A population of 80 longhorn cattle was introduced onto an island on 1 October 1960. There was no subsequent migration of longhorn cattle to or from the island. Let Pn denote the size of the longhorn cattle population on the island on 1 October n years after 1960.
(a) In this part assume that, for each integer n ≥ 0, the number of births and deaths in the year beginning 1 October n years after 1960 are 1.52Pn and 1.24Pn, respectively.
(i) Find a recurrence system satisfied by Pn.
(ii) State a closed form for Pn.
(iii) Show that this formula for Pn describes an exponential model, by identifying the value of the annual proportionate growth rate, r. (iv) Calculate the population size on 1 October 1975, according to this model.
(v) Explain briefly why this model will not be accurate in the long term.
So the recurrence system is [pic]
Pn is a geometric sequence, so
So Pn describes an exponential model, and the value of the annual proportionate growth rate is r = 28% (iv)
When n = 15, [pic]
So the population size on 1 October 1975 is 3245.
In the long term, n is very large, so the error made by the exponential model is relatively large, so this model will not be accurate in the long term.
(b) In this part, assume that the annual proportionate birth rate is 1.52, as in part (a), but that the annual proportionate death rate increases linearly with the population size P, according to the formula 1.24+ 0.0007P.
(i) Find a recurrence system for Pn.
(ii) Show that the recurrence relation is logistic, by writing it in the form Pn+1 − Pn = rPn(1 − Pn/E) and identifying the values of the parameters r and E.
You should use Mathcad file 121B1-01 in parts (iii) and (iv) below, and provide a printout of page 2 of the worksheet as amended. (iii) By making suitable amendments to a copy of the worksheet in Mathcad file 121B1-01, produce a graph showing the behaviour of the longhorn cattle population over...
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