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← 04-10 Finding Equilibria

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Showing Revision 2 created 10/24/2012 by Amara Bot.

  1. Obviously, there is an equilibrium at 0--f stays 0 once it becomes 0.
  2. This is not nothing but an equilibrium and there's two more--
  3. one is at 0.55 million tons and the other is at 1.45 million tons.
  4. The first one is obvious, but now let's look into where these two other equilibria come from.
  5. If we reach an equilibrium, the amount has to become constant.
  6. The rate of change becomes zero, which means that the gain
  7. per time has to balance the loss per time.
  8. The loss per time is constant but the gain per time depends on the current amount.
  9. Let's build this type of diagram to show how the gain depends on the amount.
  10. If you look closely, you see that this has to be a parabola.
  11. If the amount of fish is zero, we are multiplying by zero here and the gain will be zero.
  12. That's no surprise. If there is no fish, there is no growth. We know this point.
  13. When the amount of fish is equal to the maximum carrying capacity, this factor becomes zero.
  14. The product is zero again so we know this point.
  15. As a parabola is symmetric, the maximum has to occur in the middle between the zeros.
  16. This maximum sits at an amount of 1 * 10⁶ tons.
  17. We can plug this value 1 million tons into this expression
  18. and get that the gain here is 2.5 * 10⁵ tons per year, which is a little more than the loss,
  19. which is 2 * 10⁵ tons per year.
  20. We have one equilibrium here and one equilibrium there.
  21. At both of these points the gain and the loss, the gain and the loss balance each other.
  22. If you want to, you can solve this quadratic equation
  23. to find that this equilibrium sits at 0.55 * 1 million tons
  24. and this equilibrium sits at 1.45 * 1 million tons.