The Ricker Model

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The projection function for the Ricker model

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and the population dynamics

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The population dynamics for a higher value of r

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Ricker models with various values of the parameter r

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There are two equilibria  when r<2.

When r<2, 0 is an unstable equilibrium, and the carrying capacity K is a stable equilibrium. We can see this by computing

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 (1)

or by looking at a cobweb diagram

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where we see that  all population orbits converge to the carrying capacity

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 (2)

Let us now consider the case where r>2.

We look for a stable 2-cycle for r=

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 (3)

which is shown in frame 25 above.

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 (4)

We first try to solve the equation FF(P)=P,

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 (5)

Well, it seems to be beyond maple's powers to solve the equation FF(P)=P!

Maybe we can solve graphically

Let us look at some cobweb diagrams for F^2

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It looks as if there are  stable 2-cycles near 50 and 150. We approximate this 2-cycle by orbits:

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 (6)

and see that we really arrived at a 2-cycle

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 (7)

We check if it is stable

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 (8)

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 (9)

Yes, the 2-cycle  (44.88, 155.11) is stable.

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 (10)

The stable orbits of the Ricker model can be illutstrated in the bifurcation diagram

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