In this video I go over part 2 of the example which I covered in my last video on modifying the Lotka-Volterra equations for predator-prey systems by using a logistic model growth rate for the prey population. In that part I went over the equilibrium solutions, and in this part I take a look at a phase trajectory of the predator-prey, or wolf-rabbit, populations. I also sketch the possible graph of the populations of each species through time, which shows how the populations fluctuate until they stabilize at a specific level. This is a very interesting video on modeling populations of predator-prey type animals so make sure to watch this video!
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Example:
In Example 1 (
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.
d) Sketch the populations of the rabbit and wolf populations as functions of time.
Solution:
Recall the Logistic Equation (
Thus in the absence of wolves, the rabbit population stabilizes at 5000.
b) R and W are constant when R' = 0 and W' = 0:
Significance of each case:
Case 1: Both populations are zero so can't grow.
Case 2: With no wolves present, the rabbit population maxes out at 5000.
Case 3: The predator/prey interaction balances out and the populations are stable.
c) The populations of wolves and rabbits fluctuate around 64 and 1000, respectively, and eventually stabilize at those values.
d) A sketch of the rabbit and wolf populations with time is shown below:
