Finite difference schemes for a structured population model in the space of measures

Azmy S. Ackleh; Rainey Lyons; Nicolas Saintier; Azmy S. Ackleh; Rainey Lyons; Nicolas Saintier

doi:10.3934/mbe.2020039

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 747-775. doi: 10.3934/mbe.2020039

Previous Article Next Article

Research article Special Issues

Finite difference schemes for a structured population model in the space of measures

1.
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA
2.
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (1428) Pabellón I-Ciudad Universitaria-Buenos Aires-Argentina

Received: 28 July 2019 Accepted: 24 October 2019 Published: 31 October 2019

We present two finite-difference methods for approximating solutions to a structured population model in the space of non-negative Radon Measures. The first method is a first-order upwind-based scheme and the second is high-resolution method of second-order. We prove that the two schemes converge to the solution in the Bounded-Lipschitz norm. Several numerical examples demonstrating the order of convergence and behavior of the schemes around singularities are provided. In particular, these numerical results show that for smooth solutions the upwind and high-resolution methods provide a first-order and a second-order approximation, respectively. Furthermore, for singular solutions the second-order high-resolution method is superior to the first-order method.
- finite difference schemes,
- high-resolution methods,
- structured populations,
- non-negative Radon measures,
- bounded-Lipschitz norm
Citation: Azmy S. Ackleh, Rainey Lyons, Nicolas Saintier. Finite difference schemes for a structured population model in the space of measures[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 747-775. doi: 10.3934/mbe.2020039

Related Papers:

Abstract

We present two finite-difference methods for approximating solutions to a structured population model in the space of non-negative Radon Measures. The first method is a first-order upwind-based scheme and the second is high-resolution method of second-order. We prove that the two schemes converge to the solution in the Bounded-Lipschitz norm. Several numerical examples demonstrating the order of convergence and behavior of the schemes around singularities are provided. In particular, these numerical results show that for smooth solutions the upwind and high-resolution methods provide a first-order and a second-order approximation, respectively. Furthermore, for singular solutions the second-order high-resolution method is superior to the first-order method.

References

[1]	L. C. Evans, Partial Differential Equations, American Mathematical Society, 2010.
[2]	B. Perthame, Transport Equations in Biology, Birkhäuser Basel, 2007.
[3]	T Hillen and K. P. Hadeler, Hyperbolic systems and transport equations in mathematical biology, In Analysis and numerics for conservation laws, Springer, (2005), 257-279.
[4]	J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Appl. Math., 123 (2013), 141-156.
[5]	J. Metz and O. E. Diekmann, The dynamics of physiologically structured populations (lecture notes in biomathematics), Lecture Notes Biomath., (1986), 68.
[6]	J. Carrillo, R. M. Colombo, P. Gwiazda, et al., Structured populations, cell growth and measure valued balance laws, J. Differ. Equations, 252 (2012), 3245-3277.
[7]	M. Iannelli, Mathematical theory of age-structured population dynamics, Giardini editori e stampatori in Pisa, 1995.
[8]	A. S. Ackleh and B. Ma, A second-order high-resolution scheme for a juvenile-adult model of amphibians, Numer. Func. Anal. Opt., 34 (2013), 365-403.
[9]	J. Shen, C. W. Shu and M. Zhang, High resolution schemes for a hierarchical size-structured model, SIAM J. Numer. Anal., 45 (2007), 352-370.
[10]	A. S. Ackleh, J. Carter, K. Deng, et al., Fitting a structured juvenile-adult model for green tree frogs to population estimates from capture-mark-recapture field data, Bull. Math. Biol., 74 (2012), 641-665.
[11]	K. Deng and Y. Wu, Extinction and uniform strong persistence of a size-structured population model, Discrete Cont. Dyn-S, 22 (2017), 831-840.
[12]	R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, fishery investigations series ii, vol. xix, ministry of agriculture, Fisheries Food, 1-957, 1957.
[13]	W. E. Ricker, Stock and recruitment, J. Fish. Board Can., 11 (1954), 559-623.
[14]	D. Pauly, G. R. Morgan, et al., Length-based methods in fisheries research, volume 13. WorldFish, 1987.
[15]	P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differ. Equations, 248 (2010), 2703-2735.
[16]	P. Gwiazda, J. Jablonski, A. Marciniak-Czochra, et al., Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded lipschitz distance, Numer. Meth. Part. D. E., 30 (2014), 1797-1820. doi: 10.1002/num.21879
[17]	J Carrillo, P. Gwiazda and A. Ulikowska, Splitting-particle methods for structured population models: Convergence and applications, Math. Mod. Meth. Appl. S., 24 (2014), 2171-2197.
[18]	A. de Roos, Numerical methods for structured population models, Numer. Meth. Part. D. E., 4 (2005), 173-195.
[19]	J. Smoller, Shock waves and reaction-diffusion equations, volume 258. Springer Science & Business Media, 2012.
[20]	R. LeVeque, Numerical Mehtods for Conservation Laws. Springer Basel AG, 1992.
[21]	A. S. Ackleh, V. K. Chellamuthu and K. Ito, Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model, Math. Biosci. Eng., 12 (2015), 233-258.
[22]	J. Jabłoński and D. Wrzosek, Measure-valued solutions to size-structured population model of prey controlled by optimally foraging predator harvester, Math. Mod. Meth. Appl. S., 29, (2019), 1657-1689.
[23]	H. Federer, Geometric measure theory, Springer, 2014.
[24]	H. Federer, Colloquium lectures on geometric measure theory, B. Amer. Math. Soc., 84 (1978), 291-338.
[25]	R. M. Dudley, Distances of probability measures and random variables, In Selected Works of RM Dudley, Springer, (2010), 28-37.
[26]	J. H. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differ. Equations, 259 (2015), 1068-1097.
[27]	R. Fortet and E. Mourier, Convergence de la répartition empirique vers la répartition théorique, In Annales scientifiques de l'École Normale Supérieure, 70 (1953), 267-285.
[28]	A. Lasota, J. Myjak and T. Szarek, Markov operators with a unique invariant measure, J. Math. Anal. Appl., 276 (2002), 343-356.
[29]	P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space, Positivity, 22 (2017), 105-138.
[30]	P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbol. Differ. Eq., 07 (2010), 733-773.
[31]	A. S. Ackleh and R. Miller, A numerical method for a nonlinear structured population model with an indefinite growth rate coupled with the environment, Numer. Meth. Part. D. E., 35 (2019), 2348-2374.
[32]	J. L. Kelley, General Topology, D. Van Nostrand Company Inc., 1955.
[33]	C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Physics, 77 (1988), 439-471.
[34]	J. Jabłoński and A. Marciniak-Czochra, Efficient algorithms computing distances between radon measures on r, preprint, arXiv:1304.3501.
[35]	N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation^* 1, USSR Comp. Math. Math. Phys., 16 (1976), 105-119.
[36]	F. Delarue, F. Lagoutière and N. Vauchelet, Convergence order of upwind type schemes for transport equations with discontinuous coefficients, J. Math. Pures Appl., 108 (2017), 918-951.
[37]	B. A. Menge, Competition for food between two intertidal starfish species and its effect on body size and feeding, Ecology, 53 (1972), 635-644.
[38]	M. Huss, A. Gårdmark, A. Van Leeuwen, et al., Size-and food-dependent growth drives patterns of competitive dominance along productivity gradients, Ecology, 93 (2012), 847-857.
[39]	A. S. Ackleh, J. Cleveland and H. R. Thieme, Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces, J. Differ. Equations, 261 (2016), 1472-1505.
[40]	A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Differ. Equations, 217 (2005), 431-455.
[41]	P. Gwiazda, S. C. Hille, K. Łyczek, et al., Differentiability in perturbation parameter of measure solutions to perturbed transport equation, Kinet. Relat. Mod., 12 (2019), 1093-1108.
[42]	J. Skrzeczkowski, Measure solutions to perturbed structured population models-differentiability with respect to perturbation parameter, preprint arXiv:1812.01747.
[43]	A. S. Ackleh, N. Saintier and J. Skrzeczkowski, Sensitivity equations for measure-valued solutions to transport equations, Math. Biosci. Eng., 17 (2020), 514-537.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)