Abstract: : Latency is a feature of nearly all real-world biological systems, yet many models simplify analysis by assuming such delays to be negligible. Here we examine a three-dimensional host–competitor–parasite model in which a maturation delay is introduced in the parasite’s reproduction term. We analyze how this latency alters the system’s dynamics, particularly its ability to generate more complex bifurcations. In the delayed model, we identify delay-induced Hopf bifurcations, a generalized Hopf (Bautin) point obtained through a fifth-order normal form, and a codimension-two Hopf–zero bifurcation arising when a boundary transcritical condition intersects the Hopf curve.

Beyond local bifurcations, the corresponding non-delayed ODE exhibits a classical period-doubling cascade to chaos. In contrast, the delayed system produces only narrow “chaos windows,” indicating that parasite maturation latency plays a limited role in generating chaos. We also examine bistability between coexistence dynamics, such as periodic or chaotic attractors, and a boundary equilibrium, and we discuss how the introduction of delay influences the extent of this bistability.

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