# Adaptive dynamics of diverging fitness optima

## TL;DR

This paper studies how populations adapt to changing environments with multiple shifting fitness optima.

## Contribution

The novelty is analyzing populations with two diverging optimal traits that shift at different velocities.

## Key findings

- Sufficient conditions for population extinction or persistence are determined using eigenvalue problems.
- Long time solutions concentrate on lagged optima strictly behind the true shifting optima.
- When shift velocities differ, the solution concentrates on the optimum with maximum lagged fitness.

## Abstract

We study the long time behaviour of a non-local parabolic integro-differential equation modelling the evolutionary dynamics of a phenotypically-structured population in a changing environment. Such models can arise in variety of contexts from climate change to chemotherapy to the ageing body. The main novelty is that there are two locally optimal traits, each of which shifts at a possibly different linear velocity. We determine sufficient conditions to guarantee extinction or persistence of the population in terms of associated eigenvalue problems. When the population does not go extinct, we study the behaviour of long time solutions in the case of rare mutations: the long time solution concentrates as a sum of Dirac masses on a point set of "lagged optima" which are strictly behind the true shifting optima as the mutation rate goes to zero. If we further assume the shift velocities are different, we show the solution concentrates specifically on the positive lagged optimum with maximum lagged fitness. Our results imply that for populations undergoing competition in temporally changing environments, both the true optimal fitness and the required rate of adaptation for each of the diverging optimal traits determine the eventual dominance of one trait.