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Go to movie of discrete logistic 2-cycle (*r* = 2.2) case Go to movie of discrete logistic converge-to-*K* (*r* = 1.5) case

**Intraspecific competition -- discrete logistic equation
very high (chaotic) r case**

Eqn 15.1

**Embedded QuickTime Movie** (from Mathematica notebook *ChaosMovie.nb*)

**Click on the image below to start the "movie". A control bar will appear below the graph; the various buttons will allow you to stop and start it.**

Fig. 1.One-dimensional "map" of the discrete logistic (Eqn 15.1, above) withr= 3andK= 1,000. Note that the population never settles on any value or set of values. We start with anN(t) value on theX-axis, move up to the "map" (red line) to get anN(t+x) value and then take the correspondingY-value by going over to the 1:1 black line, then again move to the red line. Tracing the map is therefore a process of going back and forth from the red map line to the blackN(t+x) =N(t) 1:1 line (like a box step dance -- "up, right, down, left"). Following the map this way gives us the successive population sizes over time. Note that the trajectory moves away from theunstableequilibrium point (atK= 1,000) and toward a chaotic pattern of highly variable population sizes. Here, the "movie" runs for 99 time steps. We could go for much longer without ever repeating our steps.

**The high r value here is what drives the cycle.**

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Fig. 2.Static view of the trajectory for 99 time steps in the chaotic range from a startingNof 950._{0}

Note that values nearunstableequilibrium pointK(1,000) will spiral outward. The places where the values spend more time (and develop thicker black lines) are called "strange attractors" in chaos theory.