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: same equations, now put to work on one deliberately tiny example.
The primer gave you the equations in words; the size-structured PDE page gives them in full, with derivations and boundary conditions. In between sits a gap that is easy to underestimate: it is one thing to read \(H(a) = H_0 + \int g\,{\rm d}a\) and quite another to feel what it does. This page closes that gap the way most people actually learn a model — by pushing a handful of numbers through it.
We will follow one plant, then two, then let them shade each other, then zoom out to the landscape — using the same symbols as the primer at every step. The numbers here are toy values chosen for arithmetic, not FF16 outputs; the point is the shape of the calculation, not the figures.
A toy strategy
Give our plant three constant rates and two survival numbers. Real models make these depend on size and light (that is what the FF16 page is for) — holding them fixed just lets us do the sums 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
The primer said a plant’s whole life is three running totals of these rates (Step 2 there). With the rates held constant, each integral collapses to something you can evaluate directly.
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 \(\tilde R(a) = \tfrac{f\,S_{\rm G}}{d}\big(1 - e^{-d a}\big)\):
\[ \tilde R(a) = 10\,\big(1 - e^{-0.05\,a}\big). \]
Read across a few patch ages and the abstract integrals become a life history:
| 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 climbs steadily, its survivorship decays, and its seed tally rises fast early then flattens as fewer of its kind are left to reproduce. Three integrals, three columns.
Step 2 — from one plant to a distribution
Now germinate a second cohort five years after the first. At any later patch age \(a\) it has had five fewer years to grow, so its height is \(H_0 + g\,(a - 5)\). Line the two cohorts up at \(a = 20\):
| cohort | born at \(a_0\) | height at \(a=20\) |
|---|---|---|
| A | 0 | \(0.4 + 0.5(20) = 10.4\ {\rm m}\) |
| B | 5 | \(0.4 + 0.5(15) = 7.9\ {\rm m}\) |
Those two heights, each carrying a density, are two samples of the size-density distribution \(N(H \mid x, a)\) the primer introduced in Step 3. Add a cohort every few years and the samples fill in a curve over height — and because growth keeps carrying every cohort upward while death thins each one, the whole curve slides toward larger \(H\) and flattens as it goes. That sliding, thinning curve is what the central PDE
\[ \frac{\partial N}{\partial a} = -\,d\,N \;-\; \frac{\partial}{\partial H}\big(g\,N\big) \]
describes in one line. We have just built it by hand, one cohort at a time.
Step 3 — why following cohorts is the easy way
Watch what happens to a single cohort’s density as it rides upward. The primer’s characteristic equation (Step 4) 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
\[ \frac{{\rm d}}{{\rm d}a}\ln N = -d \qquad\Longrightarrow\qquad N(a) = N(a_0)\,e^{-d\,(a - a_0)}, \]
exactly mirroring the survival curve from Step 1 — sensible, since with no stretching or bunching the only thing changing a cohort’s density is death. The moment growth varies with height (as it does in any real strategy), the \(\partial g/\partial H\) term switches on and cohorts bunch up where growth slows and spread out where it speeds up. Seeing the clean special case first makes the general term easy to place when you meet it on the full PDE page.
The boundary condition is where new cohorts enter. At the smallest height the flux of plants must equal the arriving seedlings, \(g(H_0)\,N(H_0) = Y_x\,S_{\rm G}\) — that is what fixes the density each new cohort starts with.
Step 4 — closing the light loop, once
So far the rates were fixed. In the real model \(g\) and \(d\) depend on light, and light depends on the plants themselves. Take one turn around that loop by hand. Cohort A (10.4 m) sits above cohort B (7.9 m), so A’s leaves shade B. If the canopy above B holds a leaf area of, say, \(L = 2\), then by the primer’s Beer’s-law shading (Step 5) the light reaching B is
\[ E_a = \exp(-k_{\rm I}\,L) = \exp(-0.5 \times 2) \approx 0.37. \]
B is growing in about a third of full light. Feed that back in and B’s realised growth rate is lower than our toy \(0.5\) — so it climbs more slowly than the table in Step 1 assumed, the gap to A widens, and the shading deepens further. That mutual feedback, resolved not once but to a consistent state, is the competition the full PDE page solves properly.
Step 5 — one patch among many
Finally, zoom out. Everything above lived inside a single patch of age \(a\). The landscape holds patches of every age, weighted by how common each age is — \(P(a)\propto S_{\rm P}(a) = \exp(-\lambda a^2)\) for plant’s default rising disturbance risk (Step 6). Every landscape-level number is our per-patch quantity averaged under that weight, and the seed rain that patches produce must equal the seed rain that feeds their boundary condition — the fixed point \(Y_x = \Phi(Y_x)\). Our toy \(\tilde R\) was one seed’s contribution to that balance, from one cohort, in one patch.
Now you are ready
You have now assembled the whole model from the bottom up with real numbers: three integrals per plant, cohorts stacking into a distribution, a characteristic equation that is just those integrals viewed from the plant’s moving frame, a shading loop, and a landscape average pinned by a fixed point. The size-structured PDE page takes it from here with the full derivations:
- 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.