Optimal allocation patterns and optimal seed mass of a perennial plant

We present a novel optimal allocation model for perennial plants, in which assimilates are not allocated directly to vegetative or reproductive parts but instead go first to a storage compartment from where they are then optimally redistributed. We do not restrict considerations purely to periods favourable for photosynthesis, as it was done in published models of perennial species, but analyse the whole life period of a perennial plant. As a result, we obtain the general scheme of perennial plant development, for which annual and monocarpic strategies are special cases. We not only re-derive predictions from several previous optimal allocation models, but also obtain more information about plants’ strategies during transitions between favourable and unfavourable seasons. One of the model’s predictions is that a plant can begin to re-establish vegetative tissues from storage, some time before the beginning of favourable conditions, which in turn allows for better production potential when conditions become better. By means of numerical examples we show that annual plants with single or multiple reproduction periods, monocarps, evergreen perennials and polycarpic perennials can be studied successfully with the help of our unified model. Finally, we build a bridge between optimal allocation models and models describing trade-offs between size and the number of seeds: a modelled plant can control the distribution of not only allocated carbohydrates but also seed size. We provide sufficient conditions for the optimality of producing the smallest and largest seeds possible.

💡 Research Summary

The paper introduces a continuous‑time optimal allocation model for perennial plants that explicitly incorporates a storage compartment and covers the entire life span, including periods when photosynthesis is not possible. The plant is represented by three state variables: vegetative mass (x₁), reproductive mass (x₂), and storage mass (x₃). Photosynthetic production is a function f(x₁) of vegetative mass, while the maximal release rate from storage is g(x₃). Climate effects are modeled by ζ(t) (photosynthetic potential), µ(t) (vegetative loss), and ω(t) (storage loss). Controls v(t) (total allocation rate) and v₁(t) (allocation to vegetative tissue) determine how stored carbohydrates are redistributed. The fitness objective is the expected total seed yield, expressed as the integral of the reproductive growth rate weighted by the survival probability L(t). Using Pontryagin’s Maximum Principle, the authors derive a Hamiltonian and adjoint variables p₁(t) and p₃(t). Comparing p₁, p₃, and L yields three distinct phases: vegetative (V) when p₁ dominates, reproductive (R) when L dominates, and storage (S) when p₃ dominates. The model predicts that plants may begin rebuilding vegetative tissues from storage before favorable conditions arrive, improving subsequent growth. Special cases recover known strategies: annual plants with single or multiple reproductive periods, monocarpic (single‑reproduction) perennials, evergreen polycarpic perennials, and others. Numerical simulations illustrate how parameter changes (e.g., higher vegetative loss µ or lower storage loss ω) shift the optimal sequence of phases. The framework is then linked to the classic size‑number trade‑off in seed production. If the photosynthetic rate function f is concave (per‑unit photosynthesis declines with size), the optimal strategy is to produce many small seeds; if f is not concave, producing few large seeds is optimal. The model also allows the plant to choose germination timing, integrating dormancy and senescence effects. Overall, the study provides a unified theoretical platform that connects optimal resource allocation, phenology, and seed‑size evolution in perennial plants.