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Metering effects in population systems
1.  School of Mathematical & Natural Sciences, Arizona State University, 4701 W. Thunderbird Rd, Glendale, AZ, 85306 
2.  Mathematics Department, University of Texas at Arlington, Box 19408, Arlington, TX 760190408, United States 
References:
[1] 
A. S. Ackleh, B. G. Fitzpatrick, S. Scribner, J. J. Thibodeaux and N. Simonsen, Ecosystem modeling of college drinking: Parameter estimation and comparing models to data, Mathematical and Computer Modelling, 50 (2009), 481997. Google Scholar 
[2] 
R. P. Agarwal, D. Franco and D. ORegan, Singular boundary value problems for first and second order impulsive differential equations, Aequationes Mathematicae, 69 (2005), 8396. Google Scholar 
[3] 
E. Aguirre, T. Smith, J. Stancil and N. Davidenko, Differential equation models of neoadjuvant chemotherapeutic treatment strategies for stage III breast cancer, Biometrics Unit Technical Report BU1522M, Cornell University, 1999. Available from: http://mtbi.asu.edu/. Google Scholar 
[4] 
L. Almada, E. Camacho, R. Rodriguez, M. Thompson and L. Voss, Deterministic and smallworld network models of college drinking patterns,, 2006. Available from: , (). Google Scholar 
[5] 
D. Bainov and P. Simeonov, "Systems with Impulsive Effect: Stability, Theory and Applications,'' Ellis Horwood, Chichester, 1989. Google Scholar 
[6] 
D. Bainov and P. Simeonov, "Theory of Impulsive Differential Equations: Periodic Solutions and Applications,'' Longman, Harlow, 1993. Google Scholar 
[7] 
F. Brauer and C. CastilloChavez, "Mathematical Models in Population Biology and Epidemiology,'' Springer, New York, 2012. Google Scholar 
[8] 
N. F. Britton, "Essential Mathematical Biology,'' SpringerVerlag, 2003. Google Scholar 
[9] 
B. Brogliato, "Nonsmooth Mechanics,'' $2^{nd}$ edition, Springer, Berlin, 1999. Google Scholar 
[10] 
R. T. Bupp, D. S. Bernstein, V. S. Chellaboina and W. M. Haddad, Resetting virtual absorbers for vibration control, Journal of Vibration and Control, 6 (2000), 6183. Google Scholar 
[11] 
E. T. Camacho, "Mathematical Models of Retinal Dynamics," Ph.D. thesis, Center for Applied Mathematics, Cornell University, Ithaca, NY, 2003. Google Scholar 
[12] 
E. T. Camacho, The development and interaction of terrorist and fanatic groups, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 30863097. Google Scholar 
[13] 
E. C. Chang and C. Yap., Competitive online scheduling with level of service, Journal of Scheduling, 6 (2003), 251267. Google Scholar 
[14] 
N. P. Chau, Destabilising effect of period harvest on population dynamics, Ecological Modelling, 127 (2000), 19. Google Scholar 
[15] 
G. Chowell and H. Nishiura, Quantifying the transmission potential of pandemic influenza, Physics of Life Reviews, 5 (2008), 5077. doi: 10.1016/j.plrev.2007.12.001. Google Scholar 
[16] 
M. Chrobak, L. Epstein, J. Noga, J. Sgall, R. van Stee, T. Tich\'y and N. Vakhania, Preemptive scheduling in overloaded systems, Journal of Computer and System Sciences, 2380 (2003), 183197. Google Scholar 
[17] 
F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theoretical Population Biology, 72 (2007), 197213. Google Scholar 
[18] 
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous population, Journal of Mathematical Biology, 28 (1990), 365382. Google Scholar 
[19] 
A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Applied Mathematics Letters, 18 (2005), 729732. Google Scholar 
[20] 
D. B. Forger and D. Paydarfar, Starting, stopping, and resetting biological oscillators: In search of optimal perturbations, Journal of Theoretical Biology, 230 (2004), 521532. Google Scholar 
[21] 
S. Gao, L. Chen, J. J. Nieto and A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 60376045. Google Scholar 
[22] 
S. Gao, Z. Teng, J. J. Nieto and A. Torres, Analysis of an SIR epidemic model with pulse vaccination and distributed time delay, Journal of Biomedicine and Biotechnology, 2007, Article ID 64870, 10 pp. doi: 10.1155/2007/64870. Google Scholar 
[23] 
B. González, E. HuertaSánchez, A. OrtizNieves, T. VázquezÁlvarez and C. KribsZaleta, Am I too fat? Bulimia as an epidemic, Journal of Mathematical Psychology, 47 (2003), 515526. doi: 10.1016/j.jmp.2003.08.002. Google Scholar 
[24] 
V. Křivan, Optimal foraging and predatorprey dynamics, Theoretical Population Biology, 49 (1996), 265290. Google Scholar 
[25] 
A. R. Ives, K. Gross and V. A. A. Jansen, Periodic mortality events in predatorprey systems, Ecology, 81 (2000), 33303340. Google Scholar 
[26] 
A. Lakmeche and O. Arino, Nonlinear mathematical model of pulsed therapy of heterogeneous tumors, Nonlinear Analysis: Real World Applications, 2 (2001), 455465. Google Scholar 
[27] 
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, "Theory of Impulsive Differential Equations," World Scientific, Singapore, 1989. Google Scholar 
[28] 
W. Li and H. Huo, Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics, Journal of Computational and Applied Mathematics, 174 (2005), 227238. Google Scholar 
[29] 
J. D. Logan and W. Wolesensky, Accounting for temperature in predator functional responses, Natural Resource Modeling, 20 (2007), 549574. Google Scholar 
[30] 
R. M. Lopez, B. R. Morin and S. K. Suslov, On logistic models with timedependent coefficients and some of their applications,, , (). Google Scholar 
[31] 
L. Lu, S. Chu, S. Yeh and C. Peng, Modeling and experimental verification of a variablestiffness isolation system using a leverage mechanism, Journal of Vibration and Control, 17 (2011), 18691885. Google Scholar 
[32] 
S. Maggi and S. Rinaldi, A secondorder impact model for forest fire regimes, Theoretical Population Biology, 70 (2006), 174182. Google Scholar 
[33] 
E. S. Meadows and T. A. Badgwell, Feedback through steadystate target optimization for nonlinear model predictive control, Journal of Vibration and Control, 4 (1998), 6174. Google Scholar 
[34] 
S. Mondie, R. Lozano and J. Collado, Resetting processmodel control for unstable systems with delay, Proceedings of the 40th IEEE Conference on Decision and Control, 3 (2001), 22472252. Google Scholar 
[35] 
J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, Proceedings of the American Mathematical Society, 125 (1997), 25992604. Google Scholar 
[36] 
J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Analysis: Real World Applications, 10 (2009), 680690. Google Scholar 
[37] 
J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competition environment, Bulletin of Mathematical Biology, 58 (1996), 425447. Google Scholar 
[38] 
J. C. Panetta, A mathematical model of drug resistant: Heterogeneous tumors, Mathematical Biosciences, 147 (1998), 4161. Google Scholar 
[39] 
T. C. Reluga, Analysis of periodic growth disturbance models, Theoretical Population Biology, 66 (2004), 151161. Google Scholar 
[40] 
M. G. Roberts and B. T. Grenfell, The population dynamics of nematode infections of ruminants: Periodic perturbations as a model for management, Mathematical Medicine and Biology, 8 (1991), 8393. Google Scholar 
[41] 
M. G. Roberts and B. T. Grenfell, The population dynamics of nematode infections of ruminants: The effect of seasonally in the freeliving stages, Mathematical Medicine and Biology, 9 (1992), 2941. Google Scholar 
[42] 
M. G. Roberts and J. A. P. Heesterbeek, A simple parasite model with complicated dynamics, Journal of Mathematical Biology, 37 (1998), 272290. Google Scholar 
[43] 
A. M. Samoilenko and N. A. Perestyuk, "Impulsive Differential Equations,'' World Scientific, Singapore, 1995. Google Scholar 
[44] 
R. Scribner, A. S. Ackleh, B. G. Fitzpatrick, G. Jacquez, J. J. Thibodeaux, R. Rommel and N. Simonsen, A systems approach to college drinking: Development of a deterministic model for testing alcohol control policies, Journal of Studies on Alcohol and Drugs, 70 (2009), 805821. Google Scholar 
[45] 
B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60 (1998), 126. Google Scholar 
[46] 
D. W. Stephens and J. R. Krebs, "Foraging Theory," Princeton University Press, Princeton, 1986. Google Scholar 
[47] 
J. S. Tsai, F. Chen, S. Guo, C. Chen and L. Shieh, A novel tracker for a class of sampleddata nonlinear systems, Journal of Vibration and Control, 17 (2011), 81101. Google Scholar 
[48] 
P. Van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 2948. Google Scholar 
[49] 
A. Winfree, "The Geometry of Biological Time," $2^{nd}$ edition, Springer, New York, 2001. Google Scholar 
[50] 
J. Yan, A. Zhao and J. J. Nieto, Existence and global attractivity of positive periodic solution of periodic singlespecies impulsive LotkaVolterra systems, Mathematical and Computer Modelling, 40 (2004), 509518. Google Scholar 
[51] 
W. Zhang and M. Fan, Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Mathematical and Computer Modelling, 39 (2004), 479493. Google Scholar 
[52] 
H. Zhang, L. S. Chen and J. J. Nieto, A delayed epidemic model with stagestructure and pulses for management strategy, Nonlinear Analysis: Real World Applications, 9 (2008), 17141726. Google Scholar 
[53] 
X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Analysis: Real World Applications, 4 (2003), 639651. Google Scholar 
show all references
References:
[1] 
A. S. Ackleh, B. G. Fitzpatrick, S. Scribner, J. J. Thibodeaux and N. Simonsen, Ecosystem modeling of college drinking: Parameter estimation and comparing models to data, Mathematical and Computer Modelling, 50 (2009), 481997. Google Scholar 
[2] 
R. P. Agarwal, D. Franco and D. ORegan, Singular boundary value problems for first and second order impulsive differential equations, Aequationes Mathematicae, 69 (2005), 8396. Google Scholar 
[3] 
E. Aguirre, T. Smith, J. Stancil and N. Davidenko, Differential equation models of neoadjuvant chemotherapeutic treatment strategies for stage III breast cancer, Biometrics Unit Technical Report BU1522M, Cornell University, 1999. Available from: http://mtbi.asu.edu/. Google Scholar 
[4] 
L. Almada, E. Camacho, R. Rodriguez, M. Thompson and L. Voss, Deterministic and smallworld network models of college drinking patterns,, 2006. Available from: , (). Google Scholar 
[5] 
D. Bainov and P. Simeonov, "Systems with Impulsive Effect: Stability, Theory and Applications,'' Ellis Horwood, Chichester, 1989. Google Scholar 
[6] 
D. Bainov and P. Simeonov, "Theory of Impulsive Differential Equations: Periodic Solutions and Applications,'' Longman, Harlow, 1993. Google Scholar 
[7] 
F. Brauer and C. CastilloChavez, "Mathematical Models in Population Biology and Epidemiology,'' Springer, New York, 2012. Google Scholar 
[8] 
N. F. Britton, "Essential Mathematical Biology,'' SpringerVerlag, 2003. Google Scholar 
[9] 
B. Brogliato, "Nonsmooth Mechanics,'' $2^{nd}$ edition, Springer, Berlin, 1999. Google Scholar 
[10] 
R. T. Bupp, D. S. Bernstein, V. S. Chellaboina and W. M. Haddad, Resetting virtual absorbers for vibration control, Journal of Vibration and Control, 6 (2000), 6183. Google Scholar 
[11] 
E. T. Camacho, "Mathematical Models of Retinal Dynamics," Ph.D. thesis, Center for Applied Mathematics, Cornell University, Ithaca, NY, 2003. Google Scholar 
[12] 
E. T. Camacho, The development and interaction of terrorist and fanatic groups, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 30863097. Google Scholar 
[13] 
E. C. Chang and C. Yap., Competitive online scheduling with level of service, Journal of Scheduling, 6 (2003), 251267. Google Scholar 
[14] 
N. P. Chau, Destabilising effect of period harvest on population dynamics, Ecological Modelling, 127 (2000), 19. Google Scholar 
[15] 
G. Chowell and H. Nishiura, Quantifying the transmission potential of pandemic influenza, Physics of Life Reviews, 5 (2008), 5077. doi: 10.1016/j.plrev.2007.12.001. Google Scholar 
[16] 
M. Chrobak, L. Epstein, J. Noga, J. Sgall, R. van Stee, T. Tich\'y and N. Vakhania, Preemptive scheduling in overloaded systems, Journal of Computer and System Sciences, 2380 (2003), 183197. Google Scholar 
[17] 
F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theoretical Population Biology, 72 (2007), 197213. Google Scholar 
[18] 
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous population, Journal of Mathematical Biology, 28 (1990), 365382. Google Scholar 
[19] 
A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Applied Mathematics Letters, 18 (2005), 729732. Google Scholar 
[20] 
D. B. Forger and D. Paydarfar, Starting, stopping, and resetting biological oscillators: In search of optimal perturbations, Journal of Theoretical Biology, 230 (2004), 521532. Google Scholar 
[21] 
S. Gao, L. Chen, J. J. Nieto and A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 60376045. Google Scholar 
[22] 
S. Gao, Z. Teng, J. J. Nieto and A. Torres, Analysis of an SIR epidemic model with pulse vaccination and distributed time delay, Journal of Biomedicine and Biotechnology, 2007, Article ID 64870, 10 pp. doi: 10.1155/2007/64870. Google Scholar 
[23] 
B. González, E. HuertaSánchez, A. OrtizNieves, T. VázquezÁlvarez and C. KribsZaleta, Am I too fat? Bulimia as an epidemic, Journal of Mathematical Psychology, 47 (2003), 515526. doi: 10.1016/j.jmp.2003.08.002. Google Scholar 
[24] 
V. Křivan, Optimal foraging and predatorprey dynamics, Theoretical Population Biology, 49 (1996), 265290. Google Scholar 
[25] 
A. R. Ives, K. Gross and V. A. A. Jansen, Periodic mortality events in predatorprey systems, Ecology, 81 (2000), 33303340. Google Scholar 
[26] 
A. Lakmeche and O. Arino, Nonlinear mathematical model of pulsed therapy of heterogeneous tumors, Nonlinear Analysis: Real World Applications, 2 (2001), 455465. Google Scholar 
[27] 
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, "Theory of Impulsive Differential Equations," World Scientific, Singapore, 1989. Google Scholar 
[28] 
W. Li and H. Huo, Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics, Journal of Computational and Applied Mathematics, 174 (2005), 227238. Google Scholar 
[29] 
J. D. Logan and W. Wolesensky, Accounting for temperature in predator functional responses, Natural Resource Modeling, 20 (2007), 549574. Google Scholar 
[30] 
R. M. Lopez, B. R. Morin and S. K. Suslov, On logistic models with timedependent coefficients and some of their applications,, , (). Google Scholar 
[31] 
L. Lu, S. Chu, S. Yeh and C. Peng, Modeling and experimental verification of a variablestiffness isolation system using a leverage mechanism, Journal of Vibration and Control, 17 (2011), 18691885. Google Scholar 
[32] 
S. Maggi and S. Rinaldi, A secondorder impact model for forest fire regimes, Theoretical Population Biology, 70 (2006), 174182. Google Scholar 
[33] 
E. S. Meadows and T. A. Badgwell, Feedback through steadystate target optimization for nonlinear model predictive control, Journal of Vibration and Control, 4 (1998), 6174. Google Scholar 
[34] 
S. Mondie, R. Lozano and J. Collado, Resetting processmodel control for unstable systems with delay, Proceedings of the 40th IEEE Conference on Decision and Control, 3 (2001), 22472252. Google Scholar 
[35] 
J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, Proceedings of the American Mathematical Society, 125 (1997), 25992604. Google Scholar 
[36] 
J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Analysis: Real World Applications, 10 (2009), 680690. Google Scholar 
[37] 
J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competition environment, Bulletin of Mathematical Biology, 58 (1996), 425447. Google Scholar 
[38] 
J. C. Panetta, A mathematical model of drug resistant: Heterogeneous tumors, Mathematical Biosciences, 147 (1998), 4161. Google Scholar 
[39] 
T. C. Reluga, Analysis of periodic growth disturbance models, Theoretical Population Biology, 66 (2004), 151161. Google Scholar 
[40] 
M. G. Roberts and B. T. Grenfell, The population dynamics of nematode infections of ruminants: Periodic perturbations as a model for management, Mathematical Medicine and Biology, 8 (1991), 8393. Google Scholar 
[41] 
M. G. Roberts and B. T. Grenfell, The population dynamics of nematode infections of ruminants: The effect of seasonally in the freeliving stages, Mathematical Medicine and Biology, 9 (1992), 2941. Google Scholar 
[42] 
M. G. Roberts and J. A. P. Heesterbeek, A simple parasite model with complicated dynamics, Journal of Mathematical Biology, 37 (1998), 272290. Google Scholar 
[43] 
A. M. Samoilenko and N. A. Perestyuk, "Impulsive Differential Equations,'' World Scientific, Singapore, 1995. Google Scholar 
[44] 
R. Scribner, A. S. Ackleh, B. G. Fitzpatrick, G. Jacquez, J. J. Thibodeaux, R. Rommel and N. Simonsen, A systems approach to college drinking: Development of a deterministic model for testing alcohol control policies, Journal of Studies on Alcohol and Drugs, 70 (2009), 805821. Google Scholar 
[45] 
B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60 (1998), 126. Google Scholar 
[46] 
D. W. Stephens and J. R. Krebs, "Foraging Theory," Princeton University Press, Princeton, 1986. Google Scholar 
[47] 
J. S. Tsai, F. Chen, S. Guo, C. Chen and L. Shieh, A novel tracker for a class of sampleddata nonlinear systems, Journal of Vibration and Control, 17 (2011), 81101. Google Scholar 
[48] 
P. Van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 2948. Google Scholar 
[49] 
A. Winfree, "The Geometry of Biological Time," $2^{nd}$ edition, Springer, New York, 2001. Google Scholar 
[50] 
J. Yan, A. Zhao and J. J. Nieto, Existence and global attractivity of positive periodic solution of periodic singlespecies impulsive LotkaVolterra systems, Mathematical and Computer Modelling, 40 (2004), 509518. Google Scholar 
[51] 
W. Zhang and M. Fan, Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Mathematical and Computer Modelling, 39 (2004), 479493. Google Scholar 
[52] 
H. Zhang, L. S. Chen and J. J. Nieto, A delayed epidemic model with stagestructure and pulses for management strategy, Nonlinear Analysis: Real World Applications, 9 (2008), 17141726. Google Scholar 
[53] 
X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Analysis: Real World Applications, 4 (2003), 639651. Google Scholar 
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