We aim to analyse the concentration property, in the phenotype trait space, of
the steady state solutions of an integro-differential model representing the evolution-
ary epidemiology of plant pathogens facing host population with quantitative disease
resistance.
Our motivation comes from the fact that modeling the epidemiology and the evolu-
tion of pathogens has long been addressed through analysis of invasibility assuming
that epidemiological and evolutionary time scales are distinct. These analysis ignore
short-term evolutionary and epidemiological dynamics despite the major interests of
these dynamics in agro-ecosytems. Deriving sustainable management strategies of
resistance gene to plant pathogens is one of the many examples where one want to
make quantitative predictions far away from the epidemiological equilibrium. Up to
date, most of the modelling approaches devoted to design sustainable strategies of
resistance management tackle the case of qualitative resistance based on gene for
gene interactions that triggers host immunity to the disease. Few works consider
adaptation to quantitative resistances that reduce pathogen aggressiveness instead of
conferring immunity. Pathogen aggressiveness is determined experimentally by plant
pathologists by measuring life history traits which described the disease life cycle:
the infection efficiency, the latency time, the sporulation period and the rate of spore
production.
In this work, we used integro-differential equations to model both the epidemiologi-
cal and the evolutionary dynamics of spore-producing pathogens in a homogeneous
host population. The host population is subdivided into compartments (Susceptible
or healthy host tissue (S), Infected tissue (i) and Airborne spores (A)) by explicitly
integrating dedicated life-history traits of the pathogens as well as the life cycle of the
pathogen by taking into account the age since infection. An integral operator with
kernel m describing mutations from a pathogen strain with phenotypic value y ∈ R N
to another one with phenotypic value x ∈ R N . More specifically, we investigate the
concentration properties in the phenotype trait space of model stationary states when
the mutation kernel m depends on a small parameter ε > 0 and is highly concentrated.