Cost-effective strategies for controlling date palm mite infestations: an optimal control approach

Saleh Fahad Aljurbua; Moustafa El-Shahed; Saleh Fahad Aljurbua; Moustafa El-Shahed

doi:10.3934/math.20261058

AIMS Mathematics

2026, Volume 11, Issue 8: 26359-26396. doi: 10.3934/math.20261058

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Research article

Cost-effective strategies for controlling date palm mite infestations: an optimal control approach

Saleh Fahad Aljurbua ^{1
,
,},
Moustafa El-Shahed ^1,2

1.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia
2.
Department of Computer Science, College of Engineering and Information Technology, Onaizah Colleges, Qassim 56447, Saudi Arabia

Received: 06 April 2026 Revised: 02 August 2026 Accepted: 18 August 2026 Published: 24 August 2026
MSC : 92D25, 49J15, 49K15, 93C95, 92D40

This paper develops and analyzes a stage-structured mathematical model for the population dynamics of date palm mite infestations, incorporating density-dependent saturation in egg production and explicitly accounting for the egg, larval, nymphal, and adult stages. A population reproduction threshold is derived to characterize the transition between the pest's extinction and persistence. Stability analysis shows that the extinction equilibrium is locally and globally asymptotically stable when the reproduction threshold remains below unity. Conversely, when this threshold exceeds one, a unique positive coexistence equilibrium emerges. The model is further shown to undergo a forward bifurcation at the critical threshold, indicating a smooth transition from extinction to persistence. The global stability of the coexistence equilibrium is established using persistence theory and the Li–Muldowney geometric approach. In addition, a sensitivity analysis identifies the key biological parameters governing the long-term dynamics of the mite population. An optimal control framework is formulated to minimize the infestation while balancing the implementation costs of mechanical, biological, and chemical interventions. The necessary optimality conditions are derived using Pontryagin's maximum mrinciple. Numerical simulations support the theoretical findings, and the subsequent cost-effectiveness analysis identifies the most economically efficient control strategy. Overall, the proposed framework provides useful insights into the development of sustainable and cost-effective integrated pest management strategies for date palm protection.
- mathematical model,
- optimization problems,
- sensitivity analysis,
- Lyapunov function,
- date palm mite
Citation: Saleh Fahad Aljurbua, Moustafa El-Shahed. Cost-effective strategies for controlling date palm mite infestations: an optimal control approach[J]. AIMS Mathematics, 2026, 11(8): 26359-26396. doi: 10.3934/math.20261058

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Abstract

This paper develops and analyzes a stage-structured mathematical model for the population dynamics of date palm mite infestations, incorporating density-dependent saturation in egg production and explicitly accounting for the egg, larval, nymphal, and adult stages. A population reproduction threshold is derived to characterize the transition between the pest's extinction and persistence. Stability analysis shows that the extinction equilibrium is locally and globally asymptotically stable when the reproduction threshold remains below unity. Conversely, when this threshold exceeds one, a unique positive coexistence equilibrium emerges. The model is further shown to undergo a forward bifurcation at the critical threshold, indicating a smooth transition from extinction to persistence. The global stability of the coexistence equilibrium is established using persistence theory and the Li–Muldowney geometric approach. In addition, a sensitivity analysis identifies the key biological parameters governing the long-term dynamics of the mite population. An optimal control framework is formulated to minimize the infestation while balancing the implementation costs of mechanical, biological, and chemical interventions. The necessary optimality conditions are derived using Pontryagin's maximum mrinciple. Numerical simulations support the theoretical findings, and the subsequent cost-effectiveness analysis identifies the most economically efficient control strategy. Overall, the proposed framework provides useful insights into the development of sustainable and cost-effective integrated pest management strategies for date palm protection.

References

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