Elusive chaos: dynamics of a data-based model of flour beetle growth with time delay

The LPA model is a discrete-time map that has been well-studied and well-validated with experimental data using Tribolium castaneum. We argue that the long lifespan of adults warrants the use of a continuous-time model, and thus we propose a two-dimensional system of delay differential equations to describe Tribolium dynamics. We show that the time delay allows for the periodic behavior that characterizes flour beetle populations. We illustrate the global stability of the extinction equilibrium for $R_0<1$ . Because of nonlinearities in the model, we analyze stability of the positive equilibrium in the special case when there is no adult cannibalism of pupae, exploring the biological parameter space, and finally using a quasi-steady-state argument to reduce the model. We study bifurcations numerically and, unlike previous work, find that larvae development rate and larvae mortality affect the potential for asymptotic cyclic behavior, while other parameters (adult egg production rate per day and adult mortality) influence cycles in the transient phase. This study suggests that model structure and form play an important role in finding chaos.