Global dynamics of delay equations for populations with competition among immature individuals

We analyze a population model for two age-structured species allowing for inter- and intra-specific competition at immature life stages. The dynamics is governed by a system of Delay Differential Equations (DDEs) recently introduced by Gourley and Liu. The analysis of this model presents serious difficulties because the right-hand sides of the DDEs depend on the solutions of a system of nonlinear ODEs, and generally cannot be solved explicitly. Using the notion of strong attractor, we reduce the study of the attracting properties of the equilibria of the DDEs to the analysis of a related two-dimensional discrete system. Then, we combine some tools for monotone planar maps and planar competing Lotka–Volterra systems to describe the dynamics of the model with three different birth rate functions. We give easily verifiable conditions for global extinction of one or the two species, and for global convergence of the positive solutions to a coexistence state.