# How to simulate a vegetation stand?

Actually, I have no clue how to simulate a vegetation stand. I don't know any project either. So I had to start from the scratch. If you have some ideas to develop the model, or know some other models which have been developed for the purpose, please tell me.

The model I propose is very basic: It only tries to introduce certain characteristics that make vegetation different from ordinary individual counting models. The model is basically intended to assess questions of estimating diversity and species richness using quadrat samples on plant patches. I assumed that plants are circles and they take space: New plants tend to go to empty places, and they are not completely inside other plants.

In the following discussion, it is good note the following contrasts I make

• Expected vs. observed: Observed values are simulated values which are produced introducing sampling error into expected values. Expected value can be a floating point value and observed value can still be an integer, like generating observed number of plants as a Poisson variate for an expected number of plants.
• Plant vs. species: Plant is an "individual", or a circular patch in this case. Each plant belongs to some species.

## Stages of simulation

The simulation process has two main stages:

1. Simulate the plants. Each plant has two parameters after this stage: species and size.
2. Locate the plants: Order plants in decreasing order of size, and locate them so that the later (smaller) plants have a probabilistic tendency to fill the gaps.

## Input parameters

1. Parameter to find expected proportions of species in the stand.
2. Expected total cover in the stand. This may be 100%, or more, or less.
3. Expected average size (area occupied, cover) of a plant.
4. Parameter to describe how strong a tendency plants have to seek gaps. This can vary from 1 (no plant can be inside another), to 0 (previous plants have no effect).

## Generating plants with species identification and size

### Species abundance model

In principle, I need a species abundance model with non-closed number of species. This means that when we increase the sample size, the number of species can increase. This does not mean that observed species richness would be infinite: Most species have such low sampling probabilities that they won't occur even in largest samples. Species abundance model is continuous, but the observed numbers of plants are discrete.

I used geometric series to find the expected sampling probabilities. The parameter of geometric series gives the expected proportion of the most abundant species of all plants. The next abundant species is expected to have the same proportion of all remaining plants, etc.

### Expected total number of plants

Expected total number of plants = Expected total cover / Expected average size of plant. This is a floating point number, like other expectations.

### Observed number of plants in a species

The expected number of plants in one species is the proportion predicted by the geometric series from the Expected total number of plants. This is used as expectation in generating Poisson random number which gives the Observed number of plants in this species. This is an integer which can be zero (species does not occur in the stand). With low expected proportion, the observed number is probably zero.

### Observed size of each plant

Observed size of each plant is found as an exponentially distributed random variate, with Expected average plant size as expectation.

The observed total cover is the sum of sizes of individual plants. We have introduced two error terms: Poisson error for observed number of plants, and exponential error for observed size of the plants. So the observed total cover will have negative binomial error distribution about the expected value.

## Locating plants

After we have the plants, we have to assemble the stand by putting plants to grow somewhere. Plants are sorted, so that bigger plants will be located first.

The steps in a locating a plant are:

1. Find candidate co-ordinates for the plant.
2. Find how much previously located (bigger) plants shade the plant.
3. According to degree of shading, reduce the acceptance probability of candidate co-ordinates.
4. If acceptance probability is lower than uniform random deviate (0,1), reject the candidate co-ordinates and return to step 1.
5. Else, accept the location and go to the next plant.

Obviously locating plants can be very time-consuming, especially when there is a high number of plants (size small, cover high), and there are not much gaps left (high cover).

### Acceptance probability

First we compute how much a previously located plant overlaps with the new plant in the candidate location. Then we reduce the acceptance probability by this proportion. If there are several shading plants, their effects are added multiplicatively, like normal for probabilities. So if there are two previously located plants, one shading 60% of the new plant, another 30%, the acceptance probability is reduced to (1-0.6)(1-0.3) = 0.28.

If input parameter for shading effect is below one, the shading percentage is multiplied with that giving higher probability for accepting locations shaded by other plants.