Problem $3$: Application Problem

Objective: Apply numerical methods to solve a real$-$world application problem.

Population Growth Analysis using MATLAB $($fzero$)$

The logistic growth model describes how a population grows in an environment with limited resources. The population $P(t)$ at time $t$ is given by:

$P(t)=\frac{K}{1+\frac{K-P_0}{P_0}{e}^{-rt}}$

Given

Initial population size $P_0=10$

Carrying capacity $K=100$

Intrinsic growth rate $r=0\mathrm{.}2$

Tasks

Part $1$: Finding Time to Reach a Target Population with fzero

Define the target population $P_{\text{t}\text{a}\text{r}\text{g}\text{e}\text{t}}=50.$

Set up the function:

$f(t)=P(t)-P_{\text{t}\text{a}\text{r}\text{g}\text{e}\text{t}}$

which can be expressed as:

$f(t)=\frac{K}{1+\frac{K-P_0}{P_0}{e}^{-rt}}-P_{\text{t}\text{a}\text{r}\text{g}\text{e}\text{t}}$

Use MATLAB's fzero function to find the time $t$ when the population reaches the target size. Use an initial guess of $t_0=5.$

Display the calculated time to reach the target population.