We are tasked with solving a population model for a mouse population, which is governed by the differential equation:

$N' = 0.5N \left(1 - \frac{N}{1000}\right)$

The initial condition is $N(0) = 400$, and we are asked to use Euler's method with a step size of $h = 0.1$ to find the approximate population at $t = 0.3$.

Euler's Method:

Euler's method is a numerical technique for solving differential equations. Given the differential equation $N'(t) = f(t, N)$, the method updates the population $N$ using:

$N_{n+1} = N_n + h \cdot f(t_n, N_n)$

In this case, the function $f(t, N) = 0.5N \left(1 - \frac{N}{1000}\right)$. We'll compute the population for each time step: $t = 0.0$, $t = 0.1$, $t = 0.2$, and $t = 0.3$.

Let me calculate the population step-by-step using this method.Using Euler's method with a step size of $h = 0.1$, the approximate population values at different time points are as follows:

• At $t = 0.0$, the population $N(0.0) = 400$ (initial value).
• At $t = 0.1$, the population $N(0.1) \approx 412.0$.
• At $t = 0.2$, the population $N(0.2) \approx 424.11$.
• At $t = 0.3$, the population $N(0.3) \approx 436.32$.

Thus, the approximate population at $t = 0.3$ is 436.32.

Would you like more details or further clarification?

Here are 5 related questions for further exploration:

1. How does the accuracy of Euler's method change with smaller step sizes?
2. What happens to the population in the long run (as $t \to \infty$) based on the differential equation?
3. How does the population model change if we modify the carrying capacity from 1000 to another value?
4. Can we apply the Runge-Kutta method for a more accurate solution for the same problem?
5. How would the solution change if the initial population was 600 instead of 400?

Tip: Euler’s method is simple but may be inaccurate over larger intervals. Reducing the step size can improve accuracy.