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
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show all references

References:
[1]

SIAM J. Appl. Math., 69 (2009), 1644-1661. doi: 10.1137/080723673.  Google Scholar

[2]

Comput. Math. Appl., 64 (2012), 190-200. doi: 10.1016/j.camwa.2011.12.053.  Google Scholar

[3]

Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.  Google Scholar

[4]

Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.  Google Scholar

[5]

Second edition, Texts in Applied Mathematics, 40, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[6]

Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[7]

Math. Biosci., 205 (2007), 137-161. doi: 10.1016/j.mbs.2006.09.012.  Google Scholar

[8]

J. Differential Equations, 248 (2010), 1017-1043. doi: 10.1016/j.jde.2009.12.002.  Google Scholar

[9]

SIAM J. Control Optim., 48 (2009), 2032-2050. doi: 10.1137/080716372.  Google Scholar

[10]

AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[11]

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]

Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[13]

Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[14]

Genus, 49 (1993), 43-65. Google Scholar

[15]

Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[16]

Demography, 18 (1981), 201-216. doi: 10.2307/2061093.  Google Scholar

[17]

Journal of Mathematical Biology, 18 (1983), 201-211. doi: 10.1007/BF00276087.  Google Scholar

[18]

Springer, 1988. doi: 10.1007/978-1-4899-2055-3.  Google Scholar

[19]

Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 2000.  Google Scholar

[20]

Ecological Modeling, 193 (2006), 787-795. doi: 10.1016/j.ecolmodel.2005.08.040.  Google Scholar

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