Stability and optimization in structured population models on graphs

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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

Abstract
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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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