In this video I go over an example on the Logistic Equation for Population Growth and this time analyze a direction field for the equation. The direction field is a good way of seeing how many different solutions to the differential equation behave. The interesting characteristics of the directional field is that the solutions move from the P(t) = 0 equilibrium solution to the P(t) = K (the carrying capacity) equilibrium solution. Also the highest growth rate occurs at half the carrying capacity. The last characteristic I will prove in my next video so stay tuned for that!
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Example:
Draw a direction field for the logistic equation with k = 0.08 and carrying capacity K = 1000.
What can you deduce about the solutions?
The logistic equation is autonomous (dP/dt depends only on P, not t), so the slopes are the same along the horizontal line.
As expected the slopes are positive for 0 < P < 1000 and negative for P > 1000.
The slopes are small when P is close to 0 or 1000 (the carrying capacity).
Notice that the solutions move away from the equilibrium solution P = 0 and move toward the equilibrium solution P = 1000.
Let's use the direction field to sketch solution curves with initial populations P(0) = 100, 400, and 1300.
Notice that solution curve that start below P = 1000 are increasing and those that start above P = 1000 are decreasing.
The slopes are greatest when P ≈ 500.
- In fact we can prove that all solution curves that start below P = 500 have an inflection point when P is exactly 500.
- Recall that an inflection point is when the 2nd derivative changes signs, i.e. changes concavity.
- I will go over this in my next video, so stay tuned!
@mes are you a mathematics teacher? heh heh to be able to do all this
haha I just like doing math! (Didn't like how math was taught in school so decided to teach myself :))
Je je but it's not easy everything you do, you see that you have a lot of knowledge and that's good
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I do not love maths still you did write a great post..please follow and upvote i am also doing