The Beverton–Hold model on isolated time scales

Martin Bohner; Jaqueline Mesquita; Sabrina Streipert; Martin Bohner; Jaqueline Mesquita; Sabrina Streipert

doi:10.3934/mbe.2022544

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 11: 11693-11716. doi: 10.3934/mbe.2022544

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The Beverton–Hold model on isolated time scales

1.
Missouri S & T, Department of Mathematics and Statistics, Rolla, MO 65409-0020, USA
2.
Universidade de Brasília, Departmento de Matemática, 70910-900 Brasília, DF, Brazil
3.
McMaster University, Department of Mathematics & Statistics, Hamilton, Ont., Canada

Received: 12 July 2022 Revised: 31 July 2022 Accepted: 10 August 2022 Published: 15 August 2022

In this work, we formulate the Beverton–Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton–Holt equation. The first main theorem provides conditions for the existence of a unique $ \omega $ -periodic solution that is globally asymptotically stable, which addresses the first Cushing–Henson conjecture on isolated time scales. The second main theorem concerns the generalization of the second Cushing–Henson conjecture. It investigates the effects of periodicity by deriving an upper bound for the average of the unique periodic solution. The obtained upper bound reveals a dependence on the underlying time structure, not apparent in the classical case. This work also extends existing results for the Beverton–Holt model in the discrete and quantum cases, and it complements existing conclusions on periodic time scales. This work can furthermore guide other applications of the recently introduced periodicity on isolated time scales.
- Beverton–Holt equation,
- Cushing–Henson conjecture,
- isolated time scale,
- periodicity concept,
- periodic solutions
Citation: Martin Bohner, Jaqueline Mesquita, Sabrina Streipert. The Beverton–Hold model on isolated time scales[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11693-11716. doi: 10.3934/mbe.2022544

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Abstract

In this work, we formulate the Beverton–Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton–Holt equation. The first main theorem provides conditions for the existence of a unique $ \omega $ -periodic solution that is globally asymptotically stable, which addresses the first Cushing–Henson conjecture on isolated time scales. The second main theorem concerns the generalization of the second Cushing–Henson conjecture. It investigates the effects of periodicity by deriving an upper bound for the average of the unique periodic solution. The obtained upper bound reveals a dependence on the underlying time structure, not apparent in the classical case. This work also extends existing results for the Beverton–Holt model in the discrete and quantum cases, and it complements existing conclusions on periodic time scales. This work can furthermore guide other applications of the recently introduced periodicity on isolated time scales.

References

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