  Interactions in biology and social systems are not restricted to pairwise but
can take arbitrary sizes. Extensive studies have revealed that the
arbitrary-sized interactions significantly affect the spreading dynamics on
networked systems. Competing spreading dynamics, i.e., several epidemics spread
simultaneously and compete with each other, have been widely observed in the
real world, yet the way arbitrary-sized interactions affect competing spreading
dynamics still lacks systematic study. This study presents a model of two
competing simplicial susceptible-infected-susceptible epidemics on a
higher-order system represented by simplicial complex and analyzes the model's
critical phenomena. In the proposed model, a susceptible node can only be
infected by one of the two epidemics, and the transmission of infection to
neighbors can occur through pairwise (i.e., an edge) and high-order (e.g.,
2-simplex) interactions simultaneously. Through a mean-field (MF) theory
analysis and numerical simulations, we show that the model displays rich
dynamical behavior depending on the 2-simplex infection strength. When the
2-simplex infection strength is weak, the model's phase diagram is consistent
with the simple graph, consisting of three regions: the absolute dominant
regions for each epidemic and the epidemic-free region. With the increase of
the 2-simplex infection strength, a new phase region called the alternative
dominant region emerges. In this region, the survival of one epidemic depends
on the initial conditions. Our theoretical analysis can reasonably predict the
time evolution and steady-state outbreak size in each region. In addition, we
further explore the model's phase diagram both when the 2-simplex infection
strength is symmetrical and asymmetrical. The results show that the 2-simplex
infection strength has a significant impact on the system phase diagram.
