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.
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.
Show Hint
- +190 and -5
- +5 and -190
- -190 and +5
- -10 and +5
The Correct Option is A
Solution and Explanation
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.
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