Many processes that normally grow in proportion to their size (exponential growth), but are limited by some finite capacity are often modeled by the logistic function: P(t) = A / (1 + Be^(-kt)), where A, B, k are positive constants. What is P(0)?
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A logistic function is in the form P(t) = L / (1 + Ae^(-bt)) where L, A, and b are constants and the independent variable t is usually time; t ≥ 0. This model is useful in limited growth problems, that is, when the growth cannot go beyond a particular value for some reason.
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The logistic growth model given in Formula (1) is equivalent to $$y e^{k t}+A y=L e^{k t}$$ where $y$ is the population at time $t(t \geq 0)$ and $A, k,$ and $L$ are positive constants. Use implicit differentiation to verify that $$\begin{aligned} \frac{d y}{d t} &=\frac{k}{L} y(L-y) \\ \frac{d^{2} y}{d t^{2}} &=\frac{k^{2}}{L^{2}} y(L-y)(L-2 y) \end{aligned}$$
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