Abstract
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 or other limiting resource 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 resource 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 nutrient 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 direct coupling has stable in-phase period-2 solutions. This suggests that our model might serve to explain spatial synchrony on a larger scale.
Highlights
We introduce a new model for alternate bearing behavior
First we describe the Resource Budget Model: every year, the resource level of an individual tree increases by a constant amount called Ps
The numerical simulations of the Resource Budget Model with direct coupling for a system of two trees show that the only possible period2 solution for the trees is the out-of-phase solution (Prasad et al, 2017). We confirmed these results by performing a stability analysis of the coupled Resource Budget Model as discussed in Appendix B. These results suggest that the Resource Budget Model cannot model the spatial synchrony observed among dioecious plants, like pistachios, for which the direct coupling is the main method of interaction
Summary
First we describe the Resource Budget Model: every year, the resource level of an individual tree increases by a constant amount called Ps. If the resource level exceeds a threshold, LT , the plant will flower and bear fruits/nuts which depletes the energy reservoir of the tree. The cost of flowering is assumed to be proportional to the excess amount of resources above the threshold with a positive constant a. The cost of fruit/nut production is considered to be proportional to the cost of flowering. The Resource Budget Model is formulated as, St+1 =. St + Ps > LT where Rc is the ratio of the cost of fruit/nut production to the cost of flowering. Discussed in Appendix A, the model has a stable fixed point for m < 1. For m > 1 the system demonstrates a chaotic period-four oscillation for a very small range of the parameter, followed by a single band chaos (Prasad and Sakai, 2015)
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