×

Examples of shocks in population dynamics. (English) Zbl 1362.35186

Substantially the paper deals with a problem for a linear hyperbolic partial differential operators of first order in \(\mathbb{R}^{2}\)of the following type \[ \begin{cases}\frac{\partial u}{\partial t}(t,x)+\frac{\partial u}{\partial x}(t,x)=-\mu (x)u(t,x), \\ u(0,x)=\varphi (x), \\ u(t,0)=\int\limits_{0}^{x_{m}}\beta (y)u(t,y)dy, \end{cases} \] where the functions \(\mu ,\varphi \) and \(\beta \) are given data. The main results of the formulated problem are obtained in the framework of generalized functions of Colombeau type when the data \(\varphi \) and \(\beta \) are Sobolev-Schwartz distributions.
The problem studied is illustrated by the mathematical formulation of a model in demography.

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
92D25 Population dynamics (general)
35L04 Initial-boundary value problems for first-order hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. ARAGONA, Colombeau generalized functions on quasi-regular sets, Publicationes Mathematicae Debrecen, 68, 3-4 (2006), 371–399. · Zbl 1113.46034
[2] J. ARAGONA, A. R. G. GARCIA ANDS. O. JURIAANS, Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data, Nonlinear Analysis, 71 (2009), 5187– 5207. · Zbl 1182.46032
[3] BANQUEMONDIALE, L’IDA en action. G’erer les risques naturels, r’eduire les risques li’es au d’eveloppement, Washington D. C., 2008.
[4] J. R. BARNETT ANDM. WEBBER, Accomodating Migration to Promote Adaptation to Climate Change, World Bank Policy Research, Working Paper 5270, 2010.
[5] H. A. BIAGIONI, A nonlinear theory of Generalized Functions, Lecture Notes in Mathematics 1421, Springer-Verlag, 1990.
[6] E. CANALES, 1808-1814: D’emographie et guerre en Espagne, Annales historiques de la R’evolution franc\c{}aise, 336 (2004), 37–52.
[7] S. J. CHAPMAN, M. J. PLANK, A. JAMES ANDA. B. BASSE, A nonlinear model of age size- structured populations with applications to cell cycles, The ANZIAM Journal 49, 2 (2007), 151–169. · Zbl 1145.37042
[8] P. COLLIER, On the economic consequences of Civil War, Oxford Economic Papers, 51 (1999), 168183.
[9] J. F. COLOMBEAU, New Generalized Functions and Multiplication of Distributions, North Holland, Amsterdam, Oxford, New-York, 1984.
[10] J. F. COLOMBEAU, Elementary introduction to new generalized functions, North Holland Mathematics Studies 113, North-Holland, Amsterdam, 1984.
[11] J. M. CUSHING ANDJ. LI, Juvenile versus adult competition, Journal of Mathematical Biology 29 (1991), 457–473.
[12] A. DELCROIX, Remarks on the embedding of spaces of distributions into spaces of Colombeau gen- eralized functions, Novi Sad Journal of Mathematics, 35, 2 (2005), 27–40. · Zbl 1274.46082
[13] A. DELCROIX, V. D ’EVOUE AND’J.-A. MARTI, Generalized solutions of singular differential prob- lems. Relationship with classical solutions, Journal of Mathematical Analysis and Applications 353, 1 (2009), 386–402.
[14] A. DELCROIX, V. D ’EVOUE AND’J.-A. MARTI, Well posed problems in algebras of generalized func- tion, Applicable Analysis, 90, 11 (2011), 1747–1761.
[15] R. DILAO ANDA. LAKMECHE, On the weak solutions of the McKendrick equation: Existence of demography cycles, Mathematical Modeling of Natural Phenomena, 1, 1 (2006), 1–32.
[16] M IANNELLI, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e stampatori in Pisa 1995.
[17] M. D. JOHNSTON, C. M. EDWARDS, W. F. BODMER, P. K. MAINI ANDS. J. CHAPMAN, Mathe- matical modeling of cell population dynamics in the colonic crypt and in colorectal cancer, Proceedings of the National Academy of Sciences of the United States of America, 104, 10 (2007), 4008–4013.
[18] I. KMIT, A distributional solution to a hyperbolic problem arising in population dynamics, Electronic Journal of Differential Equations, 132 (2007), 1–23. · Zbl 1137.35334
[19] J. A. LOTKA, Th’eorie analytique des associations biologiques, Hermann, Paris, 1934 and 1939.
[20] J.-A. MARTI, Fundamental structures and asymptotic microlocalization in sheaves of generalized functions, Integral Transforms and Special Functions 6, 1-4 (1998), 223–228. · Zbl 0902.18005
[21] J.-A. MARTI,(C ,E ,P) -Sheaf structure and applications, Nonlinear theory of generalized functions (eds M. Grosser and al.), Research Notes in Mathematics, Chapman & Hall/CRC (1999), 175–186.
[22] J.-A. MARTI, Non linear Algebraic analysis of delta shock wave to Burgers’ equation, Pacific Journal of Mathematics, 210, 1 (2003), 165–187. · Zbl 1059.35122
[23] A. G. MCKENDRICK, Applications of Mathematics to Medical Problems, Proceedings of the Edinburgh Mathematical Society, 44 (1926), 98–130.
[24] NORHAYATI ANDG. C. WAKE, The solution and stability of a nonlinear age-structured population model, The ANZIAM Journal, 45 (2003), 153–165.
[25] ORGANISATION DE L’UNITE’AFRICAINE, Rapport sur le g’enocide du Rwanda. Le g’enocide qu’on aurait pu stopper, Nations Unies 2000.
[26] B. PERTHAME, S. MISCHLER, J. CLAIRAMBAULT ANDB. LAROCHE, A mathematical model of the cell cycle and its control, Rapport de recherche 4892, Institut National de Recherche en Informatique et en Automatique, 2007.
[27] L. SCHWARTZ, Th’eorie des Distributions, Hermann, Paris, 1966.
[28] D. SCHWEISGUTH, Japon: S’eisme et tsunami, quel impact sur la croissance?, Publications de l’OCFCE, Centre de recherche en ’economie de Sciences Politiques 2011.
[29] M. VERPOORTEN ANDL. BERLAGE, Economic Mobility in Rural Rwanda: A Study of the Effects of War and Genocide at the Household Level, Journal of African Economies, 3 (2007), 349–392.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.