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Understanding Population Dynamics through the Logistic Map Model

This recap discusses a simple model for population dynamics using the logistic function. Explore the different behaviors and cycles exhibited by the system based on the rate of reproduction (a parameter). Learn about attractors and bifurcation points in the system.

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Understanding Population Dynamics through the Logistic Map Model

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  1. Recap • Discussed simple model for population dynamics • New population xn+1 gotten from old xn via logistic function xn+1=axn(1-xn) • a gives rate of reproduction • simple but nonlinear • x given by iteration

  2. The Logistic Map xn+1=axn(1-xn) • After many iterations x reaches some value(s) independent of its starting value • 3 regimes: • a<1: x=0 for large n • 1<a<3: x=constant for large n • 3<a<ac: cyclic behavior • ac<a: mostly chaotic • ac=3.69.. approximately

  3. BehaviorsPeriod 1 and 2 xn a<3.0 n xn a=3.2 - 2-cycle n

  4. Behaviors Period 4 xn n a=3.53 - 4-cycle

  5. BehaviorsPeriod 8 xn n a=3.55 - 8 -cycle

  6. Period Doubling • As a is increased beyond 3.0 the system first shows 2-cycle behavior then 4-cycle, then 8 … • The period keeps doubling • Beyond some value ac=3.7.. motion is irregular (chaotic) • This period doubling route to chaos is seen frequently

  7. Why is special about the points on a cycle? • Consider fixed points. Under iteration the new value f(x) must equal the old value x x=f(x) • For a 2-cycle it must come back to x after 2 iterations y=f(x) x=f(y)=f(f(x)) • but f(x)=ax(1-x) so f(f(x))=ax(1-x)[1-ax(1-x)]

  8. Graphical solution • Fixed-pts • Solution x=f(x) corresponds to intersection of the graph y=x with y=f(x) where f(x)=ax(1-x) • Similarly, the 2 points on a 2-cycle are intersections of y=x with y=f(f(x))

  9. Why do we see only these points ? • We now understand how to find these special cycles • But why should all motions end up on one of these cycles ? • Answer: they are attractors • if we start out with some x close to some special cycle point it will end up after many iterations at the special point

  10. Example • Imagine a=2.5 x=2.5x(1-x) -> x=0.6 • Iterate with x=0.5 -> 0.6 • Iterate with x=0.7 -> 0.6 ! • Property of the logistic map for this value of a.

  11. Doubling ? • For a=2.5 see that twice iterated map x=f(f(x)) has just one intersection - see fixed point behavior. • But for a=3.3 the twice iterated map x=f(f(x)) has now 2 intersections - a 2-cycle.

  12. Bifurcation Diagram x 1 2 4 8 a

  13. Convergence • First bifurcation at a1=3.0 • Second at a2=3.449 • Third at a3=3.544 • Fourth at a4=3.567 • Gap is getting smaller … • Define d=(an+1-an)/(an+2-an+1) • Large n: d is constant • d=4.6692… Feigenbaum constant

  14. Summary • Logistic map can show variety of behaviors depending on a • regular cycles or chaos • Values on these cycles can be found by drawing graphs • These cycles are attractive • As chaotic regime is approached the period keeps doubling - infinite cycle is chaotic !

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