You have just completed your registration at OpenAire.
Before you can login to the site, you will need to activate your account.
An e-mail will be sent to you with the proper instructions.
Important!
Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version
of the site upon release.
The main purpose is to develop novel analytical techniques and provide a comprehensive qualitative analysis of global dynamics for a state-dependent feedback control system arising from biological applications including integrated pest management. The model considered consists of a planar system of differential equations with state-dependent impulsive control. We characterize the impulsive and phase sets, using the phase portraits of the planar system and the Lambert W function to define the Poincaré map for impulsive point series defined in the phase set. The existence, local and global stability of an order-1 limit cycle and sharp sufficient conditions for the global stability of the boundary order-1 limit cycle have been provided. We further examine the flip bifurcation related to the existence of an order-2 limit cycle. We show that the existence of an order-2 limit cycle implies the existence of an order-1 limit cycle. We derive sufficient conditions under which any trajectory initiating from a phase set will be free from impulsive effects after finite state-dependent feedback control actions, and we also prove that order-k (k ≥ 3) limit cycles do not exist, providing a solution to an open problem in the integrated pest management community. We then investigate multiple attractors and their basins of attraction, as well as the interior structure of a horseshoe-like attractor. We also discuss implications of the global dynamics for integrated pest management strategy. The analytical techniques and qualitative methods developed in the present paper could be widely used in many fields concerning state-dependent feedback control.
1. Tang, SY, Cheke, RA: State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50, 257-292 (2005)
2. Tang, SY, Chen, LS: Modelling and analysis of integrated pest management strategy. Discrete Contin. Dyn. Syst., Ser. B 4, 759-768 (2004)
6. Nie, LF, Teng, ZD, Hu, L: The dynamics of a chemostat model with state dependent impulsive effects. Int. J. Bifurc. Chaos 21, 1311-1322 (2011)
7. Tang, SY, Cheke, RA: Models for integrated pest control and their biological implications. Math. Biosci. 215, 115-125 (2008)
8. Tang, SY, Liang, JH, Tan, YS, Cheke, RA: Threshold conditions for interated pest management models with pesticides that have residual effects. J. Math. Biol. 66, 1-35 (2013)
10. Wei, CJ, Zhang, SW, Chen, LS: Impulsive state feedback control of cheese whey fermentation for single-cell protein production. J. Appl. Math. 2013, Article ID 354095 (2013)
11. Lou, J, Lou, YJ, Wu, JH: Threshold virus dynamics with impulsive antiretroviral drug effects. J. Math. Biol. 65, 623-652 (2012)
12. Maggioloa, F, Airoldia, M, Callegaro, A, et al.: CD4 cell-guided scheduled treatment interruptions in HIV-infected patients with sustained immunologic response to HAART. AIDS 23, 799-807 (2009)
13. Miron, RE, Smith, RJ: Modelling imperfect adherence to HIV induction therapy. BMC Infect. Dis. 10, 6 (2010)
14. Smith RJ: Adherence to antiretroviral HIV drugs: how many doses can you miss before resistance emerges? Proc. R. Soc. Lond. B, Biol. Sci. 273, 617-624 (2006)
15. Smith RJ, Schwartz, EJ: Predicting the potential impact of a cytotoxic T -lymphocyte HIV vaccine: how often should you vaccinate and how strong should the vaccine be? Math. Biosci. 212, 180-187 (2008)
16. Smith RJ, Wahl, LM: 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 (2004)
17. Yang, YP, Xiao, YN: Threshold dynamics for compartmental epidemic models with impulses. Nonlinear Anal., Real World Appl. 13, 224-234 (2012)
18. Cappuccio, A, Castiglione, F, Piccoli, B: Determination of the optimal therapeutic protocols in cancer immunotherapy. Math. Biosci. 209, 1-13 (2007)
19. Huang, MZ, Li, JX, Song, XY, Guo, HJ: Modeling impulsive injections of insulin: towards artificial pancreas. SIAM J. Appl. Math. 72, 1524-1548 (2012)
20. Panetta, JC, Adam, J: A mathematical model of cycle-specific chemotherapy. Math. Comput. Model. 22, 67-82 (1995)
21. Tang, SY, Xiao, YN: One-compartment model with Michaelis-Menten elimination kinetics and therapeutic window: an analytical approach. J. Pharmacokinet. Biopharm. 34, 807-827 (2007)
22. Tolic, IM, Mosekilde, E, Sturis, J: Modeling the insulin-glucose feedback system: the significance of pulsatile insulin secretion. J. Theor. Biol. 207, 361-375 (2000)
25. d'Onofrio, A: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci. 179, 57-72 (2002)
26. Lu, ZH, Chi, XB, Chen, LS: The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math. Comput. Model. 36, 1039-1057 (2002)
27. Fishman, S, Marcus, R: A model for spread of plant disease with periodic removals. J. Math. Biol. 21, 149-158 (1984)
28. Shulgin, B, Stone, L, Agur, Z: Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60, 1123-1148 (1998)
29. Stone, L, Shulgin, B, Agur, Z: Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math. Comput. Model. 31, 207-215 (2000)
30. Tang, SY, Xiao, YN, Cheke, RA: Dynamical analysis of plant disease models with cultural control strategies and economic thresholds. Math. Comput. Simul. 80, 894-921 (2010)
31. Tang, SY, Xiao, YN, Clancy, D: New modelling approach concerning integrated disease control and cost-effectivity. Nonlinear Anal., Theory Methods Appl. 63, 439-471 (2005)
32. Terry, AJ: Pulse vaccination strategies in a metapopulation SIR model. Math. Biosci. Eng. 7, 455-477 (2010)
33. Chacron, MJ, Pakdaman, K, Longtin, A: Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate and fire model with threshold fatigue. Neural Comput. 15, 253-278 (2003)
34. Ermentrout, GB, Kopell, N: Multiple pulse interactions and averaging in systems of coupled neural oscillators. J. Math. Biol. 29, 195-217 (1991)
36. Goel, P, Ermentrout, B: Synchrony, stability, and firing patterns in pulse-coupled oscillators. Physica D 163, 191-216 (2002)
37. Hindmarsh, JL, Rose, RM: A model of the nerve impulse using two first-order differential equations. Nature 296, 162-164 (1982)
38. Izhikevich, EM: Class 1 neural excitability, conventional synapses, weakly connected networks, and mathematical foundations of pulse-coupled models. IEEE Trans. Neural Netw. 10, 499-507 (1999)
39. Mirollo, RE, Strogatz, SH: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50, 1645-1662 (1990)
40. Nagumo, J, Arimoto, S, Yoshizawa, S: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061-2070 (1962)
41. d'Onofrio, A: On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Appl. Math. Lett. 18, 729-732 (2005)
42. Gao, SJ, Chen, LS, Teng, ZD: Impulsive vaccination of an SEIRS model with time delay and varying total population size. Bull. Math. Biol. 69, 731-745 (2007)
43. Sun, KB, Tian, Y, Chen, LS, Kasperski, A: Nonlinear modelling of a synchronized chemostat with impulsive state feedback control. Math. Comput. Model. 52, 227-240 (2010)
44. Van Lenteren, JC: Integrated pest management in protected crops. In: Integrated Pest Management. Chapman & Hall, London (1995)
45. Van Lenteren, JC, Woets, J: Biological and integrated pest control in greenhouses. Annu. Rev. Entomol. 33, 239-250 (1988)
46. Jarad, F, Abdeljawad, T, Baleanu, D: Higher order fractional variational optimal control problems with delayed arguments. Appl. Math. Comput. 218, 9234-9240 (2012)
47. Mobayen, S: Robust tracking controller for multivariable delayed systems with input saturation via composite nonlinear feedback. Nonlinear Dyn. 76, 827-838 (2014)
48. Mobayen, S: An LMI-based robust tracker for uncertain linear systems with multiple time-varying delays using optimal composite nonlinear feedback technique. Nonlinear Dyn. 80, 917-927 (2015)
49. Doha, EH, Bhrawy, AH, Baleanu, D, et al.: An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv. Differ. Equ. 2015, Article ID 15 (2015). doi:10.1186/s13662-014-0344-z
50. Bainov, DD, Simeonov, PS: Systems with Impulsive Effect: Stability, Theory and Applications. Wiley, New York (1989)
51. Benchohra, M, Henderson, J, Ntouyas, S: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)
52. Kaul, SK: On impulsive semidynamical systems. J. Math. Anal. Appl. 150, 120-128 (1990)
53. Kaul, SK: On impulsive semidynamical systems III: Lyapunov stability. In: Recent Trends in Differential Equations. World Scientific Series in Applicable Analysis, vol. 1, pp. 335-345. World Scientific, River Edge (1992)
55. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. Series in Modern Mathematics. World Scientific, Singapore (1989)
56. Melin, J: Does distribution theory contain means for extending Poincaré-Bendixson theory. J. Math. Anal. Appl. 303, 81-89 (2004)
57. Qi, JG, Fu, XL: Existence of limit cycles of impulsive differential equations with impulses as variable times. Nonlinear Anal., Theory Methods Appl. 44, 345-353 (2011)
58. Zeng, GZ, Chen, LS, Sun, LH: Existence of periodic solution of order one of planar impulsive autonomous system. J. Comput. Appl. Math. 186, 466-481 (2006)
59. Bonotto, EM: Flows of characteristic 0+ in impulsive semidynamical systems. J. Math. Anal. Appl. 332, 81-96 (2007)
60. Bonotto, EM, Federson, M: Limit sets and the Poincaré-Bendixson theorem in impulsive semidynamical systems. J. Differ. Equ. 244, 2334-2349 (2008)
61. Bonotto, EM, Federson, M: Topological conjugation and asymptotic stability in impulsive semidynamical systems. J. Math. Anal. Appl. 326, 869-881 (2007)
63. Chellaboina, VS, Bhat, SP, Haddad, WM: An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlinear Anal., Theory Methods Appl. 53, 527-550 (2003)
64. Matveev, AS, Savkin, AV: Qualitative Theory of Hybrid Dynamical Systems. Birkhäuser, Cambridge (2000)
65. Bonotto, EM: LaSalle's theorems in impulsive semidynamical systems. Nonlinear Anal., Theory Methods Appl. 71, 2291-2297 (2009)
66. Tian, Y, Sun, KB, Kasperski, A, Chen, LS: Nonlinear modelling and qualitative analysis of a real chemostat with pulse feeding. Discrete Dyn. Nat. Soc. 2010, Article ID 640594 (2010)
67. Meng, XZ, Li, ZQ: The dynamics of plant disease models with continuous and impulsive cultural control strategies. J. Theor. Biol. 266, 29-40 (2010)
68. Chen, LS: Pest control and geometric theory of semi-dynamical systems. J. Beihua Univ. Nat. Sci. 12, 1-9 (2011)
69. Jiang, GR, Lu, QS, Qian, LN: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Solitons Fractals 31, 448-461 (2007)
70. Li, YF, Xie, DL, Cui, A: Complex dynamics of a predator-prey model with impulsive state feedback control. Appl. Math. Comput. 230, 395-405 (2014)
71. Nie, LF, Peng, JG, Teng, ZD, Hu, L: Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state-dependent impulsive effects. J. Comput. Appl. Math. 224, 544-555 (2009)
72. Tian, Y, Sun, KB, Chen, LS: Modelling and qualitative analysis of a predator-prey system with state-dependent impulsive effects. Math. Comput. Simul. 82, 318-331 (2011)
73. Wei, CJ, Chen, LS: Periodic solution of prey-predator model with Beddington-DeAngelis functional response and impulsive state feedback control. J. Appl. Math. 2012, Article ID 607105 (2012)
74. Pedigo, LP, Higley, LG: A new perspective of the economic injury level concept and environmental quality. Am. Entomol. 38, 12-20 (1992)
75. Bunimovich-Mendrazitsky, S, Byrne, H, Stone, L: Mathematical model of pulsed immunotherapy for superficial bladder cancer. Bull. Math. Biol. 70, 2055-2076 (2008)
76. Bunimovich-Mendrazitsky, S, Claude Gluckman, J, Chaskalovic, J: A mathematical model of combined bacillus Calmette-Guerin (BCG) and interleukin (IL)-2 immunotherapy of superficial bladder cancer. J. Theor. Biol. 277, 27-40 (2011)
77. Panetta, JC: A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. Bull. Math. Biol. 58, 425-447 (1996)
78. Wei, HC: A numerical study of a mathematical model of pulsed immunotherapy for superficial bladder cancer. Jpn. J. Ind. Appl. Math. 30, 441-452 (2013)
79. Wei, HC, Hwang, SF, Lin, JT, Chen, TJ: The role of initial tumor biomass size in a mathematical model of periodically pulsed chemotherapy. Comput. Math. Appl. 61, 3117-3127 (2011)
80. Wei, HC, Lin, JT: Periodically pulsed immunotherapy in a mathematical model of tumor-immune interaction. Int. J. Bifurc. Chaos 23, 1-13 (2013)
81. Staccato Study Group, Swiss HIV Cohort Study: CD4 guided scheduled treatment interruption compared to continuous therapy: results of the staccato trial. Lancet 368, 459-465 (2006)
83. Mailleret, L, Lemesle, V: A note on semi-discrete modelling in the life sciences. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 367, 4779-4799 (2009)
84. Tang, SY, Xiao, YN, Wang, N, Wu, HL: Piecewise HIV virus dynamic model with CD4+ T cell count guided therapy: I. J. Theor. Biol. 308, 123-134 (2012)
85. Fleming, GF, Meropol, NJ, Rosner, GL, et al.: A phase I trial of escalating doses of trastuzumab combined with daily subcutaneous interleukin 2: report of cancer and leukemia group B 9661. Clin. Cancer Res. 8, 3718-3727 (2002)
86. INSIGHT-ESPRIT Study Group, SILCAAT Scientific Committee: Interleukin-2 therapy in patients with HIV infection. N. Engl. J. Med. 361, 1549-1559 (2009)
87. Miron, RE, Smith, RJ: Resistance to protease inhibitors in a model of HIV-1 infection with impulsive drug effects. Bull. Math. Biol. 76, 59-97 (2014)
88. Pau, AK, Tavel, JA: Therapeutic use of interleukin-2 in HIV-infected patients. Curr. Opin. Pharmacol. 2, 433-439 (2002)
89. Choh, Y, Ignacio, M, Sabelis, MW, Janssen, A: Predator-prey role reversals, juvenile experience and adult antipredator behaviour. Sci. Rep. 2, 1-6 (2012)
90. Ives, AR, Dobson, AP: Antipredator behaviour and the population dynamics of simple predator-prey systems. Am. Nat. 130, 431-447 (1987)
92. Saito, Y: Prey kills predator: counter attack success of a spider mite against its specific phytoseiid predator. Exp. Appl. Acarol. 2, 47-62 (1986)
93. Ramao-Jiliberto, R, Frodden, E, Aránguiz-Acuña, A: Pre-encounter versus post-encounter inducible defense in predator-prey systems. Ecol. Model. 200, 99-108 (2007)
95. Kuznetsov, VA, Makalkin, IA, Taylor, MA, Perelson, AS: Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56, 295-321 (1994)
96. Mukhopadhyay, B, Bhattacharyya, R: Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity. Ecol. Model. 198, 163-173 (2006)
97. Pei, YZ, Lv, YF, Li, CG: Evolutionary consequences of harvesting for a two-zooplankton one-phytoplankton system. Appl. Math. Model. 36, 1752-1765 (2012)
98. Corless, RM, Gonnet, GH, Hare, DEG, Jeffrey, DJ, Knuth, DE: On the Lambert W function. Adv. Comput. Math. 5, 329-359 (1996)
99. de Melo, W, van Strien, S: One-Dimensional Dynamics. Springer, New York (1993)
100. de Melo, W, van Strien, S: One-dimensional dynamics: the Schwarzian derivative and beyond. Bull., New Ser., Am. Math. Soc. 18, 159-162 (1988)
101. Guckenheimer, J, Holmes, P: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983)
102. Andronov, AA, Leontovich, EA, Gordan, LL, Maier, AG: Qualitative Theory of Second-Order Dynamic Systems. Wiley, New York (1973)
103. Ciesielski, K: On semicontinuity in impulsive dynamical systems. Bull. Pol. Acad. Sci., Math. 52, 71-80 (2004)
104. Ciesielski, K: On stability in impulsive dynamical systems. Bull. Pol. Acad. Sci., Math. 52, 81-91 (2004)
105. Ciesielski, K: On time reparametrizations and isomorphisms of impulsive dynamical systems. Ann. Pol. Math. 84, 1-25 (2004)
106. Zhang, ZF, Ding, TR, Huang, WZ, Dong, ZX: Qualitative Theory of Differential Equations. Translations of Mathematical Monographs, vol. 101. Am. Math. Soc., Providence (1992)
107. Simeonov, PS, Bainov, DD: Orbital stability of the periodic solutions of autonomous systems with impulse effect. Int. J. Syst. Sci. 19, 2561-2585 (1988)
108. Iooss, G: Bifurcations of Maps and Applications. North-Holland, New York (1979)