Stage-structured ratio-dependent predator-prey models revisited: When should the maturation lag result in systems' destabilization?

Time delay is often introduced into models of theoretical ecology to take into account stage structuring of real populations. The well-known characteristic benchmark of time delay models is eventual destabilization of systems' equilibria for a sufficiently large maturation/gestation time period. Here we argue, however, that some delay formulations might lack a sound biological rationale and, more importantly, the use of different delay formulations in models might result in rather different outcomes in terms of stability loss. To illustrate this idea we consider a family of predator-prey models with a ratio-dependent predator functional response with a maturation time lag of predators. In such models the functional response depends on the ratio between the predators and the available prey as opposed to a prey-dependent functional response. To describe the effects of delay, we use two different formulations from the literature: one based on the work by Beretta and Kuang (1996), which we call the conventional approach, with delay being included only into the per-capita numerical response of predator. The other formulation is the Wangersky-Cunningham (1957) approach, where delay is introduced in the overall predator numerical response. Unlike the previous studies, we focus here on deriving the explicit conditions of stability of the interior equilibrium (assuring species coexistence) in the presence of delay in terms of model parameters. We investigate three scenarios of prey growth rate parametrization: (i) the prey growth is given by the logistic function, (ii) the prey growth is subject to a strong Allee effect and (iii) there is a weak Allee effect in prey. In the latter two cases the per capita growth rate is an increasing function at low prey density. We show that the use of the two above delay formulations eventually result in completely different outcomes: with the conventional approach, the interior predator-prey equilibrium will be eventually destabilized for a supercritical time lag, whereas, implementation of the Wangersky-Cunningham approach predicts an absolute stability of the equilibrium within a large range of parameters, i.e. the system cannot be destabilized by means of delay. We find that for the models parameters, where delay-induced destabilization in the system with an Allee effect is possible (the interior equilibrium is conditionally stable), the stability loss eventually results in population collapse and extinction of both species.