The differential equation used to model population growth, under the assumption that the proportional growth rate is a decreasing linear function of population, is the logistic differential equation =rP(1-) dP dt

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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4.2

2 The differential equation used to model population growth, under the assumption that the proportional
growth rate is a decreasing linear function of population, is the logistic differential equation
dP
d = rP(1-)
where r and M are positive constants. By the change of variables, P(t) = Mx(t) we can obtain the simpler,
related differential equation
d = rx(1-x)
for x > 0.
(a) Express (1-2) in partial fraction form; hence find (1-2) dx, expressing your answer in the form
In(). (Make sure you consider the two cases x < 1 and x > 1 separately.)
(b) If x= xo> 0 when t = 0, show that the solution of the equation is
X =
1
1+(-1) e-rt
(c) Considering the two cases xo < 1 and xo > 1, determine what happens in the long run.
(d) What happens if xo = 0 or xo = 1?
Transcribed Image Text:2 The differential equation used to model population growth, under the assumption that the proportional growth rate is a decreasing linear function of population, is the logistic differential equation dP d = rP(1-) where r and M are positive constants. By the change of variables, P(t) = Mx(t) we can obtain the simpler, related differential equation d = rx(1-x) for x > 0. (a) Express (1-2) in partial fraction form; hence find (1-2) dx, expressing your answer in the form In(). (Make sure you consider the two cases x < 1 and x > 1 separately.) (b) If x= xo> 0 when t = 0, show that the solution of the equation is X = 1 1+(-1) e-rt (c) Considering the two cases xo < 1 and xo > 1, determine what happens in the long run. (d) What happens if xo = 0 or xo = 1?
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