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Go to movie of discrete logistic 2-cycle (*r* = 2.2) case Go to movie of discrete logistic chaos (*r* = 3) case

**Intraspecific competiton -- discrete logistic equation
low
r slightly oscillating approach to K case**

Eqn 15.1

**Embedded QuickTime Movie** (from Mathematica notebook *r1Movie.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= 1.5 andK= 1,000. Note that the population converges very quickly onK. 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 stair step -- "up, over, up, over"). Following the map this way gives us the successive population sizes over time. Note that the trajectory moves toward thestableequilibrium point (atK= 1,000). The last few frames involve changes so small they don't show up onscreen.

**The low r value here is what permits the convergence on K.**

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Fig. 2.Static view of the trajectory toward the stable equilibrium atK= 1,000 from a startingNof 120._{0}

Note the stair step ascent from the starting value toK. The very slight oscillations are shown by the lines spiralling inward aroundK.