American Institute of Mathematical Sciences

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

Metering effects in population systems

Erika T. Camacho 1, , Christopher M. Kribs-Zaleta 2, and  Stephen Wirkus 1,

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.
Keywords: bulimia., Metered model, drug dosing, dynamical systems, discontinuities.
Mathematics Subject Classification: Primary: 00A71, 37N25; Secondary: 92D2.
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.   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), 83.   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 BU-1522-M, (1999).   Google Scholar

[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: , ().   Google Scholar

[5]

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

[6]

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

[7]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,'', Springer, (2012).   Google Scholar

[8]

N. F. Britton, "Essential Mathematical Biology,'', Springer-Verlag, (2003).   Google Scholar

[9]

B. Brogliato, "Nonsmooth Mechanics,'', $2^{nd}$ edition, (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), 61.   Google Scholar

[11]

E. T. Camacho, "Mathematical Models of Retinal Dynamics,", Ph.D. thesis, (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), 3086.   Google Scholar

[13]

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

[14]

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

[15]

G. Chowell and H. Nishiura, Quantifying the transmission potential of pandemic influenza,, Physics of Life Reviews, 5 (2008), 50.  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), 183.   Google Scholar

[17]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models,, Theoretical Population Biology, 72 (2007), 197.   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), 365.   Google Scholar

[19]

A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission,, Applied Mathematics Letters, 18 (2005), 729.   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), 521.   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), 6037.   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 (6487).  doi: 10.1155/2007/64870.  Google Scholar

[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.  doi: 10.1016/j.jmp.2003.08.002.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, "Theory of Impulsive Differential Equations,", World Scientific, (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), 227.   Google Scholar

[29]

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

[30]

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

[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.   Google Scholar

[32]

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

[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.   Google Scholar

[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.   Google Scholar

[35]

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

[36]

J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations,, Nonlinear Analysis: Real World Applications, 10 (2009), 680.   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), 425.   Google Scholar

[38]

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

[39]

T. C. Reluga, Analysis of periodic growth disturbance models,, Theoretical Population Biology, 66 (2004), 151.   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), 83.   Google Scholar

[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.   Google Scholar

[42]

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

[43]

A. M. Samoilenko and N. A. Perestyuk, "Impulsive Differential Equations,'', World Scientific, (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), 805.   Google Scholar

[45]

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

[46]

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

[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.   Google Scholar

[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.   Google Scholar

[49]

A. Winfree, "The Geometry of Biological Time," $2^{nd}$ edition,, Springer, (2001).   Google Scholar

[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.   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), 479.   Google Scholar

[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.   Google Scholar

[53]

X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population,, Nonlinear Analysis: Real World Applications, 4 (2003), 639.   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), 481.   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), 83.   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 BU-1522-M, (1999).   Google Scholar

[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: , ().   Google Scholar

[5]

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

[6]

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

[7]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,'', Springer, (2012).   Google Scholar

[8]

N. F. Britton, "Essential Mathematical Biology,'', Springer-Verlag, (2003).   Google Scholar

[9]

B. Brogliato, "Nonsmooth Mechanics,'', $2^{nd}$ edition, (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), 61.   Google Scholar

[11]

E. T. Camacho, "Mathematical Models of Retinal Dynamics,", Ph.D. thesis, (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), 3086.   Google Scholar

[13]

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

[14]

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

[15]

G. Chowell and H. Nishiura, Quantifying the transmission potential of pandemic influenza,, Physics of Life Reviews, 5 (2008), 50.  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), 183.   Google Scholar

[17]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models,, Theoretical Population Biology, 72 (2007), 197.   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), 365.   Google Scholar

[19]

A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission,, Applied Mathematics Letters, 18 (2005), 729.   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), 521.   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), 6037.   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 (6487).  doi: 10.1155/2007/64870.  Google Scholar

[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.  doi: 10.1016/j.jmp.2003.08.002.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, "Theory of Impulsive Differential Equations,", World Scientific, (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), 227.   Google Scholar

[29]

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

[30]

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

[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.   Google Scholar

[32]

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

[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.   Google Scholar

[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.   Google Scholar

[35]

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

[36]

J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations,, Nonlinear Analysis: Real World Applications, 10 (2009), 680.   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), 425.   Google Scholar

[38]

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

[39]

T. C. Reluga, Analysis of periodic growth disturbance models,, Theoretical Population Biology, 66 (2004), 151.   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), 83.   Google Scholar

[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.   Google Scholar

[42]

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

[43]

A. M. Samoilenko and N. A. Perestyuk, "Impulsive Differential Equations,'', World Scientific, (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), 805.   Google Scholar

[45]

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

[46]

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

[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.   Google Scholar

[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.   Google Scholar

[49]

A. Winfree, "The Geometry of Biological Time," $2^{nd}$ edition,, Springer, (2001).   Google Scholar

[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.   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), 479.   Google Scholar

[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.   Google Scholar

[53]

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

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