On models of shared resource competition, coexistence and traveling waves

We investigate a two-species competition model in which both populations exploit a common standing resource. The dynamics are governed by a system of nonlinear differential equations admitting three equilibrium classes: extinction, competitive exclusion, and coexistence. Analytical conditions on the biological parameters ensuring the existence and asymptotic stability of these equilibria are derived, with particular emphasis on the coexistence equilibrium, representing the stable persistence of both species under shared-resource competition. In the corresponding reaction-diffusion model posed on an unbounded spatial domain, we further examine the stability of the coexistence equilibrium via traveling wavefronts. Using the upper-lower solution method, we establish the existence of traveling wave solutions connecting the extinction or single-dominance states to the coexistence state for a continuum of wave speeds exceeding a biologically determined minimal value, which depends explicitly on equilibrium magnitudes and other key parameters. Numerical simulations are provided to corroborate the theoretical results and to illustrate dynamic transitions from dominance to stable coexistence.