I've had trouble with a question from a recent assignment, and was unable to complete all the parts. I'd be grateful if someone could explain to me how to arrive at the correct answer for each part. I've put in the answers I arrived at for each part I answered in red, so you can let me know where I'm going wrong, or even if I'm right!
A population of cattle was introduced onto an island on 1 October 1960. There was no subsequent migration of cattle to or from the island. Let P n denote the size of the cattle population on the island on 1October n years after 1960.
a) For a), assume that, for each integer n is greater than or equal to 0, the number of births and deaths in the year beginning 1 October n years after 1960 are 1.52P n and 1.24P n, respectively.
i) Find a recurrance system satisfied by P n.
P 0=80, P n+1=1.28P n ( n=0,1,2,...)
ii) State a closed form of P n.
P n=80*(1.28) n
iii) Show that this formula for P n describes an exponential model, by identifying the value of the annual proportionate growth rate, r.
r is 0.28, so this is an exponential model as the base number of cattle is raised anually by 1+ r to the exponent n.
iv) Calculate the population size on 1 October 1975, according to this model.
Where n=15 P 15=80*(1.28) 15=3245.185537
Therefore the population on 1 October 1975 according to this model is 3245 cattle.
b) For b), assume the annual proportionate birth rate is 1.52, as in a), but that it but that the death rate increases linearly with population size P, according to the formula 1.24+0.0007P.
i) Find a reccurence system for P n.
P 0=80 P n+1-P n=(0.28-0.0007P n)P n ( n=(0,1,2,...)
ii) Show that the reccurence relation is logistic, by writing it in the form P n+1-P n= rP n(1-P n/ E) and identifying the values of the parameters r and E.
