Thursday, January 22, 2015

Chaos & Fishes Cannibalizing their Offsprings!

Ewww!! Yes that's true of some Darter, pupfish varieties and other fish species too (Filial Cannibalism)! The dynamics of the population with adults cannibalizing on their offspring can be described by the Ricker map, given by the equation:

Figure 1: Bifurcation Diagram (Click to enlarge) -Plot between R and eta. Chaos and bifurcations and period doubling observed. We observe the 2 point limit cycle is observed at R value > 7.4

Figure 2: We observe a 4-point limit cycle for values of R > 12.5, see above figure

The second iterate results in a 2-cycle periodic solution and then later this branches out and a 4-cycle periodic solution appears. Thus as R passes through a series of bifurcation values the character of the solution passes through a series of bifurcations, as period doubling of the periodic solutions.

As R increases through successive bifurcations, every even periodic solution branches into a double-periodic solution and this happens when R is such that the eigenvalue of the periodic solution passes through −1. The distance between bifurcations in R-space also gets smaller and smaller.

Bifurcation to higher periodic solutions eventually leads to chaotic behavior: Initially observe the solution oscillates between two points then it exhibits a 4-period solution then the solutions become chaotic. Again around a small window of R = 23-25 we see that the solutions exhibit regular periodic solutions after which they again become aperiodic/chaotic. So the same sequence of bifurcations get repeated and it is akin to a fractal, and the periodicity-aperiodicity-periodicity sequence is also repeated.

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