Global dynamics of a discrete two-population model for flour beetle growth

The cannibalistic behavior of <i>Tribolium</i> has been extensively researched, revealing instances of chaotic dynamics in laboratory environments for <i>Tribolium castaneum</i>. The well-established Larvae-Pupae-Adult (LPA) model has been instrumental in understanding the conditions that lead to chaos in flour beetles (genus: <i>Tribolium</i>). In response to new experimental observations showing a decline in the pupae population in <i>Tribolium confusum</i>, we proposed and analyzed a simplified two-stage Larvae-Adult (LA) model. This model integrated the pupae population within the larval group, similar to that of the original LPA model, with development transitions governed by internal rates. By applying the model to time-series data, we demonstrated its effectiveness in capturing short-term population fluctuations in <i>T. confusum</i>. We established the model's positivity and boundedness, perform stability analyses of both trivial and positive steady states, and explored bifurcations and steady-state behavior through numerical simulations. We proved global stability for the extinction and positive steady states and observed additional restrictions required for stability compared to the LPA model. Our results indicated that while chaos was a possible outcome, it was infrequent within the practical parameter ranges observed, with environmental changes related to media and nutrient alterations being more likely triggers.