  In complex ecosystems such as microbial communities, there is constant
ecological and evolutionary feedback between the residing species and the
environment occurring on concurrent timescales. Species respond and adapt to
their surroundings by modifying their phenotypic traits, which in turn alters
their environment and the resources available. To study this interplay between
ecological and evolutionary mechanisms, we develop a consumer-resource model
that incorporates phenotypic mutations. In the absence of noise, we find that
phase transitions require finely-tuned interaction kernels. Additionally, we
quantify the effects of noise on frequency dependent selection by defining a
time-integrated mutation current, which accounts for the rate at which
mutations and speciation occurs. We find three distinct phases: homogeneous,
patterned, and patterned traveling waves. The last phase represents one way in
which co-evolution of species can happen in a fluctuating environment. Our
results highlight the principal roles that noise and non-reciprocal
interactions between resources and consumers play in phase transitions within
eco-evolutionary systems.
