process is considered an 355 indirect interaction and usually implemented in the form of 356 mean-field coupling (  In this section we use diffusive coupling to investigate 373 the dynamics of a deterministic system of two coupled trees. 374 The internal dynamics of each tree is defined by equation 375 6. We usẽ , ( = 1, 2) to refer to each tree's carbon level 376 after photosynthesis and reproduction but before exchange 377 of resources. Each tree shares a fixed fraction of its carbon 378 ( ) with its neighboring tree and receives the same fraction 379 of the second tree's carbon in return. The result is a net flow 380 of carbon from one tree to another. The carbon level of each 381 tree at the beginning of year + 1 is: 382 1, +1 =̃ 1, + (̃ 2, −̃ 1, ) 2, +1 =̃ 2, + (̃ 1, −̃ 2, ).

(11)
To understand the dynamics of this system, we solve 383 equations 11 numerically to construct the orbit diagram. We 384 assume both trees have the same internal dynamics by choos-385 ing the same carbon accumulation and reproduction efficiency 386 rates. Also, similar to previous sections we simplify the 387 model by setting the carbon accumulation efficiency and re-388 productive investment rates to be equal. Therefore, the in-389 ternal dynamics of both trees only depend on one parameter, 390 . We also choose = 7 for both trees. To build the orbit 391 diagram, we follow the technique used in (Hastings, 1993). 392 We choose 20 random initial conditions for each choice of 393  ble two cycle solution is the out-of-phase solution. There-513 fore, the Resource Budget Model cannot reproduce the spa-514 tial synchrony observed among female trees of the diecious 515 plants for which root grating (direct coupling) is the main 516 interaction mechanism. Our analysis shows that our new 517 model for alternate bearing with direct coupling has stable 518 in-phase period-2 solutions for a wide range of parameters 519 and different values of coupling strength ( ). Having stable 520 in-phase solutions can be interpreted as the primary require-521 ment for the model to be used in studying spatial synchrony 522 at a larger scale.  In that case, the system will come back and settle in one of 601 the period-2 attractors inside the continuum. to be 1 = − and 2 = − − 2 . Therefore, the fixed point 614 of the system is only stable if ( + 2 ) < 1.

II) Both trees oscillating between two positive values 617
In this case, the model is the same as equation B.1. If 618 the trees are in-phase with the same amplitude, 1, = 2, , 619 there is no net flow of resources between the trees and the 620 systems will be the same as two uncoupled trees. Since for 621 an individual tree there is only a continuum of 2-period so-622 lutions when = 1, the trees will stay in-phase only if they 623 are started with equal resource levels and = 1.

646
We began our discussion assuming that 1, > 0 and 2, ≤ 647 0. Our final result for 1, is in agreement with our initial 648 assumption. But for 2, to be less than or equal 0, ( + 2 ) 649 should be greater than or equal one (( + 2 ) ≥ 1).

650
To analyze the stability of this solution, we can analyze  where 1 > 1 is the carbon accumulation rate, and is 667 the maximum capacity of the tree to accumulate carbon. We 668 can nondimensionalize equation C.1 by defining = 669 and multiplying both sides by 1 . We will have, As we can see in Figure C.2, the orbit diagram of this 675 version of the model is similar to the orbit diagram of the 676 version proposed in the main text (Figure 3a). This is an 677 affirmation of the robustness of the model.