herbaceous+vegetation

The amount of herbaceous vegetation on each patch is modeled as a univariate function. This function is implemented as an additive function, so that it can be discretized in order to calculate the amount of change in the quantity of herbaceous vegetation over a single timestep. This can be expressed as

math P_{t+1} = P{t} + \Delta P_t math

where we need to calculate the delta P. This approach will allow external factors to influence the amount of herbaceous vegetation at any time, and the resource to respond appropriately from its new starting point.

In order to represent seasonality in our model, it is necessary to keep track of the day of the year (//doy//) he form of change function depends on the time of year. If we choose a value of //doy// to serve as a cutoff between the growing season and the non-growing season, we can have different functions defining the response during different stages of the herbaceous population development. In the example below, we use a cutoff of //doy// = 175, and use a logistic function to describe the growth during the growing season (from //doy// = 1 to //doy// = 175), and an exponential function describing growth past that point. These two functions both require the parameter **//r//**, which defines the unobstructed rate of growth, and the logistic function requires the additional parameter //**K**//, which defines the carrying-capacity. We can then define the seasonal growth of the amount of herbaceous plant material using

math \Delta P_t = \begin{cases} r P_{t} (1-\frac{P_{t}}{K}) & \mbox{if } doy_t < 175\\ -r P_{t} & \mbox{otherwise} \end{cases} math

Implementing this function with a constant value of //r=0.2// and //K//=5 creates a system which, while it does not decline to zero after day 175, it decreases to a minuscule amount. This is functionally correct, as it represents the paucity of vegetation available during the winter months. However, because of the small amount present, once growth increases again, the population is not able to recover and grow fast enough to reach the carrying capacity (or any significant amount) during the following year. Because of this, we need to include in the model a term that adds an initial amount of plant material to the system, representing leaf-out, the period when buds start to break and vegetation starts to sprout. To do this, start by adding a value of one to **//P//** at //doy// = 1. Now when we run the model, the quantity of herbaceous plant is inflated to one at the beginning of the growing season.

Additional Exercises
Try adjusting the values of r and K to experiment with different responses.

Try separating the two different applications of r, so that the rate during the growing season is independent of the rate after the growing season.

Vary the length of the growing season by changing the value of //doy// that switches the growth functions (currently = 175). Can you make it so the growing season is defined by a range of //doy// values, instead of a single one (ie. growing season = 100 < //doy// < 250)?

Write subroutines that calculate the value of the //r// and //K// parameters used to calculate the delta growth values.
 * See if you can make //r// change as a function of //doy.//
 * What information is available in the Vilas county model that you developed last week that can be used to inform the calculation of the carrying capacity (//K//)? Hint - think in terms of both factor levels and continuous variables.

Next week we'll start by addressing any issues you ran into, followed by discussion about how you customized the herbaceous growth model to interact with the landuse and/or tree data.