cxf(x 0) = o, Xo E [O, l]. the positive equil ibriu m i = < r < ~' w~ test the stability ofl)/r r _Fo)r yields (r ( at 1 /1. Eval uatm g f (x) = r - 2rx ,1 f'((r - 1)/r) = -r + 2. 1, Local asymptotic stability of i = (r - 1)/r requi 12 - rJ < 1 or 1 < r < 3. 3. In addition, i = (r - 1)/r is unsta ble if r > · l At r = 1 and r -- 3 there are c ianges m the stability of the equilibria. Th to as bifurcation values. 7 Bifurcation Theory Chapter 2 Nonlinear Difference Equations, Theory, and Examples the bifurcation is called a transcritical bifurcation.
E if a < 1, both the pre and Equi libriu m (0, 0) is. redat~r beco me extmct. n a < 1, even in the absenequa tion) . for the pre~ to persi st (see the discrete stabl e if 1 < a /2 _ I < 1 1, 0) Is nonn egati ve and locally asymptotically ( Whe : thes~ ify to 1 < a < 2 and 1a + b. -:-- 1 / < 1. Thes e inequ alitie s simpl but not the preda tor. Sitive and locally asymptotic 8 1 b. -1 ', '' 1 / T + b/ < 1 + b + xy < 2. (,'\:,The y) is posit ive ~f b ibriu m satisfied If x, Y > 0 a~d. a + b < 3. ~rnng these inequ alitie s leads /1 / '' / '' / ' / / ' / / ity criteria does not hold, Whe n one of the inequalities in the local stabil in the complex plane or outsi de at least one eigenvalue eithe r lies on the unit circle ), it can be seen from the previ ous of the unit circle, IAi I 2::: 1.
An Introduction to Mathematical Biology by Linda J.S. Allen