Population growth for a species is modeled by the logistic differential equation dy…

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

Population growth for a species is modeled by the logistic differential equation

\frac{dy}{dx}=ky\left(1-\frac{y}{M}\right),\qquad k>0,\;M>0,

where

y(x)

denotes the population at time

x

k

is a growth constant, and

M

is the carrying capacity. (a) Verify directly that

y=0

and

y=M

are constant (equilibrium) solutions of the differential equation. (b) Show that the one–parameter family

y(x)=\frac{M}{1+Ae^{-kx}}

is a solution for all real constants

A\neq-1

. (c) Explain why distinct values of

A

correspond to distinct solution curves and conclude that the logistic equation possesses infinitely many non–equilibrium solutions. (d) Compare the single arbitrary constant that appears in part (b) with the number of arbitrary constants typical of a second–order differential equation. Discuss how this comparison illustrates the general relationship between order and the size of a solution family.

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