In this video I continue on to part 2 of example 1 on the predator-prey systems using the Lotka-Volterra equations which I derived in my earlier videos. In part 2 I go over parts d) and e) of the example. These parts involved looking closely at a specific phase trajectory and then from that predict the population curves for both wolves and rabbits. The data from the Hudson Bay Companies fur trading business of hares and lynxes over a 90 year period was used to compare how well the Lotka-Volterra models compare with real-world applications. In turns out that the HBC data generally does in fact follow the model developed in this example. This is a very interesting video on showing how real-life applications can be represented and modeled through mathematics, so make sure to watch this video!
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Example:
Suppose that populations of rabbits and wolves are described by the Lotka-Volterra equations (https://youtu.be/b3NCjsZhQdQ) with k = 0.08, a = 0.001, r = 0.02, and b = 0.00002.
Assume the time t is measured in months.
a) Find the constant solutions (called the equilibrium solutions) and interpret the answer.
b) Use the system of differential equations to find an expression for dW/dR.
c) Draw a direction field for the resulting differential equation in the RW-plane.
Then use that direction field to sketch some solution curves.
d) Suppose that, at some point in time, there are 1000 rabbits and 40 wolves.
Draw the corresponding solution curve and use it to describe the changes in both population levels.
e) Use part (d) to make sketches of R and W as functions of t.
Solution
a) A trivial solution is given by R = 0 and W = 0.
The other constant solution is R = 1000, W = 80.
b) We can use the Chain Rule:
d) Using the direction field from c) and starting at R = 1000 and W = 40, we get:
Starting at P0 at time t = 0, and letting t increase, do we move clockwise or counterclockwise around the phase trajectory?
If we put R = 1000 and W = 40 in the first differential equation, we get:
Since dR/dt > 0, this means that R is increasing at P0 and so we move counterclockwise around the phase trajectory.
We see that at P0, there aren't enough wolves to maintain a balance between the populations, so the rabbit population increases.
That results in more wolves and eventually there are so many wolves that the rabbits have a hard time avoiding them.
So the number of rabbits begins to decline (at P1 where we estimate that R reaches its maximum population of rabbits of about 2800.)
This means that at some later time the wolf population starts to fall (at P2, where R = 1000 and W ≈ 140.)
But this benefits the rabbits, so their population later starts to increase (at P3, where W = 80 and R ≈ 210).
As a consequence, the wolf population eventually starts to increase as well.
This happens when the populations return to their values of R = 1000 and W = 40, and the entire cycle begins again.
e) From the description in part d) on how the rabbit and wolf populations rise and fall, we can sketch the graphs of R(t) and W(t).
Suppose the points P1, P2, and P3 are reached at times t1, t2, and t3.
Then we can sketch graphs of R and W:
To make the graphs easier to compare, we draw the graphs on the same axes but with different scales for R and W:
Notice that the rabbits reach their maximum populations about a quarter of a cycle before the wolves.
An important part of the modeling process, is to interpret our mathematical conclusions as real-world predictions and to test the predictions against real data.
The Hudson’s Bay Company, which started trading in animal furs in Canada in 1670, has kept records that date back to the 1840s. The figure below shows graphs of the number of pelts of the snowshoe hare and its predator, the Canada lynx, traded by the company over a 90-year period.
You can see that the coupled oscillations in the hare and lynx populations predicted by the Lotka-Volterra model do actually occur and the period of these cycles is roughly 10 years.
Although the relatively simple Lotka-Volterra model has had some success in explaining and predicting coupled populations, more sophisticated models have also been proposed.
One way to modify the Lotka-Volterra equations is to assume that, in the absence of predators, the prey grow according to a logistic model with carrying capacity K.
Then the Lotka-Volterra equations are replaced by the system of differential equations:
Models have also been proposed to describe and predict population levels of two species that compete for the same resources or cooperate for mutual benefit.
I may cover examples on these models in later videos, stay tuned!