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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Research article Special Issues

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.

Keywords:

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

Since its initiation in 1979, the Canadian Applied and Industrial Mathematics Society — Société Canadienne de Mathématiques Appliquées et Industrielles (CAIMS–SCMAI) has gained a growing presence in industrial, mathematical, scientific, and technological circles within and outside of Canada. Its members contribute to state-of-the-art research in industry, natural sciences, medicine and health, finance, physics, engineering, and more. The annual meetings are a highlight of the year. CAIMS–SCMAI is an active member society of the International Council for Industrial and Applied Mathematics, which hosts the prestigious ICIAM Congresses every four years.

Canadian Applied and Industrial Mathematics is at the forefront of scientific and technological development. We use advanced mathematics to tackle real-world problems in science and industry and develop new theories to analyse structures that arise from the modelling of real-world problems.

Applied Mathematics has evolved from traditional applications in areas such as fluids, mechanics, and physics, to modern topics such as medicine, health, biology, data science, finance, nano-tech, etc. Its growing importance in all aspects of life, health, and management increases the need for publication venues for high-level applied and industrial mathematics. Hence CAIMS–SCMAI decided to start a scientific journal called Mathematics in Science and Industry (MSI) to add value to the discussion of applied and industrial mathematics worldwide.

Submissions to MSI in all areas of applied and industrial mathematics are welcome (https://caims.ca/mathematics_in_science_and_industry/). We offer a timely and high-quality review process, and papers are published online as open access, with the publication fee being covered by CAIMS for the first five years.

MSI is honored that leading experts in industrial and applied mathematics have offered their support as editors:

Editors in Chief:

● Thomas Hillen (University of Alberta, thillen@ualberta.ca)

● Ray Spiteri (University of Saskatchewan, spiteri@cs.usask.ca)

Associate Editors:

● Lia Bronsard (McMaster University)

● Richard Craster (Imperial College of London, UK)

● David Earn (McMaster University)

● Ronald Haynes (Memorial University)

● Jane Heffernan (York University)

● Nicholas Kevlahan (McMaster University)

● Yong-Jung Kim (KAIST, Korea)

● Mark Lewis (University of Alberta)

● Kevin J. Painter (Heriot-Watt University, UK)

● Vakhtang Putkaradze (ATCO)

● Katrin Rohlf (Ryerson University)

● John Stockie (Simon Fraser University)

● Jie Sun (Huawei, Hong Kong)

● Justin Wan (University of Waterloo)

● Michael Ward (University of British Columbia)

● Tony Ware (University of Calgary)

● Brian Wetton (University of British Columbia)

The first eight papers of MSI, presented here, are published as special issue in AIMS Mathematics. They showcase a broad representation of applied mathematics that touches the interests of Canadian researchers and our many collaborators around the world. The science that we present here is not exclusively "Canadian", but we hope that through the new journal MSI, we can contribute to scientific dissemination of knowledge and add Canadian values to the scientific discussion.

The next issue of MSI is planned for the fall of 2020 and is expected to appear again as a special issue of AIMS Mathematics.

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