A Two-Generation Discrete-Time Insect Population Model: Stability and Bifurcation Analysis

Delayed density dependence is a significant process in insect population dynamics; nonetheless, its identification and resulting dynamical implications remain subjects of substantial contention. Motivated by this controversy, we investigate a two-generation discrete-time model of the form xn+1=xnf(axn+bxn−1), where a,b>0. This model integrates both instantaneous and one-generation-lagged density dependence via a weighted amalgamation of successive population densities. Based on general assumptions regarding the diminishing recruitment function f, we delineate the local asymptotic stability of the positive equilibrium and demonstrate that the ratio ab, which quantifies the relative influence of direct versus delayed density dependence, plays a crucial role in determining stability and its structure. The stability region is maximized when a=3b; if a<3b, stability may be compromised by a Neimark–Sacker bifurcation, whereas for a>3b, it may be compromised through a period-doubling bifurcation. We also investigate strong local asymptotic stability and identify additional parameter regimes satisfying this criterion after the first expansion. Separately, we establish global asymptotic stability under explicit sufficient conditions by constructing enveloping maps. As prototypes, we consider Ricker- and logistic-type models. We implement the overarching theory, derive explicit invariant regions, establish bifurcation criteria, and obtain global stability results. The examination underscores that the balance between direct and delayed density dependence can significantly alter the qualitative behavior of insect populations.