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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![display([dynplotfilm(ricker(0.5e-1, 100), 2, 200), dynplotfilm(ricker(0.5e-1, 100), 130, 200)]); 1](images/ricker_111.gif) |
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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![[0., 6.049647464], [0., 6.049647464]](images/ricker_118.gif) |
(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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![seq(iterate(ricker(1.8, 100), P0, 200), P0 = [1, 5, 6, 150, 250]); 1](images/ricker_122.gif) |
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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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which is shown in frame 25 above.
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We first try to solve the equation FF(P)=P,
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![[0., 90.01713129], [0., 90.01713129]](images/ricker_129.gif) |
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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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![seq(iterate(proc (P) options operator, arrow; F(F(P)) end proc, P0, 300), P0 = [10, 30, 80, 180]); 1](images/ricker_133.gif) |
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(6) |
and see that we really arrived at a 2-cycle
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![orbit(F, %[1], 6); 1](images/ricker_135.gif) |
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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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, P0, 500)), P0 = [10, 20, 30, 45, 50])}; 1](images/ricker_141.gif) |
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(10) |
The stable orbits of the Ricker model can be illutstrated in the bifurcation diagram
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![pointplot([seq(seq([`+`(1.8, `*`(0.5e-2, `*`(i))), orbit(ricker(`+`(1.8, `*`(0.5e-2, `*`(i))), 100), 80, 250)[j]], j = 230 .. 250), i = 1 .. 240)], style = point, color = blue); 1](images/ricker_143.gif)
![pointplot([seq(seq([`+`(1.8, `*`(0.5e-2, `*`(i))), orbit(ricker(`+`(1.8, `*`(0.5e-2, `*`(i))), 100), 80, 250)[j]], j = 230 .. 250), i = 1 .. 240)], style = point, color = blue); 1](images/ricker_144.gif) |
