2013, 10(5&6): 1365-1379. doi: 10.3934/mbe.2013.10.1365

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 76019-0408, United States

Received  August 2012 Revised  December 2012 Published  August 2013

This study compares the effects of two types of metering (periodic resetting and periodic increments) on one variable in a dynamical system, relative to the behavior of the corresponding system with an equivalent level of constant recruitment (influx). While the level of the target population in the constant-influx system generally remains between the local extrema of the same population in the metered model, the same is not always true for other state variables in the system. These effects are illustrated by applications to models for chemotherapy dosing and for eating disorders in a school setting.
Citation: Erika T. Camacho, Christopher M. Kribs-Zaleta, Stephen Wirkus. Metering effects in population systems. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1365-1379. doi: 10.3934/mbe.2013.10.1365
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), 481-997.

[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), 83-96.

[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 BU-1522-M, Cornell University, 1999. Available from: http://mtbi.asu.edu/.

[4]

L. Almada, E. Camacho, R. Rodriguez, M. Thompson and L. Voss, Deterministic and small-world network models of college drinking patterns,, 2006. Available from: , (). 

[5]

D. Bainov and P. Simeonov, "Systems with Impulsive Effect: Stability, Theory and Applications,'' Ellis Horwood, Chichester, 1989.

[6]

D. Bainov and P. Simeonov, "Theory of Impulsive Differential Equations: Periodic Solutions and Applications,'' Longman, Harlow, 1993.

[7]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,'' Springer, New York, 2012.

[8]

N. F. Britton, "Essential Mathematical Biology,'' Springer-Verlag, 2003.

[9]

B. Brogliato, "Nonsmooth Mechanics,'' $2^{nd}$ edition, Springer, Berlin, 1999.

[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), 61-83.

[11]

E. T. Camacho, "Mathematical Models of Retinal Dynamics," Ph.D. thesis, Center for Applied Mathematics, Cornell University, Ithaca, NY, 2003.

[12]

E. T. Camacho, The development and interaction of terrorist and fanatic groups, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3086-3097.

[13]

E. C. Chang and C. Yap., Competitive online scheduling with level of service, Journal of Scheduling, 6 (2003), 251-267.

[14]

N. P. Chau, Destabilising effect of period harvest on population dynamics, Ecological Modelling, 127 (2000), 1-9.

[15]

G. Chowell and H. Nishiura, Quantifying the transmission potential of pandemic influenza, Physics of Life Reviews, 5 (2008), 50-77. doi: 10.1016/j.plrev.2007.12.001.

[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), 183-197.

[17]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theoretical Population Biology, 72 (2007), 197-213.

[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), 365-382.

[19]

A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Applied Mathematics Letters, 18 (2005), 729-732.

[20]

D. B. Forger and D. Paydarfar, Starting, stopping, and resetting biological oscillators: In search of optimal perturbations, Journal of Theoretical Biology, 230 (2004), 521-532.

[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), 6037-6045.

[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.

[23]

B. González, E. Huerta-Sánchez, A. Ortiz-Nieves, T. Vázquez-Álvarez and C. Kribs-Zaleta, Am I too fat? Bulimia as an epidemic, Journal of Mathematical Psychology, 47 (2003), 515-526. doi: 10.1016/j.jmp.2003.08.002.

[24]

V. Křivan, Optimal foraging and predator-prey dynamics, Theoretical Population Biology, 49 (1996), 265-290.

[25]

A. R. Ives, K. Gross and V. A. A. Jansen, Periodic mortality events in predator-prey systems, Ecology, 81 (2000), 3330-3340.

[26]

A. Lakmeche and O. Arino, Nonlinear mathematical model of pulsed therapy of heterogeneous tumors, Nonlinear Analysis: Real World Applications, 2 (2001), 455-465.

[27]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, "Theory of Impulsive Differential Equations," World Scientific, Singapore, 1989.

[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), 227-238.

[29]

J. D. Logan and W. Wolesensky, Accounting for temperature in predator functional responses, Natural Resource Modeling, 20 (2007), 549-574.

[30]

R. M. Lopez, B. R. Morin and S. K. Suslov, On logistic models with time-dependent coefficients and some of their applications,, , (). 

[31]

L. Lu, S. Chu, S. Yeh and C. Peng, Modeling and experimental verification of a variable-stiffness isolation system using a leverage mechanism, Journal of Vibration and Control, 17 (2011), 1869-1885.

[32]

S. Maggi and S. Rinaldi, A second-order impact model for forest fire regimes, Theoretical Population Biology, 70 (2006), 174-182.

[33]

E. S. Meadows and T. A. Badgwell, Feedback through steady-state target optimization for nonlinear model predictive control, Journal of Vibration and Control, 4 (1998), 61-74.

[34]

S. Mondie, R. Lozano and J. Collado, Resetting process-model control for unstable systems with delay, Proceedings of the 40th IEEE Conference on Decision and Control, 3 (2001), 2247-2252.

[35]

J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, Proceedings of the American Mathematical Society, 125 (1997), 2599-2604.

[36]

J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Analysis: Real World Applications, 10 (2009), 680-690.

[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), 425-447.

[38]

J. C. Panetta, A mathematical model of drug resistant: Heterogeneous tumors, Mathematical Biosciences, 147 (1998), 41-61.

[39]

T. C. Reluga, Analysis of periodic growth disturbance models, Theoretical Population Biology, 66 (2004), 151-161.

[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), 83-93.

[41]

M. G. Roberts and B. T. Grenfell, The population dynamics of nematode infections of ruminants: The effect of seasonally in the free-living stages, Mathematical Medicine and Biology, 9 (1992), 29-41.

[42]

M. G. Roberts and J. A. P. Heesterbeek, A simple parasite model with complicated dynamics, Journal of Mathematical Biology, 37 (1998), 272-290.

[43]

A. M. Samoilenko and N. A. Perestyuk, "Impulsive Differential Equations,'' World Scientific, Singapore, 1995.

[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), 805-821.

[45]

B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60 (1998), 1-26.

[46]

D. W. Stephens and J. R. Krebs, "Foraging Theory," Princeton University Press, Princeton, 1986.

[47]

J. S. Tsai, F. Chen, S. Guo, C. Chen and L. Shieh, A novel tracker for a class of sampled-data nonlinear systems, Journal of Vibration and Control, 17 (2011), 81-101.

[48]

P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.

[49]

A. Winfree, "The Geometry of Biological Time," $2^{nd}$ edition, Springer, New York, 2001.

[50]

J. Yan, A. Zhao and J. J. Nieto, Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems, Mathematical and Computer Modelling, 40 (2004), 509-518.

[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), 479-493.

[52]

H. Zhang, L. S. Chen and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for management strategy, Nonlinear Analysis: Real World Applications, 9 (2008), 1714-1726.

[53]

X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Analysis: Real World Applications, 4 (2003), 639-651.

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), 481-997.

[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), 83-96.

[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 BU-1522-M, Cornell University, 1999. Available from: http://mtbi.asu.edu/.

[4]

L. Almada, E. Camacho, R. Rodriguez, M. Thompson and L. Voss, Deterministic and small-world network models of college drinking patterns,, 2006. Available from: , (). 

[5]

D. Bainov and P. Simeonov, "Systems with Impulsive Effect: Stability, Theory and Applications,'' Ellis Horwood, Chichester, 1989.

[6]

D. Bainov and P. Simeonov, "Theory of Impulsive Differential Equations: Periodic Solutions and Applications,'' Longman, Harlow, 1993.

[7]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,'' Springer, New York, 2012.

[8]

N. F. Britton, "Essential Mathematical Biology,'' Springer-Verlag, 2003.

[9]

B. Brogliato, "Nonsmooth Mechanics,'' $2^{nd}$ edition, Springer, Berlin, 1999.

[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), 61-83.

[11]

E. T. Camacho, "Mathematical Models of Retinal Dynamics," Ph.D. thesis, Center for Applied Mathematics, Cornell University, Ithaca, NY, 2003.

[12]

E. T. Camacho, The development and interaction of terrorist and fanatic groups, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3086-3097.

[13]

E. C. Chang and C. Yap., Competitive online scheduling with level of service, Journal of Scheduling, 6 (2003), 251-267.

[14]

N. P. Chau, Destabilising effect of period harvest on population dynamics, Ecological Modelling, 127 (2000), 1-9.

[15]

G. Chowell and H. Nishiura, Quantifying the transmission potential of pandemic influenza, Physics of Life Reviews, 5 (2008), 50-77. doi: 10.1016/j.plrev.2007.12.001.

[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), 183-197.

[17]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theoretical Population Biology, 72 (2007), 197-213.

[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), 365-382.

[19]

A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Applied Mathematics Letters, 18 (2005), 729-732.

[20]

D. B. Forger and D. Paydarfar, Starting, stopping, and resetting biological oscillators: In search of optimal perturbations, Journal of Theoretical Biology, 230 (2004), 521-532.

[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), 6037-6045.

[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.

[23]

B. González, E. Huerta-Sánchez, A. Ortiz-Nieves, T. Vázquez-Álvarez and C. Kribs-Zaleta, Am I too fat? Bulimia as an epidemic, Journal of Mathematical Psychology, 47 (2003), 515-526. doi: 10.1016/j.jmp.2003.08.002.

[24]

V. Křivan, Optimal foraging and predator-prey dynamics, Theoretical Population Biology, 49 (1996), 265-290.

[25]

A. R. Ives, K. Gross and V. A. A. Jansen, Periodic mortality events in predator-prey systems, Ecology, 81 (2000), 3330-3340.

[26]

A. Lakmeche and O. Arino, Nonlinear mathematical model of pulsed therapy of heterogeneous tumors, Nonlinear Analysis: Real World Applications, 2 (2001), 455-465.

[27]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, "Theory of Impulsive Differential Equations," World Scientific, Singapore, 1989.

[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), 227-238.

[29]

J. D. Logan and W. Wolesensky, Accounting for temperature in predator functional responses, Natural Resource Modeling, 20 (2007), 549-574.

[30]

R. M. Lopez, B. R. Morin and S. K. Suslov, On logistic models with time-dependent coefficients and some of their applications,, , (). 

[31]

L. Lu, S. Chu, S. Yeh and C. Peng, Modeling and experimental verification of a variable-stiffness isolation system using a leverage mechanism, Journal of Vibration and Control, 17 (2011), 1869-1885.

[32]

S. Maggi and S. Rinaldi, A second-order impact model for forest fire regimes, Theoretical Population Biology, 70 (2006), 174-182.

[33]

E. S. Meadows and T. A. Badgwell, Feedback through steady-state target optimization for nonlinear model predictive control, Journal of Vibration and Control, 4 (1998), 61-74.

[34]

S. Mondie, R. Lozano and J. Collado, Resetting process-model control for unstable systems with delay, Proceedings of the 40th IEEE Conference on Decision and Control, 3 (2001), 2247-2252.

[35]

J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, Proceedings of the American Mathematical Society, 125 (1997), 2599-2604.

[36]

J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Analysis: Real World Applications, 10 (2009), 680-690.

[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), 425-447.

[38]

J. C. Panetta, A mathematical model of drug resistant: Heterogeneous tumors, Mathematical Biosciences, 147 (1998), 41-61.

[39]

T. C. Reluga, Analysis of periodic growth disturbance models, Theoretical Population Biology, 66 (2004), 151-161.

[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), 83-93.

[41]

M. G. Roberts and B. T. Grenfell, The population dynamics of nematode infections of ruminants: The effect of seasonally in the free-living stages, Mathematical Medicine and Biology, 9 (1992), 29-41.

[42]

M. G. Roberts and J. A. P. Heesterbeek, A simple parasite model with complicated dynamics, Journal of Mathematical Biology, 37 (1998), 272-290.

[43]

A. M. Samoilenko and N. A. Perestyuk, "Impulsive Differential Equations,'' World Scientific, Singapore, 1995.

[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), 805-821.

[45]

B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60 (1998), 1-26.

[46]

D. W. Stephens and J. R. Krebs, "Foraging Theory," Princeton University Press, Princeton, 1986.

[47]

J. S. Tsai, F. Chen, S. Guo, C. Chen and L. Shieh, A novel tracker for a class of sampled-data nonlinear systems, Journal of Vibration and Control, 17 (2011), 81-101.

[48]

P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.

[49]

A. Winfree, "The Geometry of Biological Time," $2^{nd}$ edition, Springer, New York, 2001.

[50]

J. Yan, A. Zhao and J. J. Nieto, Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems, Mathematical and Computer Modelling, 40 (2004), 509-518.

[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), 479-493.

[52]

H. Zhang, L. S. Chen and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for management strategy, Nonlinear Analysis: Real World Applications, 9 (2008), 1714-1726.

[53]

X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Analysis: Real World Applications, 4 (2003), 639-651.

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