IB Mathematics Applications & Interpretation Question Type 6: Formulating a Logistic Model Based on Descriptions Exercises

Suppose the intrinsic growth rate doubles to $r=6$ while $K=10$ and $P(0)=12$. Write the new logistic model $P(t)$.

Show that as $t\to \infty$, the logistic solution $P(t)=\frac{10}{1-\frac{1}{6}{e}^{-3t}}$ approaches the carrying capacity.

Find the horizontal asymptote of the logistic function $$P(t)=\frac{10}{1-\frac{1}{6}{e}^{-3t}}.$$

Express the solution in the form $$P(t)=\frac{10}{1+A{e}^{-3t}}$$ and determine the constant $A$ given that $P(0)=12$.

A population, $P(t)$, is modelled by the logistic function $$P(t)=\frac{10}{1+\frac{1}{6}{e}^{-3t}}$$ where $t\ge 0$ is the time in years.

Determine the time of inflection, $t_{\text{inf}}$, for this population.

Write down the logistic differential equation for a population $P(t)$ with carrying capacity 10 and intrinsic growth rate 3, and solve it to obtain the general explicit solution.

A population model is given by the function $$P(t)=\frac{10}{1-\frac{1}{6}{e}^{-3t}}$$ where $P(t)$ is the population at time $t$ in years, for $t\ge 0$.

Find the time at which the population reaches half the carrying capacity.

Using the model $P(t)=\frac{10}{1-\frac{1}{6}{e}^{-3t}}$, calculate $P(2)$ correct to three decimal places.

Determine the maximum theoretical growth rate $\left(\max \frac{dP}{dt}\right)$ and the time at which it occurs.

Solve the logistic differential equation $$\frac{dP}{dt}=3P(1-\frac{P}{10})$$ by separation of variables and verify that $$P(t)=\frac{10}{1+A{e}^{-3t}}$$ is the general solution.

Given the logistic model $P(t)=\frac{10}{1+3e^{-3t}}$, determine the carrying capacity, the intrinsic growth rate, and the initial population.

Determine the time $t$ when the population reaches $P(t)=9$ under the logistic model $$P(t)=\frac{10}{1+\frac{1}{6}{e}^{-3t}}.$$