Bifurcation structure, resonance dynamics, and stabilization in a capacity-limited honeybee–varroa map

Asifa Tassaddiq; Muhammad Sajjad Shabbir; Rabab Alharbi; Dalal Khalid Almutairi; Rizwan Ahmed; JungYoon Kim; Younhyun Jung; Jawad Khan; Asifa Tassaddiq; Muhammad Sajjad Shabbir; Rabab Alharbi; Dalal Khalid Almutairi; Rizwan Ahmed; JungYoon Kim; Younhyun Jung; Jawad Khan

doi:10.3934/math.20261319

AIMS Mathematics

2026, Volume 11, Issue 10: 33745-33767. doi: 10.3934/math.20261319

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Research article

Bifurcation structure, resonance dynamics, and stabilization in a capacity-limited honeybee–varroa map

1.
Department of Information Technology, College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudi Arabia
2.
Department of Mathematics, University of Poonch Rawalakot, Rawalakot 12350, Pakistan
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
Department of Mathematics, College of Science, Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia
5.
International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore, Pakistan
6.
Department of Game Media, College of IT Convergence, Gachon University, Seongnam, Korea
7.
School of Computing, Gachon University, Seongnam 13120, Republic of Korea

Received: 13 September 2026 Revised: 29 September 2026 Accepted: 08 October 2026 Published: 19 October 2026
MSC : 39A28, 39A30

We extend a previously studied two-dimensional honeybee–Varroa destructor census model by introducing a capacity-limited mite-removal term. For the continuous system, positivity and dissipativity are established, while for the forward Euler map, a conditional invariant simplex and coexistence equilibria are characterized. Boundary analysis distinguishes saddle-node creation of bee-only equilibria from a transcritical mite-invasion threshold. At coexistence, variation of the census interval produces nondegenerate supercritical Neimark–Sacker and period-doubling bifurcations. Two-parameter analysis identifies candidate 1:2, 1:3, and 1:4 strong resonances and their nondegeneracy indicators. An equilibrium-targeting correction provides a local stabilization boundary. Comparisons across census intervals and discretization schemes reveal scheme-dependent global dynamics. The results therefore provide a theoretical bifurcation framework, not a calibrated prediction of colony dynamics.
- nonlinear systems,
- bifurcation,
- strong resonance,
- sensitivity analysis,
- simulation
Citation: Asifa Tassaddiq, Muhammad Sajjad Shabbir, Rabab Alharbi, Dalal Khalid Almutairi, Rizwan Ahmed, JungYoon Kim, Younhyun Jung, Jawad Khan. Bifurcation structure, resonance dynamics, and stabilization in a capacity-limited honeybee–varroa map[J]. AIMS Mathematics, 2026, 11(10): 33745-33767. doi: 10.3934/math.20261319

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Abstract

We extend a previously studied two-dimensional honeybee–Varroa destructor census model by introducing a capacity-limited mite-removal term. For the continuous system, positivity and dissipativity are established, while for the forward Euler map, a conditional invariant simplex and coexistence equilibria are characterized. Boundary analysis distinguishes saddle-node creation of bee-only equilibria from a transcritical mite-invasion threshold. At coexistence, variation of the census interval produces nondegenerate supercritical Neimark–Sacker and period-doubling bifurcations. Two-parameter analysis identifies candidate 1:2, 1:3, and 1:4 strong resonances and their nondegeneracy indicators. An equilibrium-targeting correction provides a local stabilization boundary. Comparisons across census intervals and discretization schemes reveal scheme-dependent global dynamics. The results therefore provide a theoretical bifurcation framework, not a calibrated prediction of colony dynamics.

References

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