Introduction to Environmental Science E&ES 199, ASSIGNMENT I
There are 20 million people (P0) living at time t0
in the country "Howlland".and you want to predict its population growth.
At time t0 the birthrate (b0) is high
at 6%, whereas the deathrate (d0) is low at 2.5 %. Because
of improved education, changes in the social system, and free access to
mind-altering drugs, the birthrate b increases proportional to the
population number P, with the expression b=b0 + k1*P
(k1=
0.00000000035). Because of increased violent gang activity, changes in
social care systems and food shortage, the death rate d also increases
with time (proportional to P), with
d= d0 + k2*P
(k2=0.00000000055). A terrible disease sets in after 80 years, possibly an effect of the free mind-altering drugs, killing annually 1 % of Howlland's population for the rest of the period under consideration.
Questions
-
Calculate the population in yearly increments for 200 years assuming
that the birth and death rates (b and d) remain constant at b0
and d0 (simple exponential growth)
-
Calculate the population in increments of a year for 200 years assuming
that the birth and death rates adjust as given above (proportional to P)
-
Calculate the effect of the disease on the population for the 200 year
period
-
What is the carrying capacity of Howlland in Questions 1, 2 and 3. Did
the terrible disease significantly decrease the exponential population
expansion?
- Calculate the population for the same period of 200 years but now assume that a cure is found for the terrible disease after 5 years - so, the disease begins in January of the year 80 and ends in December of year 84 (5 years of 1% extra death rate by disease). How does this 5-year interval impact the K value or the final value after 200 years of growth. Compare to K with and without disease.
The basic equation for population growth is Pt=
Pt=0e^(r*t) (t is the year since t=0, we used 'N' in class (for 'number') while we use P here for 'population'). The intrinsic growth rate
r is equal to b-d, (expressed in fractions, not %, so 1.5 % is 0.015) and the first question is not that hard to solve in Excel (relatively speaking). In question 2, r is no longer constant, and we can either try to find a new expression (derive differential equation, integrate and solve) or solve the problem numerically, that is, we use little steps (such as one year) where we keep r constant for that year. The equation above can then be rewritten as Pt
= Pt-1 e^(r(t-1)*dt)
where dt is equal to the time elapsed between t and t-1 (1 year
intervals are used here). This problem can be nicely solved in Excel
with the function routine. Make 3 columns, one with time (in years), one
with r and one with P. To calculate
r, use
P values from the year before
(Excel can not use values that have not yet been calculated!). Make the
first row the t=0 values (t=0,
P=20,000,000,
r=0.035).
Use the function
rt
= b-d = (b0+k1*Pt-1)
- (d0+k2*Pt-1)
with
the constants b0, d0, k1 and k2 given above. For the disease question, calculate the r factors in a new column, using the population numbers "with disease". PLOT THE RESULTS IN EXCEL GRAPHS AND DISCUSS.
* multiplication sign ^ exponent symbol
Print up and hand in the graphs, write out the functions that you put in and discuss the limitations of the approach.
GRADING
#1 = 20 %
#2 = 30 %
#3 = 30 %
# 4 = 10%
#5 = 10 %