Growth and Decay Models in Biology -
Numbers and Percentages below are fictitious.
This assignment explores two different ways of describing growth
of populations, one using doubling time and the another using annual growth
rates. Doing the following questions will show you how the two different
descriptions or models can be interchanged - allow you to switch between
doubling times (or halving times) and annual growth or decay rates.
1. The Beluga whale population in the St.
Laurent Rivers is decreasing at a rate of 2% per year. So after t = m years, the
number left is
N(t) = N0 (1-0.02)m = N0 (0.98)m
where N0 denotes (is, represents) the initial population.
(a) Evaluate the factor (0.98)m for m = 0, 5, 10, 15, 20,
25, 30, 35 and 40 years with the aid of a calculator.
(b) How years m will it take for the factor to be close to 0.5? How many
years m (the same value) will it take for the population to decrease to half its
initial value N0? Take N0 = 400 if you wish.
(c) Solve 0.5 = (0.98)m for m using the algebraic and
computational property that ln (ax ) x ln(a) for a > 0 and x any
real number. Here ln(x) = natural logarithm of x and ax may be
computed using your calculator. Parts (a) and (b) gives the numerical method
for solving this problem.
2. (a) For several years, the Blue whale population off
an Antarctic is growing at 2.5% per year. At this rate of growth, a population
of 1000 would increase as follows.
| m = no. of years |
N(m) = population count |
| 0 |
1000 |
| 1 |
1025 = N(0)* 1.025 |
| 2 |
____ = N(1)* 1.025 |
| 3 |
____ = N(2)* 1.025 |
| 4 |
____ = N(3)* 1.025 |
| 5 |
____ = N(4)* 1.025 |
| 6 |
____ = N(5)* 1.025 |
Fill in the blank population numbers to the nearest whole number to estimate
the population, one year after another.
(b) Evaluate the formula N(m) = A*(1+i)m for m = 0, 1, 2,
3, 4, 5 and 6, assuming i =2.5% = 0.025 and A = 1000. The results should agree
with those computed one year at a time, and one year after another in part (a).
(c) Find the number of years m for which the factor (1+i)m
has a value equal to 2. Using the numerical or algebraic methods followed
earlier in question 1. The algebraic method is better - shows greater
mathematical maturity.
(d) Let n satisfy (1+i)n = 2. Compute N(p)= A*(1+i)p
for p = 0, n, 2n, 3n, 4n, assuming A = 1000, and i = 0.02= 2.5%. Do you
need to know the value of i if you are given m.
(e) Let n satisfy (1+i)n = 2. Compute A*2m/n
for m = 0, 1, 2, 3, 4, 5, 6 with A = 1000 again. Use your calculator.
(f) Let n satisfy (1+i)n = 2. Compute A*2m/nfor
m = 0, n, 2n, 3n, 4n, assuming A = 1000, and i = 0.02= 2.5%.
3. The population of ponies on a isolated island
doubles every four years for a decade or two. During that period the
population numbers N(t) = 300*2m/4when t = m
years. Show algebra implies N(t+1) = 2¼ N(t) regardless of the value
of t.
(a) Find a number i so that 21/4 =
1+i.
(b) Compute the values of (1+i)m and 2m/4for
m=0,1, 2, 3, 4 and 5.
(c) If 21/4 = 1+i, simplify
A*(1+i)m - A*2m/4
4. The population of seagulls on a isolated
island halves every four years for a decade due to a harsh environment
change. During that period the population numbers N(t) = 300*(½)m/4when
t = m years. Show algebra implies N(t+1) = (½)¼ N(t) regardless of
the value of t.
(a) Find a number i so that (½)1/4 =
1+i.
(b) Compute the values of (1+i)m and (½)m/4for
m=0,1, 2, 3, 4 and 5.
(c) If (½)1/4 = 1+i, simplify
A*(1+i)m - A*(½)m/4
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