(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. This differential equation can be solved as M P(t) 1+ ( -1) (3) M Po e-kMt 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.
(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. This differential equation can be solved as M P(t) 1+ ( -1) (3) M Po e-kMt 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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. This 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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