Simulation of a forest

The model for single-tree simulation can be applied repeatedly to produce multiple trees of different species. This can be done through parallel computing to save computation time. To form a forest, we need to model the distribution of trees in a field. We achieve this through inhomogeneous Poisson process, abbreviated as IPP, because of its flexibility and computational convenience. An IPP models the distribution of random points in space or random “events” in a time interval. It can be used to model a multitude of spatial and temporal phenomena, such as cars passing through a junction or the timing/place of animal sightings. By sampling from IPP, we can determine the number of trees as well as the positions of these trees in a 2-D field. Specifically, let D∈R2 denote the field on which the group of trees will be simulated. The random locations (i.e., (x, y) coordinates) of the trees will be denoted by S = {s_i, i = 1, …, n}. We assume that S follows an IPP with intensity function λ(s):D → R⁺, where λ(s) is a parameter that control the tree density on D. Small values of λ(s) indicate sparse regions whereas high values indicate dense regions. Given the region D and the intensity λ(s), the number of trees, n, follows a Poisson distribution with mean ∫_D λ(s)ds. To simulate S given n, we adopt a thinning approach; details can be found in [35].

The intensity function λ(s), s ∈ D is an input of the forest simulator. Its format should be specified by the user. For example, one way to specify λ(s) is to use a mixture of squared exponential (Gaussian) kernel functions, i.e., λ(s)=∑i=1pCiexp{-(sx-ai)2/hi2-(sy-bi)2/li2}, where p is the number of mixing component, (s_x, s_y) denotes the (x, y) coordinates of s, (a_i, b_i) is the center of the ith mixing component, (h_i, l_i) are positive numbers that control the standard deviation of each mixing component, and C_i is a scaling parameter. The values of (a_i, b_i), (h_i, l_i) can be determined by visualizing the location and the shape of each mixing component, and C_i can be determined by controlling the expected total number of trees in the field and the expected proportion of trees in each mixing component. In Fig 3, we demonstrate samples from IPPs with mixed squared exponential intensity functions (p = 2), where the two plots (a) and (b) correspond to λ(s) with different parameter setups.

(a) A sample corresponds to C₁ = 1.0, C₂ = 0.6, (a₁, b₁) = (5, 5), (a₂, b₂) = (15, 15), and (h₁, l₁) = (h₂, l₂) = (3, 2). (b) A sample corresponds to C₁ = 0.4, C₂ = 0.2, (a₁, b₁) = (5, 5), (a₂, b₂) = (10, 10), (h₁, l₁) = (3, 4), and (h2,l2)=(4,10).

In some cases, if there is a real experimental field that can serve as a template of the simulation and the location of each tree is known, the value of λ(s) can be estimated based on the spatial data collected from the experimental field. Various estimation methods, such as those developed by [36, 37] and [38, 39], can be used for estimating λ(s). If there are more than one plant species, the intensity estimation and simulation should be performed separately for each species. In this case, a merging step is required to pool all species into one community, and extra restrictions will be applied to ensure, for example, two trees of different species are not too close to each other.

Comparing with existing methods such as models that assume homogeneous tree distribution [40–42] and models that are based on modeling local interactions between individual plants [43–45], the IPP approach described above offers several unique features and advantages. First, as a general stochastic model, it generates flexible inhomogeneous spatial distributions for plants of different species. Second, the IPP sampling is automatic, computationally efficient, and does not require users to manually interfere with the sampling procedure. Third, if desired, the input—the intensity function—can be estimated based on a real experimental site, making it possible to mimic a real experimental site in simulation. Finally, the sampling procedure can be repeated as many times as needed, producing plant distributions with similar spatial characteristics.