Concept guide
Discrete logistic increment model: invariance of [0,K], fixed points and a short orbit
For the model G(x)=x+r x(1−x/K) with K>0 and 0≤r≤1, prove: if 0≤x≤K then 0≤G(x)≤K, so the whole orbit starting in [0,K] stays in [0,K]; 0 and K are fixed points, and for r>0 they are the only ones. Then compute x1, x2, x3 exactly for K=8, r=1, x0=2, and use an example with r=3 to show that the condition r≤1 cannot be dropped.