American Institute of Mathematical Sciences

Advanced
2015, 12(2): 311-335. doi: 10.3934/mbe.2015.12.311

Stability and optimization in structured population models on graphs

Rinaldo M. Colombo 1, and  Mauro Garavello 2,

1. 

INdAM Unit, University of Brescia, Brescia
2. 

Department of Mathematics and Applications, University of Milano-Bicocca, Via R. Cozzi, 53, 20125 Milano

Received  April 2014 Revised  July 2014 Published  December 2014

We prove existence and uniqueness of solutions, continuous dependence from the initial datum and stability with respect to the boundary condition in a class of initial--boundary value problems for systems of balance laws. The particular choice of the boundary condition allows to comprehend models with very different structures. In particular, we consider a juvenile-adult model, the problem of the optimal mating ratio and a model for the optimal management of biological resources. The stability result obtained allows to tackle various optimal management/control problems, providing sufficient conditions for the existence of optimal choices/controls.
Keywords: optimal mating ratio., management of biological resources, juvenile-adult model, Renewal equation, balance laws.
Mathematics Subject Classification: Primary: 35L50; Secondary: 92D2.
Citation: Rinaldo M. Colombo, Mauro Garavello. Stability and optimization in structured population models on graphs. Mathematical Biosciences & Engineering, 2015, 12 (2) : 311-335. doi: 10.3934/mbe.2015.12.311
References:
[1]

A. S. Ackleh and K. Deng, A nonautonomous juvenile-adult model: Well-posedness and long-time behavior via a comparison principle, SIAM J. Appl. Math., 69 (2009), 1644-1661. doi: 10.1137/080723673.  Google Scholar

[2]

A. S. Ackleh, K. Deng and X. Yang, Sensitivity analysis for a structured juvenile-adult model, Comput. Math. Appl., 64 (2012), 190-200. doi: 10.1016/j.camwa.2011.12.053.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.  Google Scholar

[4]

C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.  Google Scholar

[5]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second edition, Texts in Applied Mathematics, 40, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[6]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[7]

J. A. Carrillo, S. Cuadrado and B. Perthame, Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model, Math. Biosci., 205 (2007), 137-161. doi: 10.1016/j.mbs.2006.09.012.  Google Scholar

[8]

R. M. Colombo and G. Guerra, On general balance laws with boundary, J. Differential Equations, 248 (2010), 1017-1043. doi: 10.1016/j.jde.2009.12.002.  Google Scholar

[9]

R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050. doi: 10.1137/080716372.  Google Scholar

[10]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[11]

N. Keyfitz, The mathematics of sex and marriage, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 4: Biology and Health, University of California Press, Berkeley, Calif., 1972, 89-108.  Google Scholar

[12]

S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[13]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[14]

P. Manfredi, Logistic effects in the two-sex model with "harmonic mean" fertility function, Genus, 49 (1993), 43-65. Google Scholar

[15]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[16]

R. Schoen, The harmonic mean as the basis of a realistic two-sex marriage model, Demography, 18 (1981), 201-216. doi: 10.2307/2061093.  Google Scholar

[17]

R. Schoen, Relationships in a simple harmonic mean two-sex fertility model, Journal of Mathematical Biology, 18 (1983), 201-211. doi: 10.1007/BF00276087.  Google Scholar

[18]

R. Schoen, Modeling Multigroup Populations, Springer, 1988. doi: 10.1007/978-1-4899-2055-3.  Google Scholar

[19]

D. Serre, Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems, Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 2000.  Google Scholar

[20]

A. Sundelof and P. Aberg, Birth functions in stage structured two-sex models, Ecological Modeling, 193 (2006), 787-795. doi: 10.1016/j.ecolmodel.2005.08.040.  Google Scholar

show all references

References:
[1]

A. S. Ackleh and K. Deng, A nonautonomous juvenile-adult model: Well-posedness and long-time behavior via a comparison principle, SIAM J. Appl. Math., 69 (2009), 1644-1661. doi: 10.1137/080723673.  Google Scholar

[2]

A. S. Ackleh, K. Deng and X. Yang, Sensitivity analysis for a structured juvenile-adult model, Comput. Math. Appl., 64 (2012), 190-200. doi: 10.1016/j.camwa.2011.12.053.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.  Google Scholar

[4]

C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.  Google Scholar

[5]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second edition, Texts in Applied Mathematics, 40, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[6]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[7]

J. A. Carrillo, S. Cuadrado and B. Perthame, Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model, Math. Biosci., 205 (2007), 137-161. doi: 10.1016/j.mbs.2006.09.012.  Google Scholar

[8]

R. M. Colombo and G. Guerra, On general balance laws with boundary, J. Differential Equations, 248 (2010), 1017-1043. doi: 10.1016/j.jde.2009.12.002.  Google Scholar

[9]

R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050. doi: 10.1137/080716372.  Google Scholar

[10]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[11]

N. Keyfitz, The mathematics of sex and marriage, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 4: Biology and Health, University of California Press, Berkeley, Calif., 1972, 89-108.  Google Scholar

[12]

S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[13]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[14]

P. Manfredi, Logistic effects in the two-sex model with "harmonic mean" fertility function, Genus, 49 (1993), 43-65. Google Scholar

[15]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[16]

R. Schoen, The harmonic mean as the basis of a realistic two-sex marriage model, Demography, 18 (1981), 201-216. doi: 10.2307/2061093.  Google Scholar

[17]

R. Schoen, Relationships in a simple harmonic mean two-sex fertility model, Journal of Mathematical Biology, 18 (1983), 201-211. doi: 10.1007/BF00276087.  Google Scholar

[18]

R. Schoen, Modeling Multigroup Populations, Springer, 1988. doi: 10.1007/978-1-4899-2055-3.  Google Scholar

[19]

D. Serre, Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems, Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 2000.  Google Scholar

[20]

A. Sundelof and P. Aberg, Birth functions in stage structured two-sex models, Ecological Modeling, 193 (2006), 787-795. doi: 10.1016/j.ecolmodel.2005.08.040.  Google Scholar

[1]

Azmy S. Ackleh, Keng Deng. Stability of a delay equation arising from a juvenile-adult model. Mathematical Biosciences & Engineering, 2010, 7 (4) : 729-737. doi: 10.3934/mbe.2010.7.729
[2]

J. M. Cushing, Simon Maccracken Stump. Darwinian dynamics of a juvenile-adult model. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1017-1044. doi: 10.3934/mbe.2013.10.1017
[3]

Xianlong Fu, Dongmei Zhu. Stability analysis for a size-structured juvenile-adult population model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 391-417. doi: 10.3934/dcdsb.2014.19.391
[4]

Azmy S. Ackleh, Keng Deng, Qihua Huang. Difference approximation for an amphibian juvenile-adult dispersal mode. Conference Publications, 2011, 2011 (Special) : 1-12. doi: 10.3934/proc.2011.2011.1
[5]

József Z. Farkas, Thomas Hagen. Asymptotic behavior of size-structured populations via juvenile-adult interaction. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 249-266. doi: 10.3934/dcdsb.2008.9.249
[6]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[7]

Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271
[8]

Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033
[9]

Xing Liang, Lei Zhang. The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2055-2065. doi: 10.3934/dcdsb.2020280
[10]

Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477
[11]

Rinaldo M. Colombo, Graziano Guerra. Hyperbolic balance laws with a dissipative non local source. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1077-1090. doi: 10.3934/cpaa.2008.7.1077
[12]

Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure & Applied Analysis, 2017, 16 (2) : 629-644. doi: 10.3934/cpaa.2017031
[13]

Ryszard Rudnicki, Radoslaw Wieczorek. Does assortative mating lead to a polymorphic population? A toy model justification. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 459-472. doi: 10.3934/dcdsb.2018031
[14]

Alessia Marigo, Benedetto Piccoli. A model for biological dynamic networks. Networks & Heterogeneous Media, 2011, 6 (4) : 647-663. doi: 10.3934/nhm.2011.6.647
[15]

Linhao Xu, Marya Claire Zdechlik, Melissa C. Smith, Min B. Rayamajhi, Don L. DeAngelis, Bo Zhang. Simulation of post-hurricane impact on invasive species with biological control management. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 4059-4071. doi: 10.3934/dcds.2020038
[16]

Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018
[17]

Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
[18]

Stephan Gerster, Michael Herty. Discretized feedback control for systems of linearized hyperbolic balance laws. Mathematical Control & Related Fields, 2019, 9 (3) : 517-539. doi: 10.3934/mcrf.2019024
[19]

Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035
[20]

Ahmed Aghriche, Radouane Yafia, M. A. Aziz Alaoui, Abdessamad Tridane. Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2443-2463. doi: 10.3934/dcdss.2020194

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences