Using non-smooth models to determine thresholds for microbial pest management. (English) Zbl 1414.34039
Summary: 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.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34A37 | Ordinary differential equations with impulses |
| 34A36 | Discontinuous ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
Keywords:
microbial pest management; threshold policy; impulsive differential equations; Filippov system; global dynamicsReferences:
| [1] | Bainov D, Simeonov P (1993) Impulsive differential equations: periodic solutions and applications. CRC Press, Boca Raton · Zbl 0815.34001 |
| [2] | Bhattacharyya S, Bhattacharya D (2006) Pest control through viral disease: mathematical modeling and analysis. J Theor Biol 238:177-197 · Zbl 1445.92310 · doi:10.1016/j.jtbi.2005.05.019 |
| [3] | 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 · Zbl 1443.92149 · doi:10.1016/j.apm.2014.09.038 |
| [4] | 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 · Zbl 1353.34053 |
| [5] | Devaney R (2003) An introduction to chaotic dynamical systems. Westview Press, Boulder · Zbl 1025.37001 |
| [6] | Gao S, He Y, Chen L (2013) An epidemic model with pulses for pest management. Appl Math Comput 219:4308-4321 · Zbl 1402.92389 |
| [7] | 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 · doi:10.1007/BF02803565 |
| [8] | Huang M, Li J, Song X, Guo H (2012) Modeling impulsive injections of insulin: towards artificial pancreas. SIAM J Appl Math 72:1524-1548 · Zbl 1325.92045 · doi:10.1137/110860306 |
| [9] | 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 · doi:10.3354/ame027111 |
| [10] | 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 · Zbl 1167.34340 · doi:10.1016/j.apm.2007.11.021 |
| [11] | 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 · Zbl 1337.92180 · doi:10.1016/j.jtbi.2012.06.032 |
| [12] | 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 · Zbl 1344.92169 |
| [13] | Lakshmikantham V, Bainov D, Simeonov P (1989) Theory of impulsive differential equations. World Scientific, Singapore · Zbl 0719.34002 · doi:10.1142/0906 |
| [14] | Lenteren, JV; Dent, D. (ed.), Integrated pest management in protected crops, 311-320 (1995), London |
| [15] | Lenteren JV, Woets J (1988) Biological and integrated pest control in greenhouses. Annu Rev Entomol 33:239-250 · doi:10.1146/annurev.en.33.010188.001323 |
| [16] | Li X, Wu J (2016) Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64:63-69 · Zbl 1329.93108 · doi:10.1016/j.automatica.2015.10.002 |
| [17] | Li X, Song S, Wu J (2018) Impulsive control of unstable neural networks with unbounded time-varying delays. Sci China Inf Sci 61:012203 · doi:10.1007/s11432-017-9097-1 |
| [18] | 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 · Zbl 1309.92080 · doi:10.1016/j.mbs.2013.07.008 |
| [19] | 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 · Zbl 1365.92141 · doi:10.1016/j.jmaa.2014.09.048 |
| [20] | 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 · Zbl 1317.34109 · doi:10.1142/S0218127415500686 |
| [21] | Lou J, Lou Y, Wu J (2012) Threshold virus dynamics with impulsive antiretroviral drug effects. J Math Biol 65:623-652 · Zbl 1278.34053 · doi:10.1007/s00285-011-0474-9 |
| [22] | Nie L, Teng Z, Guo B (2013) A state dependent pulse control strategy for a SIRS epidemic system. Bull Math Biol 75:1697-1715 · Zbl 1275.92093 · doi:10.1007/s11538-013-9865-y |
| [23] | 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 · doi:10.1890/10-1180.1 |
| [24] | Peng H (2005) Wasps deliver deadly virus to crop pests. Virus Res 114:80-81 · doi:10.1016/j.virusres.2005.06.002 |
| [25] | 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 · Zbl 1332.92062 · doi:10.1007/s12591-015-0240-3 |
| [26] | Simeonov P, Bainov D (1988) Orbital stability of periodic solutions of autonomous systems with impulse effect. Int J Syst Sci 19:2561-2585 · Zbl 0669.34044 · doi:10.1080/00207728808547133 |
| [27] | 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 · Zbl 1334.92239 · doi:10.1016/j.bulm.2003.12.004 |
| [28] | 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 · Zbl 1080.92067 · doi:10.1007/s00285-004-0290-6 |
| [29] | 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 · Zbl 1256.34037 · doi:10.1016/j.na.2012.08.013 |
| [30] | 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 · Zbl 1183.92060 · doi:10.1016/j.matcom.2009.10.004 |
| [31] | 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 · Zbl 1256.34038 · doi:10.1137/110847020 |
| [32] | Tang S, Pang W, Wu J, Cheke R (2015) Global dynamics of a state-dependent feedback control system. Adv Differ Equ 2015:322 · Zbl 1422.34093 · doi:10.1186/s13662-015-0661-x |
| [33] | 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 · Zbl 1336.34072 · doi:10.1142/S0218127416500395 |
| [34] | Touboul J, Brette R (2009) Spiking dynamics of bidimensional integrate-and-fire neurons. SIAM J Appl Dyn Syst 8:1462-1506 · Zbl 1204.37019 · doi:10.1137/080742762 |
| [35] | Wang W (2006) Backward bifurcation of an epidemic model with treatment. Math Biosci 201:58-71 · Zbl 1093.92054 · doi:10.1016/j.mbs.2005.12.022 |
| [36] | Wang A, Xiao Y (2014) A Filippov system describing media effects on the spread of infectious diseases. Nonlinear Anal Hybrid Syst 11:84-97 · Zbl 1323.92215 · doi:10.1016/j.nahs.2013.06.005 |
| [37] | 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 · Zbl 1188.93038 · doi:10.1016/j.nonrwa.2009.02.027 |
| [38] | Xiao Y, Xu X, Tang S (2012) Sliding mode control of outbreaks of emerging infectious diseases. Bull Math Biol 74:2403-2422 · Zbl 1312.92043 · doi:10.1007/s11538-012-9758-5 |
| [39] | 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 · doi:10.1016/j.addr.2013.04.005 |
| [40] | 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 · Zbl 1259.92061 · doi:10.3934/mbe.2013.10.445 |
| [41] | Xiao Y, Tang S, Wu J (2015) Media impact switching surface during an infectious disease outbreak. Sci Rep 5:7838 · doi:10.1038/srep07838 |
| [42] | Yang Y, Xiao Y, Wu J (2013) Pulse HIV vaccination: feasibility for virus eradication and optimal vaccination schedule. Bull Math Biol 75:725-751 · Zbl 1273.92028 · doi:10.1007/s11538-013-9831-8 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
