Analysis of a Two‑Species Discrete Competition Model Assuming a Single‑Cycle Maturation Delay

Abstract

We introduce and analyze a discrete-time, two-species competition model that requires one cycle before newborn cohort can contribute to the growth of the population. Our formulation admits all of the outcomes of the classical discrete Lotka-Volterra like competition model: both species die out; one species wins the competition independent of the initial conditions; there is a unique coexistence fixed point that is a saddle and the winning species depends on the initial conditions; or there is a unique coexistence fixed point that is globally asymptotically stable. It also admits two novel bistable regimes. In the first regime, one stable and one unstable interior fixed point coexist with a stable and unstable boundary fixed point. In the second regime, the two boundary fixed points are saddles and there are three interior fixed points; two are attractors and the one in between them is a saddle. Using monotone dynamical systems theory, we show that every forward orbit converges to some fixed point. In the general case, where the two species’ parameters need not coincide, we derive sufficient conditions for extinction, exclusion, and unique coexistence. We also establish sufficient conditions for each of the novel bistable regimes to occur provided we know how many interior fixed points exist. In the symmetric case where certain ratios of the parameters of each species are identical, the first of the bistable regimes is ruled out and we obtain necessary and sufficient conditions for the second case. Finally, we illustrate each regime with phase portraits and bifurcation diagrams.