In this part, assume that the annual proportionate birth rate is 1.57..?

Question

In this part, assume that the proportionate birth rate is 1.57 and that the annual proportionate death rate increases linearly with the populationsize P ,
according to the formula 1.41 + 0.0005 P

(1) Find a recurrence system for Pn.

(2) Show that this 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.

would 1) pn + 1 = (0.16 + 0.0005 P)

and 2) r = 0.16 and i dnt knw how to find E

ca some1 help me ??

Answer 1

It's proportionate birth and death rates, so

P_(n+1) = P_n + 1.57 P_n - (1.41 + 0.0005 P_n)P_n

or after simplifying,

P_(n+1) = 1.16P_n - 0.0005(P_n)²

For (2),

P_(n+1) - P_n = 0.16 P_n - 0.0005(P_n)²
= 0.16P_n [1 - (0.0005/0.16)P_n]
= 0.16P_n [1 - 0.003125 P_n]
= 0.16P_n [1 - (P_n)/320]