Asymptotic formulas for $L^2$ bifurcation curves of nonlocal logistic equation of population dynamics

TL;DR

This paper derives precise asymptotic formulas for the $L^2$ bifurcation curves in a nonlocal logistic population model, revealing their shape as the $L^2$ norm of solutions grows large.

Contribution

It provides the first detailed asymptotic analysis of $L^2$ bifurcation curves for nonlocal logistic equations, enhancing understanding of their long-term behavior.

Findings

Asymptotic shape of bifurcation curves as $ o \infty$

Explicit formulas for $ifurcation$ curves in nonlocal logistic models

Insights into the stability and structure of solutions at large norms

Abstract

The one-dimensional nonlocal Kirchhoff type bifurcation problems which are derived from logistic equation of population dynamics are studied. We obtain the precise asymptotic shapes of $L^2$ bifurcation curves $\lambda = \lambda(\alpha)$ as $\alpha \to \infty$, where $\alpha:= \Vert u_\lambda \Vert_2$.