Abstract:  Forest simulation models have been proven remarkably effective at capturing the dynamics of real forests. In mathematical terms, individual-based simulators are spatial stochastic processes that predict properties of populations and communities by simulating the fate of every plant throughout its life cycle. Unfortunately, non-linear spatial stochastic processes are notoriously intractable, which limits the usefulness of forest simulators to basic scientists, and, also, they require too much computer power to be used at large scale, such as in global models; one cannot simulate every tree on the Earth. To solve the twin problems of computational intensiveness and mathematical intractability, what is needed is a way to predict a forest's community dynamics using only individual-level information, but without simulating every plant. This requires so-called macroscopic equations for variables of interest to ecologists, such as the mean density and size structure of each species and how these change though time.

In physical systems, macroscopic equations for the dynamics of fluids can be derived from stochastic models of the random collisions and transformations of individual molecules. Using similar approach we have developed a new spatial individual-based forest model that is based on a new approximation for the plasticity of crown shape. Its structure allows us to derive an accurate approximation to the individual-based model for the means of the stochastic process in a forest simulator that predicts the mean densities and size structures in the simulator using the same parameter values and functional forms, and, also, is analytically tractable. The approximation is represented by a system of Von Foerster partial differential equations coupled with an integral equation that we call the Perfect Plasticity Approximation (PPA). We have derived a series of analytical results including equilibrium abundances for trees of different crown shapes, stability conditions, transient behaviors, and coexistence conditions.