On plant growth and form

September 28, 2013

How quickly does the number of leaves on this plant increase, and why?

The photo above is of a Schlumbergera truncata (or “Christmas cactus”), which my mother (from whom I got the parent plant) calls “Woman’s tongue”. The latter might be seen as a rather sexist definition (pointy and forked..) but I guess it’s an old heritage, from when the world was younger..

Anyway, how fast do the leaves increase in number, and what drives the general form of the plant?
Say we have initially N branches, each that can generate one to three leaves (branching factor p), more or less at random.
If the branching factor were constant and identical for all branches, after one generation we would see 2N to 4N leaves; after two, 3N to 16N, or, for the j-th branch: L_j = \sum\limits_i{l_{i\,j}}.
The number of leaves l_i at each level i is seen to be \Theta(l_{i-1}) where the bound constants are \min(p) and \max(p).

In the real world, a branch has spatial extent, with each leaf occupying a volume, with no intersections or overlaps in \mathbb{R}^3. Moreover, the leaves seek and thrive in sunlight (phototropism), which conditions their orientation and size.
Therefore we could represent a plant as a vector field, with “streamlines” (nutrient direction) and “polarization” (orientation to sunlight).
Gravity, through the self-weight loading of the branches and capillary diffusion of nutrients, obviously plays a part.
We can sketch the processes at play in the following, informal manner. First some notation: for leaf k of level i, the leaf size s_{i\,k}, orientation \alpha_{i\,k}, sunlight exposure (intensity flux) \phi_{i\,k}, viability (function of exposure and distance from the root) v_{i,k}, repulsion potential (function of size and distance to nearest leaf neighbors) \rho_{i\,k}, branch node stiffness \sigma_{i\,k}, which increases as the leaf matures.
Moreover we can define the branch vitality v_b as the sum over all the parent leaves (viability upstream).
s \rightarrow \phi , \rho, \sigma (leaf size influences exposure, repulsion and stiffness)
\phi, v_b, \sigma \rightarrow s, v, \alpha (exposure, viability and stiffness influence leaf size, viability and leaf angle)
\rho \rightarrow \phi (repulsion influences exposure through position: model requires collision and ray-tracing, which require considerable computational effort)
\alpha_{i-1\,k} \rightarrow \alpha_{i\,k}
\alpha \rightarrow \phi, but also
\phi \rightarrow \alpha
As each leaf matures, it becomes larger in size, stiffer and possibly its exposure efficiency reaches a plateau.

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