A theory of demographic optimality in forests

Carbon uptake by the land is a key determinant of future climate change. Unfortunately, Dynamic Global Vegetation Models have many unknown internal parameters which leads to significant uncertainty in projections of the future land carbon sink. By contrast, observed forest inventories in both Amazonia and the USA show strikingly common tree-size distributions, pointing to a simpler modelling paradigm. The curvature of these size-distributions is related to the ratio of mortality to growth in Demographic Equilibrium Theory (DET). We extend DET to include recruitment limited by competitive exclusion from existing trees. From this, we find simultaneous maxima of tree density and biomass in terms of respectively the ratio of mortality to growth and the proportion of primary productivity allocated to reproduction, an idea we call Demographic Optimality (DO). Combining DO with the ratio of mortality to growth common to the US and Amazon forests, results in the prediction that about an eighth of productivity should be allocated to reproduction, which is broadly consistent with observations. Another prediction of the model is that seed mortality should decrease with increasing seed size, such that the advantage of having many small seeds is nullified by the higher seed mortality. Demographic Optimality is therefore consistent with the common shape of tree-size distributions seen in very different forests, and an allocation to reproduction that is independent of seed size.

1 Detailed proofs of deriving the equations for total forest properties Equilibrium forest properties such as total forest stem density N , total forest biomass M , total forest net growth G, coverage ν and net forest assimilate P can be obtained through integration of the equilibrium RED solution.
Starting from the size distribution equation (1) 1.1 Total tree density If we eliminate m using equation 3 the integral is now a standard one for an upper incomplete Gamma function Now using the equation 7 we can say which using finally becomes The properties of the upper incomplete Gamma function are such that when the first parameter is an integer it will evaluate to a finite series, which in this case is Similarly we can derive the coverage where a(m) is the tree crown area as a function of tree carbon mass a(m) = a r m mr 1/2 and a s = a(m = m s ).The total forest growth is where g(m) = g r m mr 3/4 and g s = g(m = m s ).The total forest assimilate is where p(m) = p r m mr 3/4 and p s = p(m = m s ).If the first Gamma function parameter is not an integer then the solution to each of these equations remains as a purely an upper incomplete Gamma function.Luckily, the MST allometry results in more convenient short finite series, which is much easier to use.

Mean Properties and Compact DET Equations
We can rewrite DET in terms of the mean gridbox forest properties.These tell us what the mean tree would be like if the forest had identical trees rather than the size distribution.These mean properties do not depend on the final closed-form solution or the number of trees, but are purely a function of tree traits γ, g s , a s and m s .
We can also define a mean tree seed production rate s 3 Converting µ from dry to carbon mass Define m as carbon mass of a tree and m d as dry mass of the same tree.If we assume the carbon content of the tree is approximately half its dry mass then can assume m d = 2m.Growth rate of the tree in terms of carbon mass g in terms of growth g r at reference tree mass m r is and for dry mass the growth rate g d is The growth rate of carbon mass in a tree of dry mass m d must also be half that of the dry mass growth rate, so so then substituting in equations 27 and 28 gives the carbon mass m term on the RHS is equal to half the dry mass m d and so and then we can cancel the m d terms For µ we can do the same analysis and define a µ for carbon mass and dry mass The µ value for a tree of 1 kg of dry mass is subbing in from equation 32, allows us to relate it to µ for carbon mass For RAINFOR we have µ d1 = 0.198 and ϕ = 0.75 so we get µ 1 = 0.235 4 Equations for lines of optima (nullclines) Easier to write the optima more compactly using a new variable z

Biomass exact implicit equation
This equation exactly describes the optima of biomass with respect to α.
4.2 Tree Density exact implicit equation

Biomass approximate explicit equation
We can also write the biomass optima approximately but in explicit form Figure 1: Shows the accuracy of the approximate solutions for the biomass nullcline compared to the exact solution.The more complex solution gives very accurate results even for large seed sizes.
5 Proof for equations of lines of optima

Forest Properties for Constant Tree Assimilate
To study the effect of optimum seed allocation fraction α it is useful to assume the assimilate p r is constant.This means if α varies then there is trade-off between seed production and growth.It is also useful to study how varying the seed mass m s may affect the results.To do this it is necessary to more explicitly show which terms have dependence on α and m s .
As the assimilate rate p r is fixed we need replace µ with an equivalent in terms of assimilate, this will be the mortality to assimilate ratio µ p .For a fixed reference mass m r , this is defined as where p r is the net assimilate for a tree of size m r .As growth and assim-ilate are related by g = p(1 − α) we can similarly relate µ r and µ pr So, µ s can now be written in terms of µ pr as a function of α and m s This is important as it allows the parts that remains constant (µ pr and m r ) to be isolated from both α and m s , which will vary.So, µ s is a function of both α and m s .
To write the DET equations more compactly we use a new variable z, which only varies with m s (doesn't vary with α) we can then use this to rewrite the mean forest properties in a form that separates out the α dependence We can then make the equations more compact by making the substitutions

Differentiating Coverage with respect to
Coverage (equation 62) can be differentiated fairly easily, the first step leads to noting that then finally we move denominator terms outside of the brackets (70)

Differentiating Biomass with respect to α
Starting with the biomass equation we can then differentiate it by using a modified version of the chain rule we can then substitute in the equations for ∂Z M ∂α and ∂Z ν ∂α giving 75) and then substitute in the coverage differential (equation 70) to get the biomass differential To solve the case where ∂M ∂α = 0 it is helpful to get rid of the fractions inside the square brackets.First, tidy up the coverage differential then move the denominators outside the square brackets Then replace the 1 − ν term using equation 62 and move the denominator outside the brackets From this it is possible to obtain both an exact implicit solution to the nullcline or an explicit approximation.
To find the nullcline we look for where which as we only interested in cases where M > 0, this implies

Exact Biomass Implicit Nullcline Solution
By expanding equation 81 using sympy we can obtain the exact solution as an implicit equation This is precise but unwieldy and also implicit, so we will approximate this equation with an explicit expression for α.

Explicit Biomass Approximate Nullcline Solution
To get an approximate solution we note that the for the nullcline α → 0 as z → 0. Knowing this means the equation can be approximated when z is small.
First, approximation is to say that in equation 81 as typically α << 1 then can say 1 − α ≈ 1 then collect in terms of powers of α )dm = n r m 3/4 r exp(4µ r ) y s = y(m s ) = 4µ r (m s /m r ) 1/4 = 4µ s , then we get