Families of solution curves for some non-autonomous problems

TL;DR

This paper analyzes families of positive solutions for non-autonomous boundary value problems, establishing conditions for non-intersecting solution curves and applying numerical continuation methods, with applications to logistic equations in harvesting models.

Contribution

It introduces conditions for non-intersecting solution curves and develops a numerical continuation technique for analyzing parameter-dependent solutions.

Findings

Solution curves do not intersect under certain conditions.

Maximum solution value acts as a global parameter.

Numerical continuation effectively tracks solution families.

Abstract

The paper studies families of positive solution curves for non-autonomous two-point problems \[ u"+\lambda f(u)-\mu g(x)=0, \;\; -1<x<1, \;\; u(-1)=u(1)=0 \] depending on two positive parameters $\lambda$ and $\mu$. We regard $\lambda$ as a primary parameter, giving us the solution curves, while the secondary parameter $\mu$ allows for evolution of these curves. We give conditions under which the solution curves do not intersect, and the maximum value of solutions provides a global parameter. Our primary application is to constant yield harvesting for diffusive logistic equation. We implement numerical computations of the solution curves, using continuation in a global parameter, a technique that we developed in [11].