The maths, worked through
theory · bridge
This page builds on The maths, gently. It is the stepping stone between that primer and the full size-structured PDE. Here we put the same equations to work in one deliberately small example.
The primer introduced the equations in words. The size-structured PDE page presents the full derivations and boundary conditions. This page connects the two. We will take expressions such as \(H(a) = H_0 + \int g\,{\rm d}a\) and evaluate them with a few simple numbers, so that each term has a concrete meaning before we return to the general form.
We will begin with one plant, add a second cohort, introduce shading, and finally zoom out to the landscape. At every step we use the same symbols as the primer. The numbers are toy values chosen to make the arithmetic simple, not FF16 outputs. What matters is the structure of each calculation, not the particular values.
A toy strategy
We begin by assigning three constant rates and two starting values. In a real strategy, these rates depend on size and the competitive environment; the FF16 page gives one such model. Holding them constant here lets us do each calculation by hand:
| Symbol | Meaning | Toy value |
|---|---|---|
| \(g\) | growth rate (height) | \(0.5\ {\rm m\,yr^{-1}}\) |
| \(d\) | death rate | \(0.05\ {\rm yr^{-1}}\) |
| \(f\) | fecundity | \(2\ {\rm seeds\,yr^{-1}}\) |
| \(H_0\) | height at birth | \(0.4\ {\rm m}\) |
| \(S_{\rm G}\) | germination survival | \(0.25\) |
Step 1 — one plant, as three running totals
In Step 2 of the primer, we described a plant’s life with three running totals: growth, survival, and seed production. Because the rates in this example are constant, each integral has a simple closed-form result.
Height — cumulative growth, \(H(a) = H_0 + \int_{0}^{a} g\,{\rm d}a' = H_0 + g\,a\):
\[ H(a) = 0.4 + 0.5\,a. \]
Survival — germination survival discounted by accumulated death risk, \(S_{\rm I}(a) = S_{\rm G}\,\exp(-\int_0^a d\,{\rm d}a') = S_{\rm G}\,e^{-d a}\):
\[ S_{\rm I}(a) = 0.25\,e^{-0.05\,a}. \]
Lifetime seeds so far — fecundity added up, but only over plants still alive, \(\tilde R(a) = \int_0^a f\,S_{\rm I}(a')\,{\rm d}a'\). Doing the integral of the exponential gives \(\tilde R(a) = \tfrac{f\,S_{\rm G}}{d}\big(1 - e^{-d a}\big)\):
\[ \tilde R(a) = 10\,\big(1 - e^{-0.05\,a}\big). \]
Evaluating the equations at a few patch ages gives a small life-history table:
| patch age \(a\) | height \(H\) (m) | survival \(S_{\rm I}\) | cumulative seeds \(\tilde R\) |
|---|---|---|---|
| 0 | 0.4 | 0.250 | 0.00 |
| 5 | 2.9 | 0.195 | 2.21 |
| 10 | 5.4 | 0.152 | 3.93 |
| 20 | 10.4 | 0.092 | 6.32 |
The plant grows at a steady rate, while its probability of survival declines. Cumulative seed production rises quickly at first, then more slowly because fewer plants remain alive to reproduce. Each column shows one of the three integrals.
Step 2 — from one plant to a distribution
Now introduce a second cohort five years after the first. At any later patch age \(a\), the second cohort has had five fewer years to grow, so its height is \(H_0 + g\,(a - 5)\). At patch age \(a = 20\), the two cohorts have reached different heights:
| cohort | born at \(a_0\) | height at \(a=20\) |
|---|---|---|
| A | 0 | \(0.4 + 0.5(20) = 10.4\ {\rm m}\) |
| B | 5 | \(0.4 + 0.5(15) = 7.9\ {\rm m}\) |
Each height also has an associated density, so the two cohorts give us two samples of the size-density distribution \(N(H \mid x, a)\) introduced in Step 3 of the primer. If we add cohorts at more recruitment times, these samples trace out a curve over height. Growth moves every cohort towards larger \(H\), while death reduces its density. The central PDE describes both changes at once:
\[ \frac{\partial N}{\partial a} = -\,d\,N \;-\; \frac{\partial}{\partial H}\big(g\,N\big) \]
Our two-cohort example is a sparse, hand-built version of the distribution in this equation.
Step 3 — why following cohorts is the easy way
Next, follow the density of one cohort as the cohort grows. The characteristic equation from Step 4 of the primer was
\[ \frac{{\rm d}}{{\rm d}a}\ln N = -\left(d + \frac{\partial g}{\partial H}\right). \]
Our toy growth is constant, so \(\partial g/\partial H = 0\) and the second term vanishes. The cohort’s density then obeys
\[ \frac{{\rm d}}{{\rm d}a}\ln N = -d \qquad\Longrightarrow\qquad N(a) = N(a_0)\,e^{-d\,(a - a_0)}, \]
which has the same exponential form as the survival curve in Step 1. This makes sense: when growth does not vary with height, cohorts neither spread apart nor bunch together, so only death changes their density. In a strategy where growth does vary with height, the \(\partial g/\partial H\) term accounts for that extra change. Cohorts bunch together where growth slows and spread apart where it speeds up. The full PDE page develops this general case.
We also need to specify how each new cohort enters the distribution. At the smallest height, the flow of plants must equal the arrival rate of viable seedlings, \(g(H_0)\,N(H_0) = Y_x\,S_{\rm G}\). This boundary condition determines the starting density of each cohort.
Step 4 — closing the light loop, once
So far, the rates have been fixed. In a light-competition strategy, \(g\) and \(d\) depend on light, while the light profile depends on the plants themselves. We can take one pass around this feedback loop by hand. Cohort A (10.4 m) sits above cohort B (7.9 m), so A’s leaves shade B. Suppose the leaf area above B is \(L = 2\). Using the Beer’s-law relationship from Step 5 of the primer, the light reaching B is
\[ E_a = \exp(-k_{\rm I}\,L) = \exp(-0.5 \times 2) \approx 0.37. \]
B therefore receives about one third of full light. Feeding that value back into the strategy would lower B’s realised growth rate below the toy value of \(0.5\). B would grow more slowly than Step 1 assumed, the height gap between A and B would widen, and the shading could deepen. In a full simulation, plant repeatedly updates the light environment and the rates as the patch changes. The full PDE page describes this coupled system.
Step 5 — one patch among many
Finally, zoom out from one patch to the whole landscape. The landscape contains patches of different ages, and each age is weighted by how common it is. Under plant’s default rising disturbance risk, that weight follows \(P(a)\propto S_{\rm P}(a) = \exp(-\lambda a^2)\) (Step 6 of the primer). A landscape-level result is therefore an average of the corresponding patch result over this age distribution.
The seed rain adds one more connection between scales. The seeds that patches produce must match the seed rain that feeds the boundary condition, \(Y_x = \Phi(Y_x)\). Our toy \(\tilde R\) represents one seed’s contribution to that balance, from one cohort in one patch.
Now you are ready
The toy example has now connected each mathematical layer: three integrals describe one plant; cohorts sample a size distribution; the characteristic equation follows that distribution in a moving frame; shading links the cohorts; and patch-age averaging connects one patch to the landscape. The size-structured PDE page develops the same structure formally, including:
- the PDE and its boundary condition, derived rather than assembled;
- the method of characteristics in general, with the \(\partial g/\partial H\) term carrying its full weight;
- the patch-age distribution and the Weibull appendix;
- the seed-rain fixed point for one and for many competing strategies.
Where the rates \(g\), \(d\), and \(f\) actually come from is the subject of the FF16 and assimilation & hydraulics pages; how all these integrals are stepped numerically is the implementation section.