A functional reaction-diffusion equation from climate modeling: S-shapedness of the principal branch of fixed points of the time-$1$-map

Abstract

If the seasonal cycle as well as the long response times of the climate system are taken into account, one-layer energy balance climate models give rise to parameter-dependent functional reaction-diffusion equations with 1-periodic forcing and a time delay $T\gg 1$. We show that the principal branch of fixed points of the corresponding time-1-map is S-shaped in the sense that it is a simple curve with an even number of turning points. This curve connects $(0,\mathbf {0})$ and $(\infty , \infty )$ within $(0,\infty) \times C([-T,0]\times M,(0,\infty))$, $M$ a compact, oriented Riemannian surface. The paper is a continuation of [13], where a case without time-delay was considered.