The logistic model for population growth developed by P.F. Verhulst, is given by dp = ap – bp, dt p(0) = P, > 0 where p = p(t) is the population at time t and a, b are positive constants. Show that p(t) = ар. and explain why Po # 0, (a) bp, + (a – bp, )e at ª . (b) Find the limit of p(1) when (i) (ii) t →-00.

The logistic model for population growth developed by P.F. Verhulst, is given by dp = ap – bp, dt p(0) = P, > 0 where p = p(t) is the population at time t and a, b are positive constants. Show that p(t) = ар. and explain why Po # 0, (a) bp, + (a – bp, )e at ª . (b) Find the limit of p(1) when (i) (ii) t →-00.

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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

The logistic model for population growth developed by P.F. Verhulst, is given by
dp
= ap – bp,
dt
p(0) = Po > 0
where p = p(t) is the population at time t and a, b are positive constants.
apo
Show that p(t) = bp, +(a – bp,)e™
(a)
and explain why Po # 0,
(b)
Find the limit of p(t) when
(i)
(ii)
t → -00.

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