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.  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.  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, (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.  doi: 10.1080/03605307908820117.  Google Scholar

[5]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, Second edition, (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, (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.  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.  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.  doi: 10.1137/080716372.  Google Scholar

[10]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (2006).   Google Scholar

[11]

N. Keyfitz, The mathematics of sex and marriage,, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, (1972), 89.   Google Scholar

[12]

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

[13]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge Texts in Applied Mathematics, (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.   Google Scholar

[15]

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

[16]

R. Schoen, The harmonic mean as the basis of a realistic two-sex marriage model,, Demography, 18 (1981), 201.  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.  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, (1996).   Google Scholar

[20]

A. Sundelof and P. Aberg, Birth functions in stage structured two-sex models,, Ecological Modeling, 193 (2006), 787.  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.  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.  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, (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.  doi: 10.1080/03605307908820117.  Google Scholar

[5]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, Second edition, (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, (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.  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.  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.  doi: 10.1137/080716372.  Google Scholar

[10]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (2006).   Google Scholar

[11]

N. Keyfitz, The mathematics of sex and marriage,, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, (1972), 89.   Google Scholar

[12]

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

[13]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge Texts in Applied Mathematics, (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.   Google Scholar

[15]

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

[16]

R. Schoen, The harmonic mean as the basis of a realistic two-sex marriage model,, Demography, 18 (1981), 201.  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.  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, (1996).   Google Scholar

[20]

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

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