• Previous Article
    SDE-driven modeling of phenotypically heterogeneous tumors: The influence of cancer cell stemness
  • DCDS-B Home
  • This Issue
  • Next Article
    Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients
August  2019, 24(8): 4547-4628. doi: 10.3934/dcdsb.2019156

Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness

Applied Mathematics, RheinMain University of Applied Sciences, Wiesbaden Rüsselsheim, Germany

Dedicated to Peter E. Kloeden on occasion of his 70th birthday

Received  September 2018 Revised  April 2019 Published  August 2019 Early access  June 2019

Detailed models of structured populations are spatial and involve nonlocal effects. These features lead to a broad class of population models structured by a physiological parameter and space. Our focus of interest is on the well-posedness of their initial value problems. In more detail, we specify sufficient conditions on the coefficient functions for existence, positivity, uniqueness of weak solutions and their continuous dependence on the given data. The solutions considered here have their values in $ L^p $ and, all conclusions about convergence and sequential compactness use an adaptation of the KANTOROVICH-RUBINSTEIN metric for this $ L^p $ space.

Citation: Thomas Lorenz. Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4547-4628. doi: 10.3934/dcdsb.2019156
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

A. AggarwalR. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.  doi: 10.1137/140975255.

[3]

H. W. Alt, Linear Functional Analysis. An Application-Oriented Introduction, London: Springer, 2016, Translated from the 6th German edition by Robert Nürnberg. doi: 10.1007/978-1-4471-7280-2.

[4]

H. Amann and J. Escher, Analysis. II, Birkhäuser Verlag, Basel, 2008, Translated from the 1999 German original by Silvio Levy and Matthew Cargo.

[5]

H. Amann and J. Escher, Analysis. III, Birkhäuser Verlag, Basel, 2009, Translated from the 2001 German original by Silvio Levy and Matthew Cargo. doi: 10.1007/978-3-7643-7480-8.

[6]

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.

[7]

L. T. T. An, Drug Resistance in Infectious Diseases. Modeling, Analysis and Simulation, Ph.D. thesis, Heidelberg University, Naturwissenschaftlich-Mathematische Gesamtfakultät, 2012, http://www.ub.uni-heidelberg.de/archiv/13456.

[8]

L. T. T. AnW. Jäger and M. Neuss-Radu, Systems of populations with multiple structures: Modeling and analysis, J. Dynam. Differential Equations, 27 (2015), 863-877.  doi: 10.1007/s10884-015-9469-3.

[9]

Z. Artstein and K. Prikry, Carathéodory selections and the Scorza Dragoni property, J. Math. Anal. Appl., 127 (1987), 540-547.  doi: 10.1016/0022-247X(87)90128-4.

[10]

J.-P. Aubin, Viability Theory, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991.

[11]

J.-P. Aubin, Mutational equations in metric spaces, Set-Valued Anal., 1 (1993), 3-46.  doi: 10.1007/BF01039289.

[12]

J.-P. Aubin, Mutational and Morphological Analysis, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1999, Tools for shape evolution and morphogenesis. doi: 10.1007/978-1-4612-1576-9.

[13]

B. P. Ayati, A structured-population model of proteus mirabilis swarm-colony development, J. Math. Biol., 52 (2006), 93-114.  doi: 10.1007/s00285-005-0345-3.

[14]

V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[15]

R. BorscheR. M. ColomboM. Garavello and A. Meurer, Differential equations modeling crowd interactions, J. Nonlinear Sci., 25 (2015), 827-859.  doi: 10.1007/s00332-015-9242-0.

[16]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[17]

J. A. CañizoJ. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Appl. Math., 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.

[18]

J. A. CarrilloR. M. ColomboP. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differential Equations, 252 (2012), 3245-3277.  doi: 10.1016/j.jde.2011.11.003.

[19]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[20]

R. M. Colombo and G. Guerra, Nonlocal sources in hyperbolic balance laws with applications, in Hyperbolic Problems: Theory, Numerics, Applications (eds. S. Benzoni-Gavage and D. Serre), Springer, Berlin, 2008,577–584. doi: 10.1007/978-3-540-75712-2_56.

[21]

R. M. ColomboA. Corli and M. D. Rosini, Non local balance laws in traffic models and crystal growth, ZAMM Z. Angew. Math. Mech., 87 (2007), 449-461.  doi: 10.1002/zamm.200710327.

[22]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34p. doi: 10.1142/S0218202511500230.

[23]

R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Commun. Pure Appl. Anal., 7 (2008), 1077-1090.  doi: 10.3934/cpaa.2008.7.1077.

[24]

R. M. ColomboP. Gwiazda and M. Rosińska, Optimization in structure population models through the escalator boxcar train, ESAIM Control Optim. Calc. Var., 24 (2018), 377-399.  doi: 10.1051/cocv/2017003.

[25]

R. M. Colombo and M. Lécureux-Mercier, Nonlocal crowd dynamics models for several populations, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 177-196.  doi: 10.1016/S0252-9602(12)60011-3.

[26]

E. D. Conway, Generalized solutions of linear differential equations with discontinuous coefficients and the uniqueness question for multidimensional quasilinear conservation laws, J. Math. Anal. Appl., 18 (1967), 238-251.  doi: 10.1016/0022-247X(67)90054-6.

[27]

G. CorbinA. HuntA. KlarF. Schneider and C. Surulescu, Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum, Math. Models Methods Appl. Sci., 28 (2018), 1771-1800.  doi: 10.1142/S0218202518400055.

[28]

J. M. Cushing, An Introduction to Structured Population Dynamics, vol. 71 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.

[29]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Berlin-New York, 1977, Lecture Notes in Biomathematics, Vol. 20.

[30]

A. M. de Roos, A gentle introduction to physiologically structured population models, in Structured-Population Models in Marine, Terrestrial, and Freshwater Systems (eds. S. Tuljapurkar and H. Caswell), vol. 18 of Population and Community Biology Series, Springer, Boston, MA, 1997,119–204. doi: 10.1007/978-1-4615-5973-3_5.

[31]

A. M. de RoosT. SchellekensT. Van KootenK. Van De WolfshaarD. Claessen and L. Persson, Simplifying a physiologically structured population model to a stage-structured biomass model, Theoret. Population Biol., 73 (2008), 47-62.  doi: 10.1016/j.tpb.2007.09.004.

[32]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries, vol. 22 of Advances in Design and Control, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011, Metrics, analysis, differential calculus, and optimization. doi: 10.1137/1.9780898719826.

[33]

Q. Deng and T. G. Hallam, An age structured population model in a spatially heterogeneous environment: Existence and uniqueness theory, Nonlinear Anal., 65 (2006), 379-394.  doi: 10.1016/j.na.2005.06.019.

[34]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4419-9158-4.

[35]

O. Diekmann and P. Getto, Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Differential Equations, 215 (2005), 268-319.  doi: 10.1016/j.jde.2004.10.025.

[36]

O. DiekmannM. GyllenbergH. HuangM. KirkilionisJ. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. Ⅱ. Nonlinear theory, J. Math. Biol., 43 (2001), 157-189.  doi: 10.1007/s002850170002.

[37]

O. DiekmannM. GyllenbergJ. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. I. Linear theory, J. Math. Biol., 36 (1998), 349-388.  doi: 10.1007/s002850050104.

[38] O. DiekmannH. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2013. 
[39]

C. EngwerT. HillenM. Knappitsch and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, J. Math. Biol., 71 (2015), 551-582.  doi: 10.1007/s00285-014-0822-7.

[40]

C. EngwerA. Hunt and C. Surulescu, Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings, Math. Med. Biol., 33 (2016), 435-459.  doi: 10.1093/imammb/dqv030.

[41]

C. EngwerM. Knappitsch and C. Surulescu, A multiscale model for glioma spread including cell-tissue interactions and proliferation, Math. Biosci. Eng., 13 (2016), 443-460.  doi: 10.3934/mbe.2015011.

[42]

C. EngwerC. Stinner and C. Surulescu, On a structured multiscale model for acid-mediated tumor invasion: the effects of adhesion and proliferation, Math. Models Methods Appl. Sci., 27 (2017), 1355-1390.  doi: 10.1142/S0218202517400188.

[43]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[44]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the lwr traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.

[45]

J. W. Green and F. A. Valentine, On the Arzelà-Ascoli theorem, Math. Mag., 34 (1960/1961), 199-202.  doi: 10.2307/2687984.

[46]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, vol. 34 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[47]

P. GwiazdaT. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differential Equations, 248 (2010), 2703-2735.  doi: 10.1016/j.jde.2010.02.010.

[48]

P. Hartman, Ordinary Differential Equations, vol. 38 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e: 34002)], With a foreword by Peter Bates. doi: 10.1137/1.9780898719222.

[49]

C. J. Himmelberg, Precompact contraction of metric uniformities, and the continuity of F(t, x), Rend. Sem. Mat. Univ. Padova, 50 (1973), 185–188 (1974).

[50]

C. J. Himmelberg, Correction to: "Precompact contraction of metric uniformities, and the continuity of F(t, x)" (Rend. Sem. Mat. Univ. Padova 50 (1973), 185–188 (1974)), Rend. Sem. Mat. Univ. Padova, 51 (1974), 361 (1975).

[51]

F. C. Hoppensteadt, Mathematical Methods of Population Biology, Courant Institute of Mathematical Sciences, New York University, New York, 1977.

[52]

C. C. Huang, An age-dependent population model with nonlinear diffusion in Rn, Quart. Appl. Math., 52 (1994), 377-398.  doi: 10.1090/qam/1276244.

[53]

M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Dordrecht, 2017, Models, methods and numerics. doi: 10.1007/978-94-024-1146-1.

[54]

M. Iannelli and A. Pugliese, An Introduction to Mathematical Population Dynamics, vol. 79 of Unitext, Springer, Cham, 2014, Along the trail of Volterra and Lotka, La Matematica per il 3+2. doi: 10.1007/978-3-319-03026-5.

[55]

H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.

[56]

T. Insperger and G. Stépán, Semi-discretization for Time-Delay Systems, vol. 178 of Applied Mathematical Sciences, Springer, New York, 2011, Stability and engineering applications. doi: 10.1007/978-1-4614-0335-7.

[57]

J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Math. Models Methods Appl. Sci., 22 (2012), 1150017, 25pp. doi: 10.1142/S0218202511500175.

[58]

P. E. Kloeden, C. Pötzsche and M. Rasmussen, Discrete-time nonautonomous dynamical systems, in Stability and Bifurcation Theory for Non-Autonomous Differential Equations, vol. 2065 of Lecture Notes in Math., Springer, Heidelberg, 2013, 35–102. doi: 10.1007/978-3-642-32906-7_2.

[59]

P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl., 28 (2010), 937-945.  doi: 10.1080/07362994.2010.515194.

[60]

P. E. Kloeden and T. Lorenz, Stochastic morphological evolution equations, J. Differential Equations, 251 (2011), 2950-2979.  doi: 10.1016/j.jde.2011.03.013.

[61]

P. E. Kloeden and T. Lorenz, Fuzzy differential equations without fuzzy convexity, Fuzzy Sets and Systems, 230 (2013), 65-81.  doi: 10.1016/j.fss.2012.01.012.

[62]

P. E. Kloeden and T. Lorenz, A Peano-like theorem for stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl., 31 (2013), 19-30.  doi: 10.1080/07362994.2012.727142.

[63]

P. E. Kloeden and T. Lorenz, A Peano theorem for fuzzy differential equations with evolving membership grade, Fuzzy Sets and Systems, 280 (2015), 1-26.  doi: 10.1016/j.fss.2014.12.001.

[64]

P. E. Kloeden and T. Lorenz, Nonlocal multi-scale traffic flow models: Analysis beyond vector spaces, Bull. Math. Sci., 6 (2016), 453-514.  doi: 10.1007/s13373-016-0090-5.

[65]

S. G. Krantz and H. R. Parks,, The implicit function theorem, Birkhäuser/Springer, New York, 2013, History, theory, and applications. doi: 10.1007/978-1-4614-5981-1.

[66]

A. Kucia, Scorza Dragoni type theorems, Fund. Math., 138 (1991), 197-203.  doi: 10.4064/fm-138-3-197-203.

[67]

S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0897-6.

[68]

P. Laurençot and C. Walker, An age and spatially structured population model for proteus mirabilis swarm-colony development, Math. Model. Nat. Phenom., 3 (2008), 49-77.  doi: 10.1051/mmnp:2008041.

[69]

P. A. Loeb, Real Analysis, Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-30744-2.

[70]

T. Lorenz, Mutational Analysis, vol. 1996 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, A joint framework for Cauchy problems in and beyond vector spaces. doi: 10.1007/978-3-642-12471-6.

[71]

T. Lorenz, Mutational inclusions: Differential inclusions in metric spaces, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 629-654.  doi: 10.3934/dcdsb.2010.14.629.

[72]

T. Lorenz, Differential equations for closed sets in a Banach space: survey and extension, Vietnam J. Math., 45 (2017), 5-49.  doi: 10.1007/s10013-016-0195-x.

[73]

T. Lorenz, A viability theorem for set-valued states in a Hilbert space, J. Math. Anal. Appl., 457 (2018), 1502-1567.  doi: 10.1016/j.jmaa.2017.08.011.

[74]

T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, Math. Models Methods Appl. Sci., 24 (2014), 2383-2436.  doi: 10.1142/S0218202514500249.

[75]

P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, vol. 1936 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008, Mathematical Biosciences Subseries. doi: 10.1007/978-3-540-78273-5.

[76]

J. A. J. Metz and O. Diekmann (eds.), The Dynamics of Physiologically Structured Populations, vol. 68 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1986, Papers from the colloquium held in Amsterdam, 1983. doi: 10.1007/978-3-662-13159-6.

[77]

M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.-Math. Soc. Japan (3), 24 (1942), 551–559.

[78]

B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes, Acta Appl. Math., 124 (2013), 73-105.  doi: 10.1007/s10440-012-9771-6.

[79]

B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x.

[80]

B. Ricceri and A. Villani, Separability and Scorza-Dragoni's property, Matematiche (Catania), 37 (1982), 156–161 (1985).

[81]

V. I. Smirnov, A Course of Higher Mathematics. Vol. IV [Integral Equations and Partial Differential Equations], Translated by D. E. Brown; translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964.

[82]

H. R. Thieme, Well-posedness of physiologically structured population models for daphnia magna. How biological concepts can benefit by abstract mathematical analysis, J. Math. Biol., 26 (1988), 299-317.  doi: 10.1007/BF00277393.

[83]

S. Tuljapurkar and H. Caswell, Structured-Population Models in Marine, Terrestrial, and Freshwater Systems, Population and Community Biology Series, Springer, Boston, MA, 1997. doi: 10.1007/978-1-4615-5973-3.

[84]

A. Ulikowska, An age-structured two-sex model in the space of Radon measures: well posedness, Kinet. Relat. Models, 5 (2012), 873-900.  doi: 10.3934/krm.2012.5.873.

[85]

C. Walker, Positive equilibrium solutions for age- and spatially-structured population models, SIAM J. Math. Anal., 41 (2009), 1366-1387.  doi: 10.1137/090750044.

[86]

W. Walter, Ordinary Differential Equations, vol. 182 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998, Translated from the sixth German (1996) edition by Russell Thompson, Readings in Mathematics. doi: 10.1007/978-1-4612-0601-9.

[87]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, vol. 89 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1985.

[88]

G. F. Webb, Semigroup methods in population dynamics: Proliferating cell populations, in Semigroup Theory and Applications (Trieste, 1987), vol. 116 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1989,441–449.

[89]

G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology (eds. P. Magal and S. Ruan), vol. 1936 of Lecture Notes in Math., Springer, Berlin, 2008, 1–49. doi: 10.1007/978-3-540-78273-5_1.

[90]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I, Springer-Verlag, New York, 1986, Fixed-point theorems, Translated from the German by Peter R. Wadsack. doi: 10.1007/978-1-4612-4838-5.

show all references

Dedicated to Peter E. Kloeden on occasion of his 70th birthday

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

A. AggarwalR. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.  doi: 10.1137/140975255.

[3]

H. W. Alt, Linear Functional Analysis. An Application-Oriented Introduction, London: Springer, 2016, Translated from the 6th German edition by Robert Nürnberg. doi: 10.1007/978-1-4471-7280-2.

[4]

H. Amann and J. Escher, Analysis. II, Birkhäuser Verlag, Basel, 2008, Translated from the 1999 German original by Silvio Levy and Matthew Cargo.

[5]

H. Amann and J. Escher, Analysis. III, Birkhäuser Verlag, Basel, 2009, Translated from the 2001 German original by Silvio Levy and Matthew Cargo. doi: 10.1007/978-3-7643-7480-8.

[6]

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.

[7]

L. T. T. An, Drug Resistance in Infectious Diseases. Modeling, Analysis and Simulation, Ph.D. thesis, Heidelberg University, Naturwissenschaftlich-Mathematische Gesamtfakultät, 2012, http://www.ub.uni-heidelberg.de/archiv/13456.

[8]

L. T. T. AnW. Jäger and M. Neuss-Radu, Systems of populations with multiple structures: Modeling and analysis, J. Dynam. Differential Equations, 27 (2015), 863-877.  doi: 10.1007/s10884-015-9469-3.

[9]

Z. Artstein and K. Prikry, Carathéodory selections and the Scorza Dragoni property, J. Math. Anal. Appl., 127 (1987), 540-547.  doi: 10.1016/0022-247X(87)90128-4.

[10]

J.-P. Aubin, Viability Theory, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991.

[11]

J.-P. Aubin, Mutational equations in metric spaces, Set-Valued Anal., 1 (1993), 3-46.  doi: 10.1007/BF01039289.

[12]

J.-P. Aubin, Mutational and Morphological Analysis, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1999, Tools for shape evolution and morphogenesis. doi: 10.1007/978-1-4612-1576-9.

[13]

B. P. Ayati, A structured-population model of proteus mirabilis swarm-colony development, J. Math. Biol., 52 (2006), 93-114.  doi: 10.1007/s00285-005-0345-3.

[14]

V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[15]

R. BorscheR. M. ColomboM. Garavello and A. Meurer, Differential equations modeling crowd interactions, J. Nonlinear Sci., 25 (2015), 827-859.  doi: 10.1007/s00332-015-9242-0.

[16]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[17]

J. A. CañizoJ. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Appl. Math., 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.

[18]

J. A. CarrilloR. M. ColomboP. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differential Equations, 252 (2012), 3245-3277.  doi: 10.1016/j.jde.2011.11.003.

[19]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[20]

R. M. Colombo and G. Guerra, Nonlocal sources in hyperbolic balance laws with applications, in Hyperbolic Problems: Theory, Numerics, Applications (eds. S. Benzoni-Gavage and D. Serre), Springer, Berlin, 2008,577–584. doi: 10.1007/978-3-540-75712-2_56.

[21]

R. M. ColomboA. Corli and M. D. Rosini, Non local balance laws in traffic models and crystal growth, ZAMM Z. Angew. Math. Mech., 87 (2007), 449-461.  doi: 10.1002/zamm.200710327.

[22]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34p. doi: 10.1142/S0218202511500230.

[23]

R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Commun. Pure Appl. Anal., 7 (2008), 1077-1090.  doi: 10.3934/cpaa.2008.7.1077.

[24]

R. M. ColomboP. Gwiazda and M. Rosińska, Optimization in structure population models through the escalator boxcar train, ESAIM Control Optim. Calc. Var., 24 (2018), 377-399.  doi: 10.1051/cocv/2017003.

[25]

R. M. Colombo and M. Lécureux-Mercier, Nonlocal crowd dynamics models for several populations, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 177-196.  doi: 10.1016/S0252-9602(12)60011-3.

[26]

E. D. Conway, Generalized solutions of linear differential equations with discontinuous coefficients and the uniqueness question for multidimensional quasilinear conservation laws, J. Math. Anal. Appl., 18 (1967), 238-251.  doi: 10.1016/0022-247X(67)90054-6.

[27]

G. CorbinA. HuntA. KlarF. Schneider and C. Surulescu, Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum, Math. Models Methods Appl. Sci., 28 (2018), 1771-1800.  doi: 10.1142/S0218202518400055.

[28]

J. M. Cushing, An Introduction to Structured Population Dynamics, vol. 71 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.

[29]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Berlin-New York, 1977, Lecture Notes in Biomathematics, Vol. 20.

[30]

A. M. de Roos, A gentle introduction to physiologically structured population models, in Structured-Population Models in Marine, Terrestrial, and Freshwater Systems (eds. S. Tuljapurkar and H. Caswell), vol. 18 of Population and Community Biology Series, Springer, Boston, MA, 1997,119–204. doi: 10.1007/978-1-4615-5973-3_5.

[31]

A. M. de RoosT. SchellekensT. Van KootenK. Van De WolfshaarD. Claessen and L. Persson, Simplifying a physiologically structured population model to a stage-structured biomass model, Theoret. Population Biol., 73 (2008), 47-62.  doi: 10.1016/j.tpb.2007.09.004.

[32]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries, vol. 22 of Advances in Design and Control, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011, Metrics, analysis, differential calculus, and optimization. doi: 10.1137/1.9780898719826.

[33]

Q. Deng and T. G. Hallam, An age structured population model in a spatially heterogeneous environment: Existence and uniqueness theory, Nonlinear Anal., 65 (2006), 379-394.  doi: 10.1016/j.na.2005.06.019.

[34]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4419-9158-4.

[35]

O. Diekmann and P. Getto, Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Differential Equations, 215 (2005), 268-319.  doi: 10.1016/j.jde.2004.10.025.

[36]

O. DiekmannM. GyllenbergH. HuangM. KirkilionisJ. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. Ⅱ. Nonlinear theory, J. Math. Biol., 43 (2001), 157-189.  doi: 10.1007/s002850170002.

[37]

O. DiekmannM. GyllenbergJ. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. I. Linear theory, J. Math. Biol., 36 (1998), 349-388.  doi: 10.1007/s002850050104.

[38] O. DiekmannH. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2013. 
[39]

C. EngwerT. HillenM. Knappitsch and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, J. Math. Biol., 71 (2015), 551-582.  doi: 10.1007/s00285-014-0822-7.

[40]

C. EngwerA. Hunt and C. Surulescu, Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings, Math. Med. Biol., 33 (2016), 435-459.  doi: 10.1093/imammb/dqv030.

[41]

C. EngwerM. Knappitsch and C. Surulescu, A multiscale model for glioma spread including cell-tissue interactions and proliferation, Math. Biosci. Eng., 13 (2016), 443-460.  doi: 10.3934/mbe.2015011.

[42]

C. EngwerC. Stinner and C. Surulescu, On a structured multiscale model for acid-mediated tumor invasion: the effects of adhesion and proliferation, Math. Models Methods Appl. Sci., 27 (2017), 1355-1390.  doi: 10.1142/S0218202517400188.

[43]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[44]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the lwr traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.

[45]

J. W. Green and F. A. Valentine, On the Arzelà-Ascoli theorem, Math. Mag., 34 (1960/1961), 199-202.  doi: 10.2307/2687984.

[46]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, vol. 34 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[47]

P. GwiazdaT. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differential Equations, 248 (2010), 2703-2735.  doi: 10.1016/j.jde.2010.02.010.

[48]

P. Hartman, Ordinary Differential Equations, vol. 38 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e: 34002)], With a foreword by Peter Bates. doi: 10.1137/1.9780898719222.

[49]

C. J. Himmelberg, Precompact contraction of metric uniformities, and the continuity of F(t, x), Rend. Sem. Mat. Univ. Padova, 50 (1973), 185–188 (1974).

[50]

C. J. Himmelberg, Correction to: "Precompact contraction of metric uniformities, and the continuity of F(t, x)" (Rend. Sem. Mat. Univ. Padova 50 (1973), 185–188 (1974)), Rend. Sem. Mat. Univ. Padova, 51 (1974), 361 (1975).

[51]

F. C. Hoppensteadt, Mathematical Methods of Population Biology, Courant Institute of Mathematical Sciences, New York University, New York, 1977.

[52]

C. C. Huang, An age-dependent population model with nonlinear diffusion in Rn, Quart. Appl. Math., 52 (1994), 377-398.  doi: 10.1090/qam/1276244.

[53]

M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Dordrecht, 2017, Models, methods and numerics. doi: 10.1007/978-94-024-1146-1.

[54]

M. Iannelli and A. Pugliese, An Introduction to Mathematical Population Dynamics, vol. 79 of Unitext, Springer, Cham, 2014, Along the trail of Volterra and Lotka, La Matematica per il 3+2. doi: 10.1007/978-3-319-03026-5.

[55]

H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.

[56]

T. Insperger and G. Stépán, Semi-discretization for Time-Delay Systems, vol. 178 of Applied Mathematical Sciences, Springer, New York, 2011, Stability and engineering applications. doi: 10.1007/978-1-4614-0335-7.

[57]

J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Math. Models Methods Appl. Sci., 22 (2012), 1150017, 25pp. doi: 10.1142/S0218202511500175.

[58]

P. E. Kloeden, C. Pötzsche and M. Rasmussen, Discrete-time nonautonomous dynamical systems, in Stability and Bifurcation Theory for Non-Autonomous Differential Equations, vol. 2065 of Lecture Notes in Math., Springer, Heidelberg, 2013, 35–102. doi: 10.1007/978-3-642-32906-7_2.

[59]

P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl., 28 (2010), 937-945.  doi: 10.1080/07362994.2010.515194.

[60]

P. E. Kloeden and T. Lorenz, Stochastic morphological evolution equations, J. Differential Equations, 251 (2011), 2950-2979.  doi: 10.1016/j.jde.2011.03.013.

[61]

P. E. Kloeden and T. Lorenz, Fuzzy differential equations without fuzzy convexity, Fuzzy Sets and Systems, 230 (2013), 65-81.  doi: 10.1016/j.fss.2012.01.012.

[62]

P. E. Kloeden and T. Lorenz, A Peano-like theorem for stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl., 31 (2013), 19-30.  doi: 10.1080/07362994.2012.727142.

[63]

P. E. Kloeden and T. Lorenz, A Peano theorem for fuzzy differential equations with evolving membership grade, Fuzzy Sets and Systems, 280 (2015), 1-26.  doi: 10.1016/j.fss.2014.12.001.

[64]

P. E. Kloeden and T. Lorenz, Nonlocal multi-scale traffic flow models: Analysis beyond vector spaces, Bull. Math. Sci., 6 (2016), 453-514.  doi: 10.1007/s13373-016-0090-5.

[65]

S. G. Krantz and H. R. Parks,, The implicit function theorem, Birkhäuser/Springer, New York, 2013, History, theory, and applications. doi: 10.1007/978-1-4614-5981-1.

[66]

A. Kucia, Scorza Dragoni type theorems, Fund. Math., 138 (1991), 197-203.  doi: 10.4064/fm-138-3-197-203.

[67]

S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0897-6.

[68]

P. Laurençot and C. Walker, An age and spatially structured population model for proteus mirabilis swarm-colony development, Math. Model. Nat. Phenom., 3 (2008), 49-77.  doi: 10.1051/mmnp:2008041.

[69]

P. A. Loeb, Real Analysis, Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-30744-2.

[70]

T. Lorenz, Mutational Analysis, vol. 1996 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, A joint framework for Cauchy problems in and beyond vector spaces. doi: 10.1007/978-3-642-12471-6.

[71]

T. Lorenz, Mutational inclusions: Differential inclusions in metric spaces, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 629-654.  doi: 10.3934/dcdsb.2010.14.629.

[72]

T. Lorenz, Differential equations for closed sets in a Banach space: survey and extension, Vietnam J. Math., 45 (2017), 5-49.  doi: 10.1007/s10013-016-0195-x.

[73]

T. Lorenz, A viability theorem for set-valued states in a Hilbert space, J. Math. Anal. Appl., 457 (2018), 1502-1567.  doi: 10.1016/j.jmaa.2017.08.011.

[74]

T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, Math. Models Methods Appl. Sci., 24 (2014), 2383-2436.  doi: 10.1142/S0218202514500249.

[75]

P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, vol. 1936 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008, Mathematical Biosciences Subseries. doi: 10.1007/978-3-540-78273-5.

[76]

J. A. J. Metz and O. Diekmann (eds.), The Dynamics of Physiologically Structured Populations, vol. 68 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1986, Papers from the colloquium held in Amsterdam, 1983. doi: 10.1007/978-3-662-13159-6.

[77]

M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.-Math. Soc. Japan (3), 24 (1942), 551–559.

[78]

B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes, Acta Appl. Math., 124 (2013), 73-105.  doi: 10.1007/s10440-012-9771-6.

[79]

B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x.

[80]

B. Ricceri and A. Villani, Separability and Scorza-Dragoni's property, Matematiche (Catania), 37 (1982), 156–161 (1985).

[81]

V. I. Smirnov, A Course of Higher Mathematics. Vol. IV [Integral Equations and Partial Differential Equations], Translated by D. E. Brown; translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964.

[82]

H. R. Thieme, Well-posedness of physiologically structured population models for daphnia magna. How biological concepts can benefit by abstract mathematical analysis, J. Math. Biol., 26 (1988), 299-317.  doi: 10.1007/BF00277393.

[83]

S. Tuljapurkar and H. Caswell, Structured-Population Models in Marine, Terrestrial, and Freshwater Systems, Population and Community Biology Series, Springer, Boston, MA, 1997. doi: 10.1007/978-1-4615-5973-3.

[84]

A. Ulikowska, An age-structured two-sex model in the space of Radon measures: well posedness, Kinet. Relat. Models, 5 (2012), 873-900.  doi: 10.3934/krm.2012.5.873.

[85]

C. Walker, Positive equilibrium solutions for age- and spatially-structured population models, SIAM J. Math. Anal., 41 (2009), 1366-1387.  doi: 10.1137/090750044.

[86]

W. Walter, Ordinary Differential Equations, vol. 182 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998, Translated from the sixth German (1996) edition by Russell Thompson, Readings in Mathematics. doi: 10.1007/978-1-4612-0601-9.

[87]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, vol. 89 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1985.

[88]

G. F. Webb, Semigroup methods in population dynamics: Proliferating cell populations, in Semigroup Theory and Applications (Trieste, 1987), vol. 116 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1989,441–449.

[89]

G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology (eds. P. Magal and S. Ruan), vol. 1936 of Lecture Notes in Math., Springer, Berlin, 2008, 1–49. doi: 10.1007/978-3-540-78273-5_1.

[90]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I, Springer-Verlag, New York, 1986, Fixed-point theorems, Translated from the German by Peter R. Wadsack. doi: 10.1007/978-1-4612-4838-5.

[1]

Qihua Huang, Hao Wang. A toxin-mediated size-structured population model: Finite difference approximation and well-posedness. Mathematical Biosciences & Engineering, 2016, 13 (4) : 697-722. doi: 10.3934/mbe.2016015

[2]

Peixuan Weng, Xiao-Qiang Zhao. Spatial dynamics of a nonlocal and delayed population model in a periodic habitat. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 343-366. doi: 10.3934/dcds.2011.29.343

[3]

Chris Cosner, Andrew L. Nevai. Spatial population dynamics in a producer-scrounger model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1591-1607. doi: 10.3934/dcdsb.2015.20.1591

[4]

G. Buffoni, S. Pasquali, G. Gilioli. A stochastic model for the dynamics of a stage structured population. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 517-525. doi: 10.3934/dcdsb.2004.4.517

[5]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229

[6]

Manas Bhatnagar, Hailiang Liu. Well-posedness and critical thresholds in a nonlocal Euler system with relaxation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5271-5289. doi: 10.3934/dcds.2021076

[7]

Yuanxian Hui, Genghong Lin, Jianshe Yu, Jia Li. A delayed differential equation model for mosquito population suppression with sterile mosquitoes. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4659-4676. doi: 10.3934/dcdsb.2020118

[8]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315

[9]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

[10]

Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071

[11]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735

[12]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[13]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036

[14]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241

[15]

A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469

[16]

Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527

[17]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure and Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219

[18]

Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637

[19]

Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737

[20]

Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4231-4253. doi: 10.3934/dcdsb.2021225

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (287)
  • HTML views (224)
  • Cited by (0)

Other articles
by authors

[Back to Top]