Lecture 2- January 7, 2008

1. Natural capital: growth processes and management

biological renewable resources such as trees, fish, grass that grow to replace what is used

growth rate is (to a rough approximation) dependent on the size of the population relative to the resources needed for that population to grow

soil moisture and nutrients for trees or grass

food for fish

change in the population during a growth phase would be (roughly) equal to the number_of_individuals * growth_rate

where growth rate is the number of new individuals per individual in the population per time

demographic transition - human growth rates are around 0% to 2% per year

animal populations grow much faster

question for management is - how many trees or fish can we harvest and still keep the population productive?

 

2. First thead of models - describing the growth and harvest of a natural resource

"Exponential" growth - I prefer to call it a constant ratio of growth population at time 0 =100 increases by 10% over the time period population at time 1 = 110 increases at the same rate over the next time period

population at time 2 = 110 + .1*110 = 121

population(3) = population(2) + .1 * population(2)

population(4) = population(3) + .1 * population(3)

and continue iterating

 

3. Multiple representations - example 1

short time step or small growth rate

verbal

each week the population of rats increases by 0.02

initial population is 100 rats

the population over the first 40 weeks more than doubles (up to about 220 rats)

graphical (data from the rat counter)

 

Table

time	rats
0	100
1	102
2	104.04

 

Algebra

2 different equations to describe this

iterative -

fit the curve to an equation

show this in Excel

time, rats, delta, growth_rate

Simulation model in STELLA (build in class)

stocks = rats

flow = new rats

constant = growth_rate

positive feedback

value of these simulations in Excel or STELLA - sensitivity to time step (delta) or growth_rate

 

4. Example 2: fewer animals, larger jumps

1. description of reproduction

Guinea pigs on an island

rich in detail with attention to individuals and population levels

may not be explicit enough

geneology - diagram

 

2. chart of population growth

from this data -

0, 2
4, 2
8, 6
12, 6
16, 18
20, 18
24, 54

Excel example

generate several versions in class

 

3. algebraic representation

fit the data to an exponential equation

doesn't fit the observations as well

 

4. generate a similar population growth with a simulation

a. iterative, time steps, each row in a spreadsheet

pop(at time 2) = population(at time 1) + new rodents

new rodents = (population(at time 1)/2) * 4 per litter

create spreadsheet and chart

can use this approach to examine sources of variability

for example if there is only 3 per litter or as many as 5 per litter

b. "systems" version

build model in STELLA, simulate with time

need to translate discrete time step numbers to growth rates per month

4 per every 8 months = .5 per month

this is not exaclty true because of compounding in exponential functions

 

5. individual based models

keep track or simulate data on each individual or space in the territory