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

Samantha J. Brozak; Kamrun N. Keya; Denise Dengi; Sophia Peralta; John D. Nagy; Yang Kuang; Samantha J. Brozak; Kamrun N. Keya; Denise Dengi; Sophia Peralta; John D. Nagy; Yang Kuang

doi:10.3934/mbe.2025072

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 8: 1980-1998. doi: 10.3934/mbe.2025072

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Research article

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

1.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe 85287, USA
2.
Department of Mathematics, University of California Berkeley, Berkeley 94720, USA
3.
Department of Life Sciences, Scottsdale Community College, Scottsdale 85256, USA

^† These authors contributed equally.
Academic Editor: Gail Wolkowicz

Received: 29 April 2025 Revised: 27 May 2025 Accepted: 09 June 2025 Published: 25 June 2025

The cannibalistic behavior of Tribolium has been extensively researched, revealing instances of chaotic dynamics in laboratory environments for Tribolium castaneum. The well-established Larvae-Pupae-Adult (LPA) model has been instrumental in understanding the conditions that lead to chaos in flour beetles (genus: Tribolium). In response to new experimental observations showing a decline in the pupae population in Tribolium confusum, 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 T. confusum. 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.
- flour beetles,
- discrete map,
- matrix model,
- LPA model,
- Tribolium,
- population model
Citation: Samantha J. Brozak, Kamrun N. Keya, Denise Dengi, Sophia Peralta, John D. Nagy, Yang Kuang. Global dynamics of a discrete two-population model for flour beetle growth[J]. Mathematical Biosciences and Engineering, 2025, 22(8): 1980-1998. doi: 10.3934/mbe.2025072

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Abstract

The cannibalistic behavior of Tribolium has been extensively researched, revealing instances of chaotic dynamics in laboratory environments for Tribolium castaneum. The well-established Larvae-Pupae-Adult (LPA) model has been instrumental in understanding the conditions that lead to chaos in flour beetles (genus: Tribolium). In response to new experimental observations showing a decline in the pupae population in Tribolium confusum, 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 T. confusum. 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.

References

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