Journal of Mathematical Biology

, Volume 78, Issue 5, pp 1389–1424 | Cite as

Using non-smooth models to determine thresholds for microbial pest management

  • Aili Wang
  • Yanni XiaoEmail author
  • Robert SmithEmail author


Releasing infectious pests could successfully control and eventually maintain the number of pests below a threshold level. To address this from a mathematical point of view, two non-smooth microbial pest-management models with threshold policy are proposed and investigated in the present paper. First, we establish an impulsive model with state-dependent control to describe the cultural control strategies, including releasing infectious pests and spraying chemical pesticide. We examine the existence and stability of an order-1 periodic solution, the existence of order-k periodic solutions and chaotic phenomena of this model by analyzing the properties of the Poincaré map. Secondly, we establish and analyze a Filippov model. By examining the sliding dynamics, we investigate the global stability of both the pseudo-equilibria and regular equilibria. The findings suggest that we can choose appropriate threshold levels and control intensity to maintain the number of pests at or below the economic threshold. The modelling and control outcomes presented here extend the results for the system with impulsive interventions at fixed moments.


Microbial pest management Threshold policy Impulsive differential equations Filippov system Global dynamics 

Mathematics Subject Classification

34A37 34A38 



The authors are grateful to two anonymous reviewers, whose comments greatly improved the manuscript. AW was supported by the National Natural Science Foundation of China (NSFC, 11801013), the Shaanxi Education Department (16JK1047) and funding from the Baoji University of Arts and Sciences (ZK16048). YX was supported by the National Natural Science Foundation of China (NSFC, 11571273 and 11631012) and Fundamental Research Funds for the Central Universities (GK 08143042). RS? was supported by an NSERC Discovery Grant. For citation purposes, note that the question mark in “Smith?” is part of his name.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceBaoji University of Arts and SciencesBaojiPeople’s Republic of China
  2. 2.Department of Applied MathematicsXi’an Jiaotong UniversityXi’anPeople’s Republic of China
  3. 3.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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