{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T01:55:19Z","timestamp":1778637319298,"version":"3.51.4"},"reference-count":43,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T00:00:00Z","timestamp":1600214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP200100345"],"award-info":[{"award-number":["DP200100345"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A standard reaction\u2013diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction\u2013subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction\u2013diffusion equations. In this paper, we formulate clear examples of reaction\u2013subdiffusion systems, based on; equal birth and death rate dynamics, Fisher\u2013Kolmogorov, Petrovsky and Piskunov (Fisher\u2013KPP) equation dynamics, and Fitzhugh\u2013Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction\u2013diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.<\/jats:p>","DOI":"10.3390\/e22091035","type":"journal-article","created":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T10:30:12Z","timestamp":1600252212000},"page":"1035","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Time Fractional Fisher\u2013KPP and Fitzhugh\u2013Nagumo Equations"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6688-9346","authenticated-orcid":false,"given":"Christopher N.","family":"Angstmann","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, UNSW, Sydney 2052 NSW, Australia"}]},{"given":"Bruce I.","family":"Henry","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, UNSW, Sydney 2052 NSW, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,16]]},"reference":[{"key":"ref_1","first-page":"114","article-title":"Diffusion and ecological problems: Mathematical models","volume":"10","author":"Okubo","year":"1980","journal-title":"Biomathematics"},{"key":"ref_2","unstructured":"Britton, N.F. (1986). Reaction-Diffusion Equations and Their Applications to Biology, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (2003). Mathematical Biology. II Spatial Models and Biomedical Applications, Springer.","DOI":"10.1007\/b98869"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1109\/MCS.2009.932926","article-title":"Modeling and analysis of mass-action kinetics","volume":"29","author":"Chellaboina","year":"2009","journal-title":"IEEE Control Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1111\/j.1469-1809.1937.tb02153.x","article-title":"The Wave of Advance of Advantageous Genes","volume":"7","author":"Fisher","year":"1937","journal-title":"Ann. Eugen."},{"key":"ref_6","first-page":"1","article-title":"A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem","volume":"1","author":"Kolmogorov","year":"1937","journal-title":"Bull. Mosc. Univ. Math. Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1002\/andp.19053220806","article-title":"On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat","volume":"17","author":"Einstein","year":"1905","journal-title":"Ann. Der Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"R848","DOI":"10.1103\/PhysRevE.51.R848","article-title":"Fractional master equations and fractal time random walks","volume":"51","author":"Hilfer","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4191","DOI":"10.1103\/PhysRevE.53.4191","article-title":"Stochastic foundations of fractional dynamics","volume":"53","author":"Compte","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1063\/1.1704269","article-title":"Random walks on lattices II","volume":"6","author":"Montroll","year":"1965","journal-title":"J. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"562494","DOI":"10.1155\/2011\/562494","article-title":"On Riemann-Liouville and Caputo Derivatives","volume":"2011","author":"Li","year":"2011","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/S0378-4371(99)00469-0","article-title":"Fractional reaction-diffusion","volume":"276","author":"Henry","year":"2000","journal-title":"Phys. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"031116","DOI":"10.1103\/PhysRevE.74.031116","article-title":"Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations","volume":"74","author":"Henry","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0377-0427(00)00288-0","article-title":"Wright functions as scale-invariant solutions of the diffusion wave equation","volume":"118","author":"Gorenflo","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"031102","DOI":"10.1103\/PhysRevE.73.031102","article-title":"Reaction-subdiffusion equations","volume":"73","author":"Sokolov","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1016\/j.physleta.2007.06.044","article-title":"Progagating fronts in reaction-transport systems with memory","volume":"371","author":"Yadav","year":"2007","journal-title":"Phys. Letts. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"021111","DOI":"10.1103\/PhysRevE.77.021111","article-title":"Anomalous subdiffusion with multispecies linear reaction dynamics","volume":"77","author":"Langlands","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"061130","DOI":"10.1103\/PhysRevE.77.061130","article-title":"Anomalous reaction-transport processes: The dynamics beyond the law of mass action","volume":"77","author":"Campos","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"011128","DOI":"10.1103\/PhysRevE.78.011128","article-title":"Front propagation in A + B \u2192 2A reaction under subdiffusion","volume":"78","author":"Froemberg","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"011117","DOI":"10.1103\/PhysRevE.81.011117","article-title":"Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts","volume":"81","author":"Fedotov","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"031115","DOI":"10.1103\/PhysRevE.81.031115","article-title":"Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous-time random walks","volume":"81","author":"Abad","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"061123","DOI":"10.1103\/PhysRevE.82.061123","article-title":"Reaction-subdiffusion model of morphogen gradient formation","volume":"82","author":"Yuste","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1051\/mmnp\/20138202","article-title":"Continuous time random walks with reactions forcing and trapping","volume":"8","author":"Angstmann","year":"2013","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1051\/mmnp\/201611102","article-title":"Mathematical modelling of sub-diffusion reaction systems","volume":"11","author":"Nepomnyashchy","year":"2016","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"032111","DOI":"10.1103\/PhysRevE.102.032111","article-title":"Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains","volume":"102","author":"Abad","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3847","DOI":"10.1016\/j.cnsns.2010.02.007","article-title":"On the solutions of time-fractional reaction-diffusion equations","volume":"15","author":"Rida","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4376","DOI":"10.1007\/s40314-018-0579-5","article-title":"A class of efficient difference method for time fractional reaction-dffusion equation","volume":"37","author":"Zhang","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apnum.2019.01.007","article-title":"The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: Numerical analysis","volume":"140","author":"Li","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1515\/nleng-2018-0057","article-title":"A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses","volume":"8","author":"Prakash","year":"2019","journal-title":"Nonlinear Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-019-1009-z","article-title":"A numerical approach for a class of time-fractional reaction-diffusion equation through exponential B-spline method","volume":"39","author":"Kanth","year":"2020","journal-title":"Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"L17403","DOI":"10.1029\/2008GL034899","article-title":"Tempered anomalous diffusion in heterogeneous systems","volume":"35","author":"Meerschaert","year":"2008","journal-title":"Geophys. Res. Letts."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.jcp.2014.04.024","article-title":"Tempered fractional calculus","volume":"293","author":"Sabzikar","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Mathai, A.M., and Haubold, H.J. (2008). Mittag-Leffler Functions and Fractional Calculus. Special Functions for Applied Scientists, Springer.","DOI":"10.1007\/978-0-387-75894-7"},{"key":"ref_35","first-page":"1","article-title":"Computation of the Mittag-Leffler function E\u03b1,\u03b2(z) and its derivative","volume":"5","author":"Gorenflo","year":"2002","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_36","first-page":"395","article-title":"The G and H functions as symmetrical Fourier kernels","volume":"98","author":"Fox","year":"1961","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_37","first-page":"227","article-title":"On the G-function","volume":"26","author":"Meijer","year":"1946","journal-title":"Mathematics"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1016\/j.jcp.2015.11.053","article-title":"From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations","volume":"307","author":"Angstmann","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1007\/BF01025990","article-title":"Statics and dynamics of a diffusion-limited reaction: Anomalous kinetics, non-equilibrium self-ordering, and a dynamic transition","volume":"60","author":"Burschka","year":"1990","journal-title":"J. Stat. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1090\/S0002-9947-1984-0760971-6","article-title":"Stability of the travelling wave solution of the Fitzhugh-Nagumo system","volume":"286","author":"Jones","year":"1984","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1082","DOI":"10.1016\/j.camwa.2015.06.031","article-title":"Pattern formation in the FitzHugh-Nagumo model","volume":"70","author":"Zheng","year":"2015","journal-title":"Comput. Math. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0006-3495(61)86902-6","article-title":"Impulse and physiological states in theoretical models of nerve membrane","volume":"1","author":"Fitzhugh","year":"1961","journal-title":"Biophys. J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2061","DOI":"10.1109\/JRPROC.1962.288235","article-title":"An active pulse transmission line stimulating nerve axon","volume":"50","author":"Nagumo","year":"1962","journal-title":"Proc. IRE"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/1035\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:10:22Z","timestamp":1760177422000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/1035"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,16]]},"references-count":43,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["e22091035"],"URL":"https:\/\/doi.org\/10.3390\/e22091035","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,16]]}}}