Question

The numbers of rabbits (R) and their predators, foxes (F), in an ecosystem are modelled by the Lotka-Volterra equations as follows: \[ \frac{dR}{dt} = 2R - 0.01 R F \] \[ \frac{dF}{dt} = -F + 0.005 R F \] where the time is measured in months. If there are currently 100 rabbits and 10 foxes, the number of rabbits is changing at the rate of _________ per month and the number of foxes is changing at the rate of ________ foxes per month.

• A. +190 and -5
• B. +5 and -190
• C. -190 and +5
• D. -10 and +5

Hint:

Understanding predator-prey dynamics through models like Lotka-Volterra can be crucial for managing ecosystems effectively.

Answer 1

The Correct Option is A

Step 1: Calculate the Rate of Change for Rabbits (dR/dt).
Substitute \( R = 100 \) and \( F = 10 \) into the rabbit equation: \[ \frac{dR}{dt} = 2(100) - 0.01(100)(10) = 200 - 10 = 190 \] The rabbit population is increasing at a rate of 190 rabbits per month. 
Step 2: Calculate the Rate of Change for Foxes (dF/dt).
Substitute \( R = 100 \) and \( F = 10 \) into the fox equation: \[ \frac{dF}{dt} = -10 + 0.005(100)(10) = -10 + 5 = -5 \] The fox population is decreasing at a rate of 5 foxes per month.