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When is the logistic growth model and exponential model equivalent?

Logistic Growth Model:

$P(1+t)=(P(t)*(-\frac b N )+1+b)*P(t)$

Where $b$ is the birth rate and $N$ is the Carrying Capacity.

Exponential models example the SI model (susceptible and infected)

$S(1+t)=S(t)-(\frac B M*I(t)*S(t))$

$I(1+t)=I(t)+(\frac B M*I(t)*S(t))$

Seeing that the SI model is a continuous loop, where the population is changing in the amount of susceptible and infected, but the total population size remains constant, I think the logistic growth model would be equivalent to the SI model when the population size remains constant, i.e. when the number of births equals the number of deaths or when the population reaches carrying capacity.

Any thoughts on the question? It would be greatly appreciated.

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The SI model and Logistic Growth model are equivalent when $n(t)$ approaches $N$ i.e. when $n(t)=N$. So $n(t+1)= N$
Since total population doesn't change
$S(t+1) + I(t+1) = S(t) + I(t) = n(t) = N = n(t+1)$

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