Branching Phenomena in Fluid Dynamics and Chemical Reaction-Diffusion Theory

Bifurcation in a predator-prey model with diffusion: Rotating waves

“…Note that it follows from the implicit function theorem (cf. [9]) that (4.21) has a unique curve of solutions in a neighbourhood of (0, c 0 , ε 0 ) if •G(·, c 0 , ε 0 ) is Fréchet-differentiable near 0 and • the linear operator L 0 ∈ Lin(Y) defined by L 0 y := y − ∂ 1G (0, c 0 , ε 0 )y is continuously invertible.…”

“…With these notations we can formulate the basic result on the bifurcation of nontrivial solutions based on a theorem proved in [2] which is a consequence of the implicit function theorem described in [9]. Theorem 5.2 Suppose thatG : Y × (R\{0}) × R → Y is k-times continuously differentiable in an open neighbourhood of (0, c 0 , ε 0 ) and assume that…”