Optimal control of a discrete-time plant–herbivore/pest model with bistability in fluctuating environments

Sunmi Lee; Chang Yong Han; Minseok Kim; Yun Kang; Sunmi Lee; Chang Yong Han; Minseok Kim; Yun Kang

doi:10.3934/mbe.2022237

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 5: 5075-5103. doi: 10.3934/mbe.2022237

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Optimal control of a discrete-time plant–herbivore/pest model with bistability in fluctuating environments

1.
Department of Applied Mathematics, Kyung Hee University, Yongin, 17104, South Korea
2.
Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

Received: 18 August 2021 Revised: 25 February 2022 Accepted: 03 March 2022 Published: 17 March 2022

Motivated by regulating/eliminating the population of herbivorous pests, we investigate a discrete-time plant–herbivore model with two different constant control strategies (removal versus reduction), and formulate the corresponding optimal control problems when its dynamics exhibits varied types of bi-stability and fluctuating environments. We provide basic analysis and identify the critical factors to characterize the optimal controls and the corresponding plant–herbivore dynamics such as the control upper bound (the effectiveness level of the implementation of control measures) and the initial conditions of the plant and herbivore. Our results show that optimal control could be easier when the model has simple dynamics such as stable equilibrium dynamics under constant environment or the model exhibits chaotic dynamics under fluctuating environments. Due to bistability, initial conditions are important for optimal controls. Regardless of with or without fluctuating environments, initial conditions taken from the near the boundary makes optimal control easier. In general, the pest is hard to be eliminated when the control upper bound is not large enough. However, as the control upper bound is increased or the initial conditions are chosen from near the boundary of the basin of attractions, the pest can be manageable regardless of the fluctuating environments.
- discrete plant–herbivore system,
- discrete optimal control theory,
- bi-stability,
- fluctuating environments
Citation: Sunmi Lee, Chang Yong Han, Minseok Kim, Yun Kang. Optimal control of a discrete-time plant–herbivore/pest model with bistability in fluctuating environments[J]. Mathematical Biosciences and Engineering, 2022, 19(5): 5075-5103. doi: 10.3934/mbe.2022237

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Abstract

Motivated by regulating/eliminating the population of herbivorous pests, we investigate a discrete-time plant–herbivore model with two different constant control strategies (removal versus reduction), and formulate the corresponding optimal control problems when its dynamics exhibits varied types of bi-stability and fluctuating environments. We provide basic analysis and identify the critical factors to characterize the optimal controls and the corresponding plant–herbivore dynamics such as the control upper bound (the effectiveness level of the implementation of control measures) and the initial conditions of the plant and herbivore. Our results show that optimal control could be easier when the model has simple dynamics such as stable equilibrium dynamics under constant environment or the model exhibits chaotic dynamics under fluctuating environments. Due to bistability, initial conditions are important for optimal controls. Regardless of with or without fluctuating environments, initial conditions taken from the near the boundary makes optimal control easier. In general, the pest is hard to be eliminated when the control upper bound is not large enough. However, as the control upper bound is increased or the initial conditions are chosen from near the boundary of the basin of attractions, the pest can be manageable regardless of the fluctuating environments.

References

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