Culling System
Maths Help reccurence relationships….?
A fictitious population of adult crocodiles in a region had size 2400 on 01 aug 2008 and was increasing at the rate of appx 10.7% per year. Because of the danger to humans, it was proposed that there should be an annual cull of 260 of these crocodiles just before 01 August. Let A”n” denote the size of the population “n” years after 01 Aug 2008.
It is suggested that the effect on the population size could be modelled by the recurrence system
A”o” = 2400, A”n+1″ = 1.107A”n” – 260 (n = 0,1,2…)
i) Explain clearly how the recurrence relation in this recurrence system is obtained from the information given at the beginning of the question
ii) use the recurrence system to find the population size predicted by the model for 01 Aug 2010 (2 years after 2008)
iii) Find a closed form for the sequence (4 Signif Fig)
iV) Use the closed form to work out the population on 01 Aug 2020.
Help!!!! Thanks (NO STUPID ANSWERS_ I am loosing sleep over this!!!
First, let’s define some variables to keep the equations tidy.
- A(n) is the population of crocodiles on 01 August after n years, where counting starts from n = 0 in 2008. The initial population A(0) is 2400.
- the growth rate j is the proportional increase in the number of crocodiles each year in the absence of culling. If the percentage increase is 10.7% pa, j is equal to 1 + 10.7/100 = 1.107. So without culling, the number would increase from 2400 to j x 2400 in the first complete year.
- the number of crocodiles culled each year is represented by the constant C
(i) so the population would increase by breeding in the first year from A(0) = 2400 to j x A(0); just before the year’s end C crocodiles would by culled, leaving a number A(1) where
A(1) = j.A(0) – C
At the start of year 1 there are A(1) crocodiles which increases to j x A(1) by the end of that year, when another C are culled, leaving
A(2) = j.A(1) – C
and this process continues every year. For year (n + 1) we may write the recurrence relation as
A(n + 1) = j.A(n) – C
(ii) Putting some numbers into these equations, we have
A(1) = 1.107 x 2400 – 260 = 2397
rounding up to the nearest integer, since there must be an whole number of live crocodiles. With A(1) as the new figure for the population, the number at the end of the next year 2010 (n = 2) will be
A(2) = j x A(1) – 260 = 1.107 x 2397 – 260 = 2393.
(iii) To obtain a closed form for A(n), we write the expressions for a number of years, substituting explicitly the expression for the previous year ie
A(1) = j.A(0) – C
A(2) = j.A(1) – C = j.[j.A(0) - C] – C
A(3) = j.A(2) – C = j.{j.[j.A(0) - C] – C} – C
and so on. We find that this enables us to divide the equation into one term for the effects of breeding and another for those from culling. These may be generalised to give the equation
A(n) = A(0).j^n – C.(j^(n-1) + j^(n-2) + … j² + 1)
The series in the second term is readily summed to give
A(n) = A(0).j^n – C.(j^n – 1)/(j – 1)
and this enables isolated values to be calculated without requiring those for all the intervening years. In each case the number should be rounded to the nearest integer, for the reason given above, and this will have 4 significant figures (unless the population falls below 1000).
(iv) on August 01, 2020, when n = 12, this equation produces
A(12) = 2400,(1.107)^12 – 260.(1.107^12 – 1)/(1.107 – 1) = 2329.
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