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

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Abstract

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.

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References

  • Bainov D, Simeonov P (1993) Impulsive differential equations: periodic solutions and applications. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Bhattacharyya S, Bhattacharya D (2006) Pest control through viral disease: mathematical modeling and analysis. J Theor Biol 238:177–197

    Article  MathSciNet  Google Scholar 

  • Cai Y, Zhao C, Wang W, Wang J (2015) Dynamics of a Leslie–Gower predator–prey model with additive allee effect. Appl Math Model 39:2092–2106

    Article  MathSciNet  Google Scholar 

  • Chong N, Dionne B, Smith R? (2016) An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza. J Math Biol 73:751–784

  • Devaney R (2003) An introduction to chaotic dynamical systems. Westview Press, Boulder

    MATH  Google Scholar 

  • Gao S, He Y, Chen L (2013) An epidemic model with pulses for pest management. Appl Math Comput 219:4308–4321

    MathSciNet  MATH  Google Scholar 

  • Gastrich M, Leigh-Bell J, Gobler C, Anderson O, Wilhelm S (2004) Viruses as potential regulators of regional brown tide blooms caused by the alga Aureococcus anophagefferens. Estuaries 27:112–119

    Article  Google Scholar 

  • Huang M, Li J, Song X, Guo H (2012) Modeling impulsive injections of insulin: towards artificial pancreas. SIAM J Appl Math 72:1524–1548

    Article  MathSciNet  MATH  Google Scholar 

  • Jacquet S, Heldal M, Iglesias-Rodriguez D, Larsen A, Wilson W, Brabak G (2002) Flow cytometric analysis of an Emiliana huxleyi bloom terminated by viral infection. Aquat Microb Ecol 27:111–124

    Article  Google Scholar 

  • Jiao J, Chen L, Cai S (2009) Impulsive control strategy of a pest management SI model with nonlinear incidence rate. Appl Math Model 33:555–563

    Article  MathSciNet  MATH  Google Scholar 

  • Kar T, Ghorai A, Jana S (2012) Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide. J Theor Biol 310:187–198

    Article  MathSciNet  MATH  Google Scholar 

  • Křivan V, Lewis M, Bentz B, Bewick S, Lenhart S, Liebhold A (2016) A dynamical model for bark beetle outbreaks. J Theor Biol 407:25–37

  • Lakshmikantham V, Bainov D, Simeonov P (1989) Theory of impulsive differential equations. World Scientific, Singapore

    Book  Google Scholar 

  • Lenteren JV (1995) Integrated pest management in protected crops. In: Dent D (ed) Integrated pest management. Chapman & Hall, London, pp 311–320

    Google Scholar 

  • Lenteren JV, Woets J (1988) Biological and integrated pest control in greenhouses. Annu Rev Entomol 33:239–250

    Article  Google Scholar 

  • Li X, Wu J (2016) Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64:63–69

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Song S, Wu J (2018) Impulsive control of unstable neural networks with unbounded time-varying delays. Sci China Inf Sci 61:012203

    Article  Google Scholar 

  • Liang J, Tang S, Nieto J, Cheke R (2013) Analytical methods for detecting pesticide switches with evolution of pesticide resistance. Math Biosci 245:249–257

    Article  MathSciNet  MATH  Google Scholar 

  • Liang J, Tang S, Cheke R, Wu J (2015) Models for determining how many natural enemies to release inoculatively in combinations of biological and chemical control with pesticide resistance. J Math Anal Appl 422:1479–1503

    Article  MathSciNet  MATH  Google Scholar 

  • Liu B, Liu W, Tao F, Cong J (2015) A dynamical analysis of a piecewise smooth pest control SI model. Int J Bifurc Chaos 25:1550068

    Article  MathSciNet  MATH  Google Scholar 

  • Lou J, Lou Y, Wu J (2012) Threshold virus dynamics with impulsive antiretroviral drug effects. J Math Biol 65:623–652

    Article  MathSciNet  MATH  Google Scholar 

  • Nie L, Teng Z, Guo B (2013) A state dependent pulse control strategy for a SIRS epidemic system. Bull Math Biol 75:1697–1715

    Article  MathSciNet  MATH  Google Scholar 

  • O’Rourke M, Jones L (2011) Analysis of landscape-scale insect pest dynamics and pesticide use: an empirical and modeling study. Ecol Appl 21:3199–3210

    Article  Google Scholar 

  • Peng H (2005) Wasps deliver deadly virus to crop pests. Virus Res 114:80–81

    Article  Google Scholar 

  • Sasmal S, Bhowmick A, Al-Khaled K, Bhattacharya S, Chattopadhyay J (2016) Interplay of functional responses and weak allee effect on pest control via viral infection or natural predator: an eco-epidemiological study. Differ Equ Dyn Syst 24:21–50

    Article  MathSciNet  MATH  Google Scholar 

  • Simeonov P, Bainov D (1988) Orbital stability of periodic solutions of autonomous systems with impulse effect. Int J Syst Sci 19:2561–2585

    Article  MathSciNet  MATH  Google Scholar 

  • Smith R, Wahl L (2004) Distinct effects of protease and reverse transcriptase inhibition in an immunological model of HIV-1 infection with impulsive drug effects. Bull Math Biol 66:1259–1283

    Article  MathSciNet  MATH  Google Scholar 

  • Tang S, Cheke R (2005) State-dependent impulsive models of integrated pest management (ipm) strategies and their dynamics consequences. J Math Biol 50:257–292

    Article  MathSciNet  MATH  Google Scholar 

  • Tang S, Liang J (2013) Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge. Nonlinear Anal Theor Methods Appl 76:165–180

    Article  MathSciNet  MATH  Google Scholar 

  • Tang S, Xiao Y, Cheke R (2010) Dynamical analysis of plant disease models with cultural control strategies and economic thresholds. Math Comput Simul 80:894–921

    Article  MathSciNet  MATH  Google Scholar 

  • Tang S, Liang J, Xiao Y, Cheke R (2012) Sliding bifurcations of Filippov two stage pest control models with economic thresholds. SIAM J Appl Math 72:1061–1080

    Article  MathSciNet  MATH  Google Scholar 

  • Tang S, Pang W, Wu J, Cheke R (2015) Global dynamics of a state-dependent feedback control system. Adv Differ Equ 2015:322

    Article  MathSciNet  MATH  Google Scholar 

  • Tang B, Xiao Y, Tang S, Cheke R (2016) A feedback control model of comprehensive therapy for treating immunogenic tumours. Int J Bifurc Chaos 26:1650039

    Article  MathSciNet  MATH  Google Scholar 

  • Touboul J, Brette R (2009) Spiking dynamics of bidimensional integrate-and-fire neurons. SIAM J Appl Dyn Syst 8:1462–1506

    Article  MathSciNet  MATH  Google Scholar 

  • Wang W (2006) Backward bifurcation of an epidemic model with treatment. Math Biosci 201:58–71

    Article  MathSciNet  MATH  Google Scholar 

  • Wang A, Xiao Y (2014) A Filippov system describing media effects on the spread of infectious diseases. Nonlinear Anal Hybrid Syst 11:84–97

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Chen L, Nieto J (2010) The dynamics of an epidemic model for pest control with impulsive effect. Nonlinear Anal Real World Appl 11:1374–1386

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao Y, Xu X, Tang S (2012) Sliding mode control of outbreaks of emerging infectious diseases. Bull Math Biol 74:2403–2422

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao Y, Miao H, Tang S, Wu H (2013a) Modeling antiretroviral drug responses for HIV-1 infected patients using differential equation models. Adv Drug Deliv Rev 65:940–953

    Article  Google Scholar 

  • Xiao Y, Zhao T, Tang S (2013b) Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Math Biosci Eng 10:445–461

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao Y, Tang S, Wu J (2015) Media impact switching surface during an infectious disease outbreak. Sci Rep 5:7838

    Article  Google Scholar 

  • Yang Y, Xiao Y, Wu J (2013) Pulse HIV vaccination: feasibility for virus eradication and optimal vaccination schedule. Bull Math Biol 75:725–751

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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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Authors and Affiliations

  1. School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, 721013, People’s Republic of China

    Aili Wang

  2. Department of Applied Mathematics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Yanni Xiao

  3. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Robert Smith

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  1. Aili Wang
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  2. Yanni Xiao
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  3. Robert Smith
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Correspondence to Yanni Xiao or Robert Smith.

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Wang, A., Xiao, Y. & Smith, R. Using non-smooth models to determine thresholds for microbial pest management. J. Math. Biol. 78, 1389–1424 (2019). https://doi.org/10.1007/s00285-018-1313-z

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  • DOI: https://doi.org/10.1007/s00285-018-1313-z

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