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Go to movie of discrete logistic chaos (*r* = 3) case Go to movie of discrete logistic converge-to-*K* (*r* = 1.5) case

**Intraspecific competiton -- discrete logistic equation
high r 2-point cycle case**

Eqn 15.1

**Embedded QuickTime Movie** (from Mathematica notebook *DiscLogAnimTraj03.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= 2.2 andK= 1,000. Note that the population converges on a two-point cycle. 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 an oscillation betweentwo other stable equilibrium points-- one at 497 and the other at 1,503. Once it gets there it will oscillate endlessly between these two values.

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

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Fig. 2.Static view of the trajectory toward the 2-point stable equilibrium from a startingNof 950._{0}

Note the rectangular outward spiral from the starting value (near theunstableequilibrium pointK) to the stable oscillatory points, 497 and 1503. The upper left corner is at {497, 1503} while the lower right is {1503, 497}. These are the two places the red map curve crosses the line of slope -1 (it also crosses atK).