Transcribed Image Text: (2) A common growth model for a population P(t) is so-called logistic growth: dP — k. Р. (М-

Excerpt
that for large times, the general solution (3) yields lim;→∞ P(t) = M (c) Now solve a special case of the logistic equation with k = 3, M = 2 and Po = 1. Use separation by variables and for the P-integral use partial fractions.

Transcribed Image Text: (2) A common growth model for a population P(t) is so-called logistic growth:
dP
— k. Р. (М- Р)
dt
(2)
For small times the growth dP/dtwill be roughly proportional to P, but for later times, the growth will
slow down because M – P will approach zero. In the logistic equation the constant k is called the growth
factor and M is called the carrying capacity. Ts differential equation can be solved as
M
P(t)
(3)
(* -1)
M
1+
Ро
kMt
e
where P(0) = Po-
(a) Show that when we insert t = 0 in the general solution (3), we indeed find P(0) = Po.
(b) Show that for large times, the general solution (3) yields lim;→∞ P(t) = M
(c) Now solve a special case of the logistic equation with k = 3, M = 2 and Po = 1. Use separation by
variables and for the P-integral use partial fractions.

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