Question

The logistic growth model formula is given by

〖ye〗^kt+Ay=〖Le〗^kt

Where y I the population at time t(t≥0) and A,K and L are positive constants.

Use implicit differentiations to verify that

dy/dx=k/L y(L-y)


(d^2 y)/〖dt〗^2 =k^2/L^2 y(L-y)(L-2y)

Answer 1

The logistic growth model formula is given

$$ye^{kT}+Ay=Le^{kT}$$

where $y$ is the population at time $t (t\geq0)$ and $A, k, L$ are positive constants

Use implicit differentiations to verify that

(a)

$${dy\over dt}={k\over L}y(L-y)$$

$${dy\over dt}e^{kT}+kye^{kT}+A{dy\over dt}=kLe^{kT}$$

$${dy\over dt}(e^{kT}+A)=ke^{kT}(L-y)$$

$$e^{kT}+A={Le^{kT}\over y}$$

Then

$${dy\over dt}={ke^{kT}(L-y)\over Le^{kT}}y$$

$${dy\over dt}={k\over L}y(L-y)$$

(b)

$${dy^2 \over dt^2 }={k^2 \over L^2 }y(L-y)(L-2y)$$

$${dy^2 \over dt^2 }={d \over dt }({k\over L}y(L-y))={k\over L}({dy\over dt}(L-y)-y{dy\over dt})=$$

$$={k\over L}{dy\over dt}(L-2y)={k\over L}({k\over L})y(L-y)(L-2y)=$$

$$={k^2 \over L^2 }y(L-y)(L-2y)$$