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

Stability and optimization in structured population models on graphs

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.
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[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.

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