Global dynamical analysis of plant-disease models with nonlinear impulsive cultural control strategy

Tingting Zhao; Robert J. Smith?; Tingting Zhao; Robert J. Smith?

doi:10.3934/mbe.2019353

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7022-7056. doi: 10.3934/mbe.2019353

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Global dynamical analysis of plant-disease models with nonlinear impulsive cultural control strategy

Tingting Zhao ¹,
Robert J. Smith? ^{2
,
,}

1.
School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, P.R. China
2.
Department of Mathematics and Faculty of Medicine, The University of Ottawa, 150 Louis-Pasteur Pvt, Ottawa, ON, K1N 6N5, Canada

Received: 07 May 2019 Accepted: 17 July 2019 Published: 01 August 2019

To eradicate plant diseases and maintain the number of infected plants below an economic threshold, two impulsive plant-disease models with periodic and state-dependent nonlinear cultural control are established. We focus on saturated nonlinear roguing (identifying and removing infected plants), with three situations for healthy plants: constant replanting, proportional replanting and proportional incidental removal. The global dynamics of the model with periodic impulsive effects are investigated. We establish conditions for the existence and stability of the disease-free periodic solution, the existence of a positive periodic solution and permanence. Latin Hypercube Sampling is used to perform a sensitivity analysis on the threshold of disease extinction to determine the significance of each parameter. Disease extinction will follow from increasing the harvesting rate, increasing the intervention period or decreasing the replanting number. The global behavior of the model with an economic threshold is established, including the existence and global stability of periodic solutions. The results imply that the control methods have an important effect on disease development, and the density-dependent parameter can decelerate extinction and accelerate the growth of healthy plants. These findings suggest that we can successfully eradicate the disease or maintain the infections below a certain level under suitable control measures. The analytic methods developed here provide a general framework for exploring plant-disease models with nonlinear impulsive control strategies.
- plant diseases,
- economic threshold,
- integrated disease management,
- fold bifurcation,
- impulsive differential equations,
- persistence
Citation: Tingting Zhao, Robert J. Smith?. Global dynamical analysis of plant-disease models with nonlinear impulsive cultural control strategy[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7022-7056. doi: 10.3934/mbe.2019353

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Abstract

To eradicate plant diseases and maintain the number of infected plants below an economic threshold, two impulsive plant-disease models with periodic and state-dependent nonlinear cultural control are established. We focus on saturated nonlinear roguing (identifying and removing infected plants), with three situations for healthy plants: constant replanting, proportional replanting and proportional incidental removal. The global dynamics of the model with periodic impulsive effects are investigated. We establish conditions for the existence and stability of the disease-free periodic solution, the existence of a positive periodic solution and permanence. Latin Hypercube Sampling is used to perform a sensitivity analysis on the threshold of disease extinction to determine the significance of each parameter. Disease extinction will follow from increasing the harvesting rate, increasing the intervention period or decreasing the replanting number. The global behavior of the model with an economic threshold is established, including the existence and global stability of periodic solutions. The results imply that the control methods have an important effect on disease development, and the density-dependent parameter can decelerate extinction and accelerate the growth of healthy plants. These findings suggest that we can successfully eradicate the disease or maintain the infections below a certain level under suitable control measures. The analytic methods developed here provide a general framework for exploring plant-disease models with nonlinear impulsive control strategies.

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