  Particles that are catalytically-active and chemotactic can interact through
the concentration fields upon which they act, which in turn may lead to
wide-scale spatial self-organization. When these active particles interact
through several fields, these interactions gain an additional structure, which
can result in new forms of collective behaviour. Here, we study a mixture of
active species which catalyze the conversion of a substrate chemical into a
product chemical, and chemotax in concentration gradients of both substrate and
product. Such species develop non-reciprocal, specific interactions that we
coarse-grain into attractive and repulsive, which can lead to a potentially
complex interaction network. We consider the particular case of a metabolic
cycle of three species, each of which interacts with itself and both other
species in the cycle. We find that the stability of a cycle of species that
only chemotax in gradients of their substrate is piloted by a set of two
parameter-free conditions, which we use to classify the low number of
corresponding interaction networks. In the more general case of substrate- and
product-chemotactic species, we can derive a set of two high-dimensional
stability conditions, which can be used to classify the stability of all the
possible interaction networks based on the self- and pair-interaction motifs
they contain. The classification scheme that we introduce can help guide future
studies on the dynamics of complex interaction networks and explorations of the
corresponding large parameter spaces in such metabolically active complex
systems.
