Longhorn Cattle Population
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) …show more content…
(v)
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 the first 40 years, as predicted by this …show more content…
(v) Describe the long-term behaviour of the population as predicted by the model, justifying your answer without reference to
Mathcad.not if nomathcad manually accepted please
Solution:
(i)
[pic]
So the recurrence system is [pic]
(ii)
[pic]
[pic]
So [pic]
So r = 0.28, E = 400
So
[pic]
(iii)
The table from n=1 to n=40 is below:
|n |Pn |
|0 |80.00 |
|1 |97.92 |
|2 |118.63 |
|3 |141.99 |
|4 |167.63 |
|5 |194.90 |
|6 |222.88 |
|7 |250.52 |
|8 |276.73 |
|9 |300.61 |
|10 |321.52 |
|11 |339.19 |
|12 |353.63 |
|13 |365.10
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) …show more content…
(v)
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 the first 40 years, as predicted by this …show more content…
(v) Describe the long-term behaviour of the population as predicted by the model, justifying your answer without reference to
Mathcad.not if nomathcad manually accepted please
Solution:
(i)
[pic]
So the recurrence system is [pic]
(ii)
[pic]
[pic]
So [pic]
So r = 0.28, E = 400
So
[pic]
(iii)
The table from n=1 to n=40 is below:
|n |Pn |
|0 |80.00 |
|1 |97.92 |
|2 |118.63 |
|3 |141.99 |
|4 |167.63 |
|5 |194.90 |
|6 |222.88 |
|7 |250.52 |
|8 |276.73 |
|9 |300.61 |
|10 |321.52 |
|11 |339.19 |
|12 |353.63 |
|13 |365.10