A host-pathogen coevolution model. Part I: Run straight for your life

TL;DR

This paper introduces a new mathematical model for host-pathogen coevolution, demonstrating perpetual evolutionary chase scenarios and analyzing their behaviors through simulations and rigorous mathematical proofs.

Contribution

It presents a novel PDE-based model capturing chase dynamics in host-pathogen coevolution and proves the existence of traveling wave solutions.

Findings

Emergence of chase scenarios in simulations

Identification of traveling pulse solutions

Periodic phenotypic distributions in coevolution

Abstract

In this study, we propose a novel model describing the coevolution between hosts and pathogens, based on a non-local partial differential equation formalism for populations structured by phenotypic traits. Our objective with this model is to illustrate scenarios corresponding to the evolutionary concept of ''Chase Red Queen scenario'', characterized by perpetual evolutionary chases between hosts and pathogens. First, numerical simulations show the emergence of such scenarios, depicting the escape of the host (in phenotypic space) pursued by the pathogen. We observe two types of behaviors, depending on the assumption about the presence of a phenotypic optimum for the host: either the formation of traveling pulses moving along a straight line with constant speed and constant profiles, or stable phenotypic distributions that periodically rotate along a circle in the phenotypic space. Through rigorous perturbation techniques and careful application of the implicit function theorem in rather intricate function spaces, we demonstrate the existence of the first type of behavior, namely traveling pulses moving with constant speed along a straight line. Just as the Lotka-Volterra models have revealed periodic dynamics without the need for environmental forcing, our work shows that, from the pathogen's point of view, various trajectories of mobile optima can emerge from coevolution with a host species.