We present a structured coagulation-fragmentation model which describes the population dynamics of oceanic phytoplankton. This model is formulated on the space of Radon measures equipped with the bounded Lipschitz norm and unifies the study of the discrete and continuous coagulation-fragmentation models. We prove that the model is well-posed and show it can reduce down to the classic discrete and continuous coagulation-fragmentation models. To understand the interplay between the physical processes of coagulation and fragmentation and the biological processes of growth, reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary solutions are absolutely continuous under some conditions on model parameters. We develop a semi-discrete approximation scheme which conserves mass and prove its convergence to the unique weak solution. We then use the scheme to perform numerical simulations for the model.
Keywords: Coagulation-fragmentation equations, structured populations, non-negative Radon measures, Bounded-Lipschitz norm, semi-discrete schemes, conservation of Mass
@article{M2AN_2021__55_5_2473_0,
author = {Ackleh, Azmy S. and Lyons, Rainey and Saintier, Nicolas},
title = {A structured coagulation-fragmentation equation in the space of radon measures: {Unifying} discrete and continuous models},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2473--2501},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {5},
doi = {10.1051/m2an/2021061},
mrnumber = {4330732},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021061/}
}
TY - JOUR AU - Ackleh, Azmy S. AU - Lyons, Rainey AU - Saintier, Nicolas TI - A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 2473 EP - 2501 VL - 55 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021061/ DO - 10.1051/m2an/2021061 LA - en ID - M2AN_2021__55_5_2473_0 ER -
%0 Journal Article %A Ackleh, Azmy S. %A Lyons, Rainey %A Saintier, Nicolas %T A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 2473-2501 %V 55 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021061/ %R 10.1051/m2an/2021061 %G en %F M2AN_2021__55_5_2473_0
Ackleh, Azmy S.; Lyons, Rainey; Saintier, Nicolas. A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 2473-2501. doi: 10.1051/m2an/2021061
[1] , Parameter estimation in a structured algal coagulation-fragmentation model. Nonlinear Anal.: Theory Methods App. 28 (1997) 837–854. | MR | Zbl | DOI
[2] and , A nonautonomous juvenile-adult model: well-posedness and long-time behavior via a comparison principle. SIAM J. Appl. Math. 69 (2009) 1644–1661. | MR | Zbl | DOI
[3] and , Modeling aggregation and growth processes in an algal population model: analysis and computations. J. Math. Biol. 35 (1997) 480–502. | MR | Zbl | DOI
[4] and , A model for the interaction of phytoplankton aggregates and the environment: approximation and parameter estimation. Inverse Prob. Sci. Eng. 26 (2017) 152–182. | MR | DOI
[5] and , Well-posedness for a system of transport and diffusion equations in measure spaces. J. Math. Anal. App. 492 (2020) 1–32. | MR | DOI
[6] and , Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete Continuous Dyn. Syst. Ser. B 26 (2021) 1469. | MR | DOI
[7] , and , Existence-uniqueness and monotone approximation for a phytoplankton-zooplankton aggregation model. Z. Angew. Math. Phys. 57 (2006) 733–749. | MR | Zbl | DOI
[8] , , , , and , 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. | MR | Zbl | DOI
[9] , and , Finite difference schemes for a structured population model in the space of measures. Math. Biosci. Eng. MBE 17 (2020) 747–775. | MR | DOI
[10] and , The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation. J. Stat. Phys. 61 (1990) 203–234. | MR | Zbl | DOI
[11] and , Coagulation, fragmentation and growth processes in a size structured population. Discrete Continuous Dyn. Syst.-Ser. B 11 (2009) 563–585. | MR | Zbl | DOI
[12] and , On the dynamics of exploited fish populations. In: Fisheries and Food. Vol. XIX of Fishery Investigations Series II. Ministry of Agriculture (1957) 1–957.
[13] and , Note on the kinetics of systems manifesting simultaneous polymerization-depolymerization phenomena. J. Phys. Chem. 49 (1945) 77–80. | DOI
[14] and , Convergence of a finite volume scheme for coagulation-fragmentation equations. Math. Comput. 77 (2008) 851–882. | MR | Zbl | DOI
[15] and , Particle aggregation. Ann. Rev. Mar. Sci. 1 (2009) 65–90. | DOI
[16] , Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance. J. Stat. Phys. 129 (2007) 1–26. | MR | Zbl | DOI
[17] , and , Measure solutions for some models in population dynamics. Acta Appl. Math. 123 (2013) 141–156. | MR | Zbl | DOI
[18] , , and , Structured populations, cell growth and measure valued balance laws. J. Differ. Equ. 252 (2012) 3245–3277. | MR | Zbl | DOI
[19] and , Stably coalescent stochastic froths. Adv. Appl. Probab. 31 (1999) 199–219. | MR | Zbl | DOI
[20] , , and , Relative entropy method for measure solutions of the growth-fragmentation equation. SIAM J. Math. Anal. 50 (2018) 5811–5824. | MR | DOI
[21] and , Extinction and uniform strong persistence of a size-structured population model. Discrete Continuous Dyn. Syst. Ser. B 22 (2017) 831–840. | MR | DOI
[22] , Distances of probability measures and random variables. In: Selected Works of RM Dudley. Springer (2010) 28–37. | MR | DOI
[23] and , Approximative solution of the continuous coagulation-fragmentation equation. Stochastic Anal. App. 18 (2000) 921–948. | MR | Zbl | DOI
[24] , and , Mild solutions to a measure-valued mass evolution problem with flux boundary conditions. J. Differ. Equ. 259 (2015) 1068–1097. | MR | DOI
[25] , Geometric Measure Theory. Springer (2014). | Zbl | MR
[26] , Colloquium lectures on geometric measure theory. Bull. Am. Math. Soc. 84 (1978) 291–338. | MR | Zbl | DOI
[27] and , Convergence de la Répartition Empirique Vers la Répartition Théorique. Annales scientifiques de l’École Normale Supérieure 70 (1953) 267–285. | MR | Zbl | Numdam | DOI
[28] , and , The continuous coagulation equation with multiple fragmentation. J. Math. Anal. App. 374 (2011) 71–87. | MR | Zbl | DOI
[29] , and , Weak solutions to the continuous coagulation equation with multiple fragmentation. Nonlinear Anal. Theory Methods App. 75 (2012) 2199–2208. | MR | Zbl | DOI
[30] and , Structured population equations in metric spaces. J. Hyperbolic Differ. Equ. 07 (2010) 733–773. | MR | Zbl | DOI
[31] , and , A Nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients. J. Differ. Equ. 248 (2010) 2703–2735. | MR | Zbl | DOI
[32] , and , Measures under the flat norm as ordered normed vector space. Positivity 22 (2017) 105–138. | MR | DOI
[33] , Analytical solutions of Smoluchowski’s coagulation equation. J. Phys. A 25 (1992) 3763–3771. | MR | Zbl | DOI
[34] and , Efficient algorithms computing distances between Radon measures on ℝ. Preprint (2013). | arXiv
[35] and , Measure-valued solutions to size-structured population model of prey controlled by optimally foraging predator harvester. Math. Models Methods Appl. Sci. 29 (2019) 1657–1689. | MR | DOI
[36] , A model of the formation of marine algal flocs by physical coagulation processes. Deep Sea Res. Part A. Oceanogr. Res. Papers 37 (1990) 1197–1211. | DOI
[37] and , Effect of coagulation on nutrient and light limitation of an algal bloom. Limnol. Oceanogr. 37 (1992) 77–89. | DOI
[38] , Existence of getting solutions for coagulation-fragmentation equations. Commun. Math. Phys. 194 (1998) 541–567. | MR | Zbl | DOI
[39] and , Numerical simulation of solutions and moments of the Smoluchowski coagulation equation., Preprint (2013). | arXiv
[40] , Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math. Models Methods Appl. Sci. 27 (2004) 703–721. | MR | Zbl | DOI
[41] , and , Markov operators with a unique invariant measure. J. Math. Anal. App. 276 (2002) 343–356. | MR | Zbl | DOI
[42] , On a class of continuous coagulation–fragmentation equations. J. Differ. Equ. 167 (2000) 245–274. | MR | Zbl | DOI
[43] and , Global existence for the discrete diffusive coagulation–fragmentation equations in . Rev. Mat. Iberoam. 18 (2002) 731–745. | MR | Zbl | DOI
[44] and , From the discrete to the continuous coagulation fragmentation equations. Proc. R. Soc. Edinburgh Sect. A: Math. 132 (2002) 1219–1248. | MR | Zbl | DOI
[45] and , On coalescence equations and related models, edited by , and . In: Modeling and Computational Methods for Kinetic Equations. Boston, Birkhäuser (2004) 321–356. | MR | Zbl | DOI
[46] , and , A high order positivity preserving DG method for coagulation–fragmentation equations. SIAM J. Sci. Comput. 41 (2019) 448–465. | MR | DOI
[47] , and , An existence and uniqueness result for a coagulation and multiple-fragmentation equation. SIAM J. Math. Anal. 28 (1997) 1173–1190. | MR | Zbl | DOI
[48] , A scalar transport equation. Trans. Am. Math. Soc. 85 (1957) 547–560. | MR | Zbl | DOI
[49] , Zur Allgemeinen Theorie der Raschen Koagulation. Fortschrittsberichte über Kolloide und Polymere 27 (1928) 223–250. | DOI
[50] , Smoluchowski’s coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9 (1999) 78–109. | MR | Zbl | DOI
[51] and , Length-Based Methods in Fisheries Research. WorldFish 13 (1987).
[52] , Stock and recruitment. J. Fisheries Board Can. 11 (1954) 559–623. | DOI
[53] and , Fragmentation-coagulation models of phytoplankton. Bull. Polish Acad. Sci. Math. 54 (2006) 175–191. | MR | Zbl | DOI
[54] , and , Adomian decomposition method for solving fragmentation and aggregation population balance equations. J. Appl. Math. Comput. 48 (2014) 265–292. | MR | Zbl | DOI
[55] , Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Phys. Z. 17 (1916) 557–571, 585–599.
[56] , A global existence theorem for the general coagulation–fragmentation equation with unbounded kernels. Math. Methods Appli. Sci. 11 (1989) 627–648. | MR | Zbl | DOI
Cité par Sources :
