Set up and solve the exponential growth-decay equation and the logistic equation and interpret their solutions

What this dot point is asking

SCSA wants you to set up these two models from a description, solve them (logistic via separation and partial fractions), and interpret the parameters and long-term behaviour.

The exponential model

When the rate of change is proportional to the current amount, $\tfrac{dy}{dt} = ky$. Separating variables and integrating gives

where $y_0$ is the initial value and $k$ is the constant of proportionality. Positive $k$ models unrestricted growth; negative $k$ models decay such as radioactive half-life. This model has no upper bound, which is unrealistic over long times.

The logistic model

Real populations are limited by resources. The logistic equation adds a braking factor:

where $M$ is the carrying capacity. When $P$ is small the bracket is near $1$ and growth is nearly exponential; as $P$ approaches $M$ the bracket approaches $0$ and growth slows to a halt.

Equilibria and the S-curve

Solving the logistic equation uses separation of variables with a partial-fraction decomposition of $\tfrac{1}{P(1 - P/M)}$, leading to a solution of the form $P = \tfrac{M}{1 + A e^{-kt}}$.