September 2017, 10(3): 669-688. doi: 10.3934/krm.2017027

Finite range method of approximation for balance laws in measure spaces

Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

* Corresponding author: Piotr Gwiazda

Received  April 2016 Revised  July 2016 Published  December 2016

In the following paper we reconsider a numerical scheme recently introduced in [10]. The method was designed for a wide class of size structured population models with a nonlocal term describing the birth process. Despite its numerous advantages it features the exponential growth in time of the number of particles constituting the numerical solution. We introduce a new algorithm free from this inconvenience. The improvement is based on the application the Finite Range Approximation to the nonlocal term. We prove the convergence of the derived method and provide the rate of its convergence. Moreover, the results are illustrated by numerical simulations applied to various test cases.

Citation: Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
References:
[1]

L. M. AbiaO. Angulo and J. C. Lopez-Marcos, Numerical schemes for a size-structured cell population model with equal fission, Mathematical and Computer Modelling, 50 (2009), 653-664. doi: 10.1016/j.mcm.2009.05.023.

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, Journal of Differential Equations, 217 (2005), 431-455. doi: 10.1016/j.jde.2004.12.013.

[3]

A. S. AcklehB. G. Fitzpatrick and H. R. Thieme, Rate distributions and survival of the fittest: A formulation on the space of measures, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 917-928. doi: 10.3934/dcdsb.2005.5.917.

[4]

A. L. BertozziT. KolokonikovH. Sun and D. Uminsky, Stability of ring patterns arising from 2d particle interactions, Physical Review E, 84 (2011).

[5]

C. K. Birdsal and A. B. Langdon, Plasma Physics Via Computer Simulation, McGraw-Hill, New York, 1985.

[6]

A. BrannstromL. Carlsson and D. Simpson, On the convergence of the escalator boxcar train, SIAM J. Numer. Anal., 51 (2013), 3213-3231. doi: 10.1137/120893215.

[7]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford Leture Series in Mathematics and its Applications vol. 20, Oxford University Press, 2000.

[8]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539. doi: 10.1142/S0218202511005131.

[9]

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

[10]

J.A. CarrilloP. Gwiazda and A. Ulikowska, Splitting particle methods for structured population models: Convergence and applications, Math. Models Methods Appl. Sci., 24 (2014), 2171-2197. doi: 10.1142/S0218202514500183.

[11]

R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009), 733-753. doi: 10.3934/dcds.2009.23.733.

[12]

G. H. Cottet and P. A. Raviart, Particle methods for the one-dimensional Vlasov-Poisson equations, SIAM J. Numer. Anal., 21 (1984), 52-76. doi: 10.1137/0721003.

[13]

A. M. de Roos, Numerical methods for structured population models: The escalator boxcar train, Numerical Methods for Partial Differential Equations, 4 (1988), 173-195. doi: 10.1002/num.1690040303.

[14]

A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Monographs in Population Biology 51, Princeton University Press, Princeton, 2013.

[15]

O. Diekmann and Ph. 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.

[16]

M. R. D'Orsogna, Y. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006).

[17]

J. EversS. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015), 1068-1097. doi: 10.1016/j.jde.2015.02.037.

[18]

K. Ganguly and H. D. Victory Jr, On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal., 26 (1989), 249-288. doi: 10.1137/0726015.

[19]

J. GoodmanT. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the 2-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 415-430. doi: 10.1002/cpa.3160430305.

[20]

P. GwiazdaJ. JabƚońskiA. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Numerical Methods for Partial Differential Equations, 30 (2014), 1797-1820. doi: 10.1002/num.21879.

[21]

P. GwiazdaG. Jamróz and A. Marciniak-Czochra, Models of discrete and continuous cell differentiation in the framework of transport equation, SIAM Journal on Mathematical Analysis, 44 (2012), 1103-1133. doi: 10.1137/11083294X.

[22]

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

[23]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010), 733-773. doi: 10.1142/S021989161000227X.

[24]

F. H. Harlow, The particle-in-cell computing method for fluid dynamics, Methods in computational physics, 3 (1964), 319-343.

[25]

D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (BGK) equation, SIAM J. Numer. Anal., 33 (1996), 2099-2119. doi: 10.1137/S0036142994266856.

[26]

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.

[27]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.

[28]

P. A. Raviart, An analysis of particle methods, Numerical Methods in Fluid Dynamics, Lecture Notes in Math. , Springer, Berlin, 1127 (1985), 243-324. doi: 10.1007/BFb0074532.

[29]

E. Tadmor, A review of numerical methods for nonlinear partial differential equations, Bulletin of the American Mathematical Society, 49 (2012), 507-554. doi: 10.1090/S0273-0979-2012-01379-4.

[30]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate studies in mathematics, American Mathematical Society, 2003. doi: 10.1007/b12016.

[31]

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

[32]

M. Westdickenberg and J. Wilkening, Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations, M2AN Math. Model. Numer. Anal., 44 (2010), 133-166. doi: 10.1051/m2an/2009043.

show all references

References:
[1]

L. M. AbiaO. Angulo and J. C. Lopez-Marcos, Numerical schemes for a size-structured cell population model with equal fission, Mathematical and Computer Modelling, 50 (2009), 653-664. doi: 10.1016/j.mcm.2009.05.023.

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, Journal of Differential Equations, 217 (2005), 431-455. doi: 10.1016/j.jde.2004.12.013.

[3]

A. S. AcklehB. G. Fitzpatrick and H. R. Thieme, Rate distributions and survival of the fittest: A formulation on the space of measures, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 917-928. doi: 10.3934/dcdsb.2005.5.917.

[4]

A. L. BertozziT. KolokonikovH. Sun and D. Uminsky, Stability of ring patterns arising from 2d particle interactions, Physical Review E, 84 (2011).

[5]

C. K. Birdsal and A. B. Langdon, Plasma Physics Via Computer Simulation, McGraw-Hill, New York, 1985.

[6]

A. BrannstromL. Carlsson and D. Simpson, On the convergence of the escalator boxcar train, SIAM J. Numer. Anal., 51 (2013), 3213-3231. doi: 10.1137/120893215.

[7]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford Leture Series in Mathematics and its Applications vol. 20, Oxford University Press, 2000.

[8]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539. doi: 10.1142/S0218202511005131.

[9]

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

[10]

J.A. CarrilloP. Gwiazda and A. Ulikowska, Splitting particle methods for structured population models: Convergence and applications, Math. Models Methods Appl. Sci., 24 (2014), 2171-2197. doi: 10.1142/S0218202514500183.

[11]

R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009), 733-753. doi: 10.3934/dcds.2009.23.733.

[12]

G. H. Cottet and P. A. Raviart, Particle methods for the one-dimensional Vlasov-Poisson equations, SIAM J. Numer. Anal., 21 (1984), 52-76. doi: 10.1137/0721003.

[13]

A. M. de Roos, Numerical methods for structured population models: The escalator boxcar train, Numerical Methods for Partial Differential Equations, 4 (1988), 173-195. doi: 10.1002/num.1690040303.

[14]

A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Monographs in Population Biology 51, Princeton University Press, Princeton, 2013.

[15]

O. Diekmann and Ph. 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.

[16]

M. R. D'Orsogna, Y. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006).

[17]

J. EversS. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015), 1068-1097. doi: 10.1016/j.jde.2015.02.037.

[18]

K. Ganguly and H. D. Victory Jr, On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal., 26 (1989), 249-288. doi: 10.1137/0726015.

[19]

J. GoodmanT. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the 2-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 415-430. doi: 10.1002/cpa.3160430305.

[20]

P. GwiazdaJ. JabƚońskiA. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Numerical Methods for Partial Differential Equations, 30 (2014), 1797-1820. doi: 10.1002/num.21879.

[21]

P. GwiazdaG. Jamróz and A. Marciniak-Czochra, Models of discrete and continuous cell differentiation in the framework of transport equation, SIAM Journal on Mathematical Analysis, 44 (2012), 1103-1133. doi: 10.1137/11083294X.

[22]

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

[23]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010), 733-773. doi: 10.1142/S021989161000227X.

[24]

F. H. Harlow, The particle-in-cell computing method for fluid dynamics, Methods in computational physics, 3 (1964), 319-343.

[25]

D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (BGK) equation, SIAM J. Numer. Anal., 33 (1996), 2099-2119. doi: 10.1137/S0036142994266856.

[26]

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.

[27]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.

[28]

P. A. Raviart, An analysis of particle methods, Numerical Methods in Fluid Dynamics, Lecture Notes in Math. , Springer, Berlin, 1127 (1985), 243-324. doi: 10.1007/BFb0074532.

[29]

E. Tadmor, A review of numerical methods for nonlinear partial differential equations, Bulletin of the American Mathematical Society, 49 (2012), 507-554. doi: 10.1090/S0273-0979-2012-01379-4.

[30]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate studies in mathematics, American Mathematical Society, 2003. doi: 10.1007/b12016.

[31]

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

[32]

M. Westdickenberg and J. Wilkening, Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations, M2AN Math. Model. Numer. Anal., 44 (2010), 133-166. doi: 10.1051/m2an/2009043.

Figure 1.  Function $f$ and its Finite Range Approximation
Figure 2.  Function $f^{\varepsilon}$ and its approximation $\bar f^{\varepsilon}$
Figure 3.  Order of convergence
Table 1.  The error of the FRA method for $\varepsilon=0.1$
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.1)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $0.0008865973$ -
$5.0000\cdot 10^{-2}$ $0.0005886514$ 0.59086543
$2.5000\cdot 10^{-2}$ $0.0003434678$ 0.77723882
$1.2500\cdot 10^{-2}$ $0.0002668756$ 0.36400752
$6.2500\cdot 10^{-3}$ $0.0002400045$ 0.15310595
$3.1250\cdot 10^{-3}$ $0.0002258416$ 0.08775007
$1.5625\cdot 10^{-3}$ $0.0002187803$ 0.04582869
$7.8125\cdot 10^{-4}$ $0.0002154824$ 0.02191237
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.1)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $0.0008865973$ -
$5.0000\cdot 10^{-2}$ $0.0005886514$ 0.59086543
$2.5000\cdot 10^{-2}$ $0.0003434678$ 0.77723882
$1.2500\cdot 10^{-2}$ $0.0002668756$ 0.36400752
$6.2500\cdot 10^{-3}$ $0.0002400045$ 0.15310595
$3.1250\cdot 10^{-3}$ $0.0002258416$ 0.08775007
$1.5625\cdot 10^{-3}$ $0.0002187803$ 0.04582869
$7.8125\cdot 10^{-4}$ $0.0002154824$ 0.02191237
Table 2.  The error of the FRA method for $\varepsilon=0.0125$
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.0125)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}})/ \log 2$
$1.0000\cdot 10^{-1}$ $6.702793\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.255311\cdot 10^{-4}$ 0.6554979
$2.5000\cdot 10^{-2}$ $1.757573\cdot 10^{-4}$ 1.2756799
$1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 0.9135408
$6.2500\cdot 10^{-3}$ $5.658711\cdot 10^{-5}$ 0.7214983
$3.1250\cdot 10^{-3}$ $4.055238\cdot 10^{-5}$ 0.4806869
$1.5625\cdot 10^{-3}$ $3.332588\cdot 10^{-5}$ 0.2831438
$7.8125\cdot 10^{-4}$ $2.973577\cdot 10^{-5}$ 0.1644434
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.0125)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}})/ \log 2$
$1.0000\cdot 10^{-1}$ $6.702793\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.255311\cdot 10^{-4}$ 0.6554979
$2.5000\cdot 10^{-2}$ $1.757573\cdot 10^{-4}$ 1.2756799
$1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 0.9135408
$6.2500\cdot 10^{-3}$ $5.658711\cdot 10^{-5}$ 0.7214983
$3.1250\cdot 10^{-3}$ $4.055238\cdot 10^{-5}$ 0.4806869
$1.5625\cdot 10^{-3}$ $3.332588\cdot 10^{-5}$ 0.2831438
$7.8125\cdot 10^{-4}$ $2.973577\cdot 10^{-5}$ 0.1644434
Table 3.  The error of the FRA method for $\varepsilon=0.0015625$
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.0015625)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $6.616854\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.139056\cdot 10^{-4}$ 0.6768439
$2.5000\cdot 10^{-2}$ $1.570582\cdot 10^{-4}$ 1.3980025
$1.2500\cdot 10^{-2}$ $7.418803\cdot 10^{-5}$ 1.0820407
$6.2500\cdot 10^{-3}$ $3.649949\cdot 10^{-5}$ 1.0820407
$3.1250\cdot 10^{-3}$ $1.883518\cdot 10^{-5}$ 0.9544464
$1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 0.8669200
$7.8125\cdot 10^{-4}$ $6.277910\cdot 10^{-6}$ 0.7181537
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.0015625)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $6.616854\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.139056\cdot 10^{-4}$ 0.6768439
$2.5000\cdot 10^{-2}$ $1.570582\cdot 10^{-4}$ 1.3980025
$1.2500\cdot 10^{-2}$ $7.418803\cdot 10^{-5}$ 1.0820407
$6.2500\cdot 10^{-3}$ $3.649949\cdot 10^{-5}$ 1.0820407
$3.1250\cdot 10^{-3}$ $1.883518\cdot 10^{-5}$ 0.9544464
$1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 0.8669200
$7.8125\cdot 10^{-4}$ $6.277910\cdot 10^{-6}$ 0.7181537
Table 4.  The error of the FRA method for $\varepsilon=\Delta t$
$\Delta t$ ${\rm{Err}}(1,\Delta t,\Delta t)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,2\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $8.865973\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.797352\cdot 10^{-4}$ 0.8860407
$2.5000\cdot 10^{-2}$ $1.991736\cdot 10^{-4}$ 1.2682122
$1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 1.0939823
$6.2500\cdot 10^{-3}$ $4.477800\cdot 10^{-5}$ 1.0591816
$3.1250\cdot 10^{-3}$ $2.162411\cdot 10^{-5}$ 1.0501496
$1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 1.0661308
$7.8125\cdot 10^{-4}$ $4.763338\cdot 10^{-6}$ 1.1164650
$\Delta t$ ${\rm{Err}}(1,\Delta t,\Delta t)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,2\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $8.865973\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.797352\cdot 10^{-4}$ 0.8860407
$2.5000\cdot 10^{-2}$ $1.991736\cdot 10^{-4}$ 1.2682122
$1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 1.0939823
$6.2500\cdot 10^{-3}$ $4.477800\cdot 10^{-5}$ 1.0591816
$3.1250\cdot 10^{-3}$ $2.162411\cdot 10^{-5}$ 1.0501496
$1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 1.0661308
$7.8125\cdot 10^{-4}$ $4.763338\cdot 10^{-6}$ 1.1164650
[1]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233

[2]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056

[3]

L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203

[4]

Agnieszka Ulikowska. An age-structured two-sex model in the space of radon measures: Well posedness. Kinetic & Related Models, 2012, 5 (4) : 873-900. doi: 10.3934/krm.2012.5.873

[5]

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

[6]

Kazuhiro Ishige. On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 521-546. doi: 10.3934/dcds.1995.1.521

[7]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[8]

Simona Fornaro, Stefano Lisini, Giuseppe Savaré, Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1675-1707. doi: 10.3934/dcds.2012.32.1675

[9]

Bruno Buonomo, Deborah Lacitignola. On the stabilizing effect of cannibalism in stage-structured population models. Mathematical Biosciences & Engineering, 2006, 3 (4) : 717-731. doi: 10.3934/mbe.2006.3.717

[10]

Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049

[11]

Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627

[12]

Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299

[13]

M. M. Rao. Integration with vector valued measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5429-5440. doi: 10.3934/dcds.2013.33.5429

[14]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943

[15]

Andrea Tosin, Paolo Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Networks & Heterogeneous Media, 2011, 6 (3) : 561-596. doi: 10.3934/nhm.2011.6.561

[16]

Yingli Pan, Ying Su, Junjie Wei. Bistable waves of a recursive system arising from seasonal age-structured population models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 511-528. doi: 10.3934/dcdsb.2018184

[17]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

[18]

Kathryn Lindsey, Rodrigo Treviño. Infinite type flat surface models of ergodic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5509-5553. doi: 10.3934/dcds.2016043

[19]

Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77

[20]

David Cowan. Rigid particle systems and their billiard models. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 111-130. doi: 10.3934/dcds.2008.22.111

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (5)
  • HTML views (2)
  • Cited by (0)

[Back to Top]