In this video I go over another example on Predator-Prey systems and this time modify the Lotka-Volterra equations by modeling the prey population by using a logistic model, while keeping the predator model the same. In part 1 of this example I go over parts a) and b) which involve finding the equilibrium solutions, as well as showing that in the absence of the predators, the prey population simply approaches the carrying capacity, as expected using the logistic model. In the next part of the example I go over the graphing phase trajectories and solutions curves for the predator and prey populations (represented by rabbits and wolves), so stay tuned for that!
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Example:
In Example 1 (https://youtu.be/dkQYqI4FXxA), we used Lotka-Volterra equations to model populations of rabbits and wolves.
Let's modify those equations as follows:
a) According to these equations, what happens to the rabbit population in the absence of wolves?
b) Find all the equilibrium solutions and explain their significance.
c) The figure shows the phase trajectory that starts at the point (1000, 40).
Describe what eventually happens to the rabbit and wolf populations.
Reveal spoiler
d) Sketch the populations of the rabbit and wolf populations as functions of time.
Solution:
Recall the Logistic Equation (https://youtu.be/FMFTLa8URDg):
Thus in the absence of wolves, the rabbit population stabilizes at 5000.
b) R and W are constant when R' = 0 and W' = 0:
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