Growth and Decay Models in Biology -
Growth and Decay Models in Biology offers numerical exercises
in the forward and backward use of the compound growth and decay
model A = P[1+ii]m. The backward use employs m-th roots
and logarithms. The numbers in the exercises are illustrative only.
Numbers and Percentages below are fictitious - illustrative only.
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.
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m = no. of years
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N(m) = population count
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0
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1000
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1
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1025 = N(0)* 1.025
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2
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____ = N(1)* 1.025
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3
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____ = N(2)* 1.025
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4
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____ = N(3)* 1.025
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5
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____ = N(4)* 1.025
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6
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____ = N(5)* 1.025
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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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