Dynamical analysis of an integrated pest management predator–prey model with weak Allee effect

References

• K. Baisad and S. Moonchai, Analysis of stability and Hopf bifurcation in a fractional Gauss-type predatorprey model with Allee effect and Holling type-III functional response, Adv. Differ. Eqs. 2018 (2018), p. 82. doi: 10.1186/s13662-018-1535-9
   Google Scholar
• H.J. Barclay, Models for pest control using predator release habitat management and pesticide release in combination, J. Appl. Ecol. 19 (1982), pp. 337–348. doi: 10.2307/2403471
   Web of Science ®Google Scholar
• W.B. Bian, X. Liu, and J. Geng, Research progress on pests controlling mechanism by natural enemies and its evaluation methods, China Plant Protect. 36 (2016), pp. 17–23.
   Google Scholar
• E. Bonotto and M. Federson, Limit sets and the poincaré bendixson theorem in impulsive semidynamical systems, J. Differ. Eqs. 244 (2008), pp. 2334–2349. doi: 10.1016/j.jde.2008.02.007
   Web of Science ®Google Scholar
• E. Bonotto and M. Federson, Poisson stability for impulsive semidynamical systems, Nonlinear Anal. 71 (2009), pp. 6148–6156. doi: 10.1016/j.na.2009.06.008
   Web of Science ®Google Scholar
• E. Bonotto and J. Grulha, Lyapunov stability of closed sets in impulsive semidynamical systems, Electron. J. Differential Equations 78 (2010), pp. 1–18.
   Google Scholar
• G. Buffoni, M. Groppi, and C. Soresina, Dynamics of predatorprey models with a strong Allee effect on the prey and predator-dependent trophic functions, Nonlinear Anal. Real World Appl. 30 (2016), pp. 143–169. doi: 10.1016/j.nonrwa.2015.12.001
   Web of Science ®Google Scholar
• L.S. Chen, Pest control and geometric theory of semi-continuous dynamical system, J. Beihua Univ. 12 (2011), pp. 1–9.
   Google Scholar
• L.S. Chen, Theory and application of semi-continuous dynamic system, J. Yulin Normal Univ. 34(2) (2012), pp. 2–10.
   Google Scholar
• X.X. Chen, S.X. Ren, and F. Zhang, Mechanism of pest management by natural enemies and their sustainable utilization, Chin. J. Appl. Entomol. 50 (2013), pp. 9–18.
   Google Scholar
• F. Courchamp, J. Berec, and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, New York, 2008.
   Google Scholar
• H.J. Guo, X.Y. Song, and L.S. Chen, Qualitative analysis of a korean pine forest model with impulsive thinning measure, Appl. Math. Comput. 234 (2014), pp. 203–213.
   Web of Science ®Google Scholar
• G.R. Jiang, Q.S. Lu, and L.N. Qian, Complex dynamics of a Holling type II preypredator system with state feedback control, Chaos Solitons Fractals 31 (2007), pp. 448–461. doi: 10.1016/j.chaos.2005.09.077
   Web of Science ®Google Scholar
• J.J. Jiao and L.S. Chen, Stabilization periodic solution of a state-dependent impulsive single population model subject to Allee effect, Math. Appl. 25 (2012), pp. 413–418.
   Google Scholar
• D.M. Johnson, A.M. Liebhold, P.C. Tobin, and O.N. Bjrnstad, Allee effects and pulsed invasion by the gypsymoth, Nature 444 (2006), pp. 361–363. doi: 10.1038/nature05242
   PubMed Web of Science ®Google Scholar
• D.Flores Jos and Gonzlez-Olivares Eduardo, Dynamics of a predatorprey model with Allee effect on prey and ratio-dependent functional response, Ecol. Complexity 18 (2014), pp. 59–66. doi: 10.1016/j.ecocom.2014.02.005
   Web of Science ®Google Scholar
• A.M. Kramer, B. Dennis, A.M. Liebhold, and J.M. Drake, The evidence for Allee effects, Popul. Ecol. 51 (2009), pp. 341–354. doi: 10.1007/s10144-009-0152-6
   Web of Science ®Google Scholar
• X. Lai, S. Liu, and R. Lin, Rich dynnamical behaviours for predator–prey model with weak Allee effect, Appl. Anal. 89 (2010), pp. 1271–1292. doi: 10.1080/00036811.2010.483557
   Web of Science ®Google Scholar
• X.M. Li and Y. Liao, Study on evaluation of hemiptera predatory insects, J. Anhui Agri. Sci. 39 (2011), pp. 10297–10298.
   Google Scholar
• L.F. Nie, Z.D. Teng, L Hu, and J.G. Peng, Qualitative analysis of a modified LeslieGower and Holling-type II predatorprey model with state dependent impulsive effects, Nonlinear Anal. Real World Appl. 11 (2010), pp. 1364–1373. doi: 10.1016/j.nonrwa.2009.02.026
   Web of Science ®Google Scholar
• Y.H. Peng and T.H. Zhang, Turing instability and pattern induced by cross-diffusion in a predator–prey system with Allee effect, Appl. Math. Comput. 275 (2016), pp. 1–12.
   Web of Science ®Google Scholar
• R. Peshin and A.K. Dhawan, Integrated Pest Management: Innovation-Development Process, Springer-Verlag, New York, 2009.
   Google Scholar
• L.N. Qian, Q.S. Lu, Q.G. Meng, and Z.S. Feng, Dynamical behaviors of a preypredator system with impulsive control, J. Math. Anal. Appl. 363 (2010), pp. 345–356. doi: 10.1016/j.jmaa.2009.08.048
   Web of Science ®Google Scholar
• F. Rao and Y. Kang, The complex dynamics of a diffusive preypredator model with an Allee effect in prey, Ecol. Complexity 28 (2016), pp. 123–144. doi: 10.1016/j.ecocom.2016.07.001
   Web of Science ®Google Scholar
• P.S. Simeonov and D.D. Bainov, Orbital stability of periodic solutions of autonomous systems with impulse effect, Int. J. Syst. Sci. 19 (1989), pp. 2561–2585. doi: 10.1080/00207728808547133
   Web of Science ®Google Scholar
• K.B. Sun, T.H. Zhang, and Y. Tian, Theoretical study and control optimization of an integrated pest management predator–prey model with power growth rate, Math. Biosci. 279 (2016), pp. 13–26. doi: 10.1016/j.mbs.2016.06.006
   PubMed Web of Science ®Google Scholar
• K.B. Sun, T.H. Zhang, and Y. Tian, Dynamics analysis and control optimization of a pest management predator–prey model with an integrated control strategy, Appl. Math. Comput. 292 (2017), pp. 253–271.
   Web of Science ®Google Scholar
• S.Y. Tang and R.A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol. 50 (2005), pp. 257–292. doi: 10.1007/s00285-004-0290-6
   PubMed Web of Science ®Google Scholar
• S.Y. Tang and L.S. Chen, Modelling and analysis of integrated pest management strategy, Discrete Contin. Dyn. Syst. Ser. B 4 (2004), pp. 759–768. doi: 10.3934/dcdsb.2004.4.759
   Web of Science ®Google Scholar
• S.Y. Tang, B. Tang, and A.L. Wang, Holling II predator–prey impulsive semi-dynamic model with complex poincaré map, Nonlinear Dyn. 81 (2015), pp. 1–11. doi: 10.1007/s11071-015-1936-1
   Web of Science ®Google Scholar
• S.Y. Tang, Y.N. Xiao, L.S. Chen, and R.A. Cheke, Integrated pest management models and their dynamical behaviour, Bull. Math. Biol. 67 (2005), pp. 115–135. doi: 10.1016/j.bulm.2004.06.005
   PubMed Web of Science ®Google Scholar
• S.Y. Tang, Y.N. Xiao, L.S. Chen, and R.A. Cheke, Integrated pest management models and their dynamical behaviour, Bull. Math. Biol. 67 (2010), pp. 115–135. doi: 10.1016/j.bulm.2004.06.005
   Google Scholar
• Y. Tian, K.B. Sun, and L.S. Chen, Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system, Int. J. Biomath. 7 (2014), pp. 1450018.
   Web of Science ®Google Scholar
• Ünal Ufuktepe, Sinan Kapçak, and Olcay Akman, Stability and invariant manifold for a predator–prey model with Allee effect, Adv Differ. Eqs. 2013 (2013), pp. 348. doi: 10.1186/1687-1847-2013-348
   Google Scholar
• United Nations, Concise Report on the World Population Situation in 2014, New York, 2014.
   Google Scholar
• J.C. Van Lenteren, Envirnomental manipulation advantageous to natural enemies of pests, in Integrated Pest Management, V. Delucchi, ed., Parasitis, Geneva, 1987, pp. 123–166.
   Google Scholar
• J.C. Van Lenteren, Integrated pest management in protected crops, in Integrated Pest Management, D. Dent, ed., Chapman Hall, London, 1995, pp. 311–320.
   Google Scholar
• J.C. Van Lenteren, Integrated pest management in protected crops, in Integrated Pest Management, D. Dent, ed., Chapman Hall, London, 1995, pp. 311–320.
   Google Scholar
• J.M. Wang, H.D. Cheng, and X.Z. Meng, Geometrical analysis and control optimization of a predator–prey model with multi state dependent impulse, Adv. Differ. Eqs. 2017 (2017), p. 252. doi: 10.1186/s13662-017-1300-5
   Google Scholar
• W.M. Wang, Y.N. Zhu, Y.L. Cai, and W.J. Wang, Dynamical complexity induced by Allee effect in a predatorprey model, Nonlinear Anal. Real World Appl. 16 (2014), pp. 103–119. doi: 10.1016/j.nonrwa.2013.09.010
   Web of Science ®Google Scholar
• Y.N. Xiao and F. Van Dan Bosch, The dynamics of an ecoepidemic model with biological control, Ecol. Model. 168 (2003), pp. 203–214. doi: 10.1016/S0304-3800(03)00197-2
   Web of Science ®Google Scholar
• J. Xu, Y. Tian, H. Guo, and X. Song, Dynamical analysis of a pest management LeslieGower model with ratio-dependent functional response, Nonlinear Dyn. 81 (2018), pp. 705–720. doi:10.1007/s11071-018-4219-9
   Google Scholar
• J. Yang and S.Y. Tang, Holling type II predatorprey model with nonlinear pulse as state-dependent feedback control, J. Comput. Appl. Math. 291 (2016), pp. 225–241. doi: 10.1016/j.cam.2015.01.017
   Web of Science ®Google Scholar
• X.W. Yu, S.L. Yuan, and T.H Zhang, Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Commun. Nonlinear Sci. Numer. Simul. 59 (2018), pp. 359–374. doi: 10.1016/j.cnsns.2017.11.028
   Web of Science ®Google Scholar
• T.H. Zhang, X. Liu, X.Z. Meng, and T.Q. Zhang, Spatio-temporal dynnamics near the steady state of a planktonic system, Comput. Math. Appl. 75 (2018), pp. 4490–4504. doi: 10.1016/j.camwa.2018.03.044
   Web of Science ®Google Scholar
• T.Q. Zhang, W.B. Ma, X.Z. Meng, and T.H. Zhang, Periodic solution of a prey-predator model with nonlinear state feedback control, Appl. Math. Comput. 266 (2015), pp. 95–107.
   Web of Science ®Google Scholar
• T.Q. Zhang, W.B. Ma, X.Z. Meng, and Tonghua Zhang, Periodic solution of a preypredator model with nonlinear state feedback control, Appl. Math. Comput. 266 (2015), pp. 95–107.
   Web of Science ®Google Scholar
• T.H. Zhang, T.Q. Zhang, and X.Z. Meng, Spatio-temporal dynamics near the steady state of a planktonic system, Comput. Math. Appl. 75 (2018), pp. 4490–4504. doi: 10.1016/j.camwa.2018.03.044
   Web of Science ®Google Scholar