Convergence to equilibrium for positive solutions of some mutation-selection model

TL;DR

This paper investigates the long-term behavior of positive solutions in a mutation-selection model with Neumann boundary conditions, establishing conditions for existence, uniqueness, and stability of steady states, including in cases of weak competition kernels.

Contribution

It provides the first comprehensive analysis of the global stability and existence conditions for positive steady states in mutation-selection models with general kernels.

Findings

Existence of a unique globally stable positive steady state in blind competition cases.

Necessary and sufficient conditions for positive steady states with general kernels.

Stability results for models with small perturbations in the competition kernel.

Abstract

In this paper we are interested in the long time behaviour of the positive solutions of the mutation selection model with Neumann Boundary condition: $$ \frac{\partial u(x,t)}{dt}=u\left[r(x)-\int_{\O}K(x,y)|u|^{p}(y)\,dy\right]+\nabla\cdot\left(A(x)\nabla u(x)\right),\qquad \text{in}\quad \R^+\times\O$$ where $\O\subset \R^N$ is a bounded smooth domain, $k(.,.) \in C(\bar \O \times C(\bar\O), \R), p\ge 1$ and $A(x)$ is a smooth elliptic matrix. In a blind competition situation, i.e $K(x,y)=k(y)$, we show the existence of a unique positive steady state which is positively globally stable. That is, the positive steady state attracts all the possible trajectories initiated from any non negative initial datum. When $K$ is a general positive kernel, we also present a necessary and sufficient condition for the existence of a positive steady states. We prove also some stability result on the dynamic of the equation when the competition kernel $K$ is of the form $K(x,y)=k_0(y)+\eps k_1(x,y)$. That is, we prove that for sufficiently small $\eps$ there exists a unique steady state, which in addition is positively asymptotically stable. The proofs of the global stability of the steady state essentially rely on non-linear relative entropy identities and an orthogonal decomposition. These identities combined with the decomposition provide us some a priori estimates and differential inequalities essential to characterise the asymptotic behaviour of the solutions.