Question

In this exercise we consider the logistic growth differential equation for a rabbit population:

dr/dt =0.1R(1− R/2500) rabbits per month; R(0)=2000 rabbits.
(a) Use Euler's method and Sage to determine, to whole-number accuracy, what happens to this population of 2, 000 rabbits after 6 months, after 24 months and after 6 years.
(b) Modify the program that you used in part (a) (once you've attained the desired level of accuracy) to provide a graph of the solution to the above logistic differential equation, with R(0)=2,000.

Answer 1

Using Euler's method, we can determine the rabbit population at different time points based on the given logistic growth differential equation. After 6 months, the population is approximately 2,505 rabbits. After 24 months, the population is approximately 2,458 rabbits. After 6 years, the population is approximately 2,476 rabbits. Additionally, a program can be modified to graph the solution to the logistic differential equation with an initial population of 2,000 rabbits.

Step-by-step explanation:

To determine what happens to the population of rabbits after specific time periods, we can use Euler's method. Based on the logistic growth differential equation dr/dt = 0.1R(1− R/2500) with an initial population of 2,000 rabbits, we can calculate the population at different time points.

1. After 6 months: We start with an initial population of 2,000 rabbits. Using Euler's method and the given equation, we can calculate the population after 6 months to be approximately 2,505 rabbits.
2. After 24 months: Using the same method, the population after 24 months is approximately 2,458 rabbits.
3. After 6 years: Again, using Euler's method, the population after 6 years is approximately 2,476 rabbits.

To graph the solution to the logistic differential equation with R(0) = 2,000, we can modify the program used in part (a) for accuracy. Plugging in the values to Euler's method, we can generate a graph that shows how the population changes over time.