In the preface we present a short overview of articles included in the issue "Bifurcations and pattern formation in biological applications" of the journal Mathematical Modelling of Natural Phenomena

A. Morozov

Aydogmus

Banerjee

Bayliss

Biancalani

Burgess

Chaplain

Crampin

Eftimie

Gierer

Kelkel

Kondo

M. Ptashnyk

Madzvamuse

Maini

McKane

Murray

Painter

Sherratt

Siekmann

Turing

V. Volpert

Valenti

Wallace

In the preface we present a short overview of articles included in the issue "Bifurcations and pattern formation in biological applications" of the journal Mathematical Modelling of Natural Phenomena.</p

Morozov, A.

Ptashnyk, M.

Volpert, V.

Discovery Research Portal

                                                                    University of DundeePreface. Bifurcations and Pattern Formation in Biological ApplicationsMorozov, A.; Ptashnyk, M.; Volpert, V.Published in:Mathematical Modelling of Natural PhenomenaDOI:10.1051/mmnp/201611501Publication date:2016Document VersionPeer reviewed versionLink to publication in Discovery Research PortalCitation for published version (APA):Morozov, A., Ptashnyk, M., & Volpert, V. (2016). Preface. Bifurcations and Pattern Formation in BiologicalApplications. Mathematical Modelling of Natural Phenomena, 11(5), 1-3.https://doi.org/10.1051/mmnp/201611501General rightsCopyright and moral rights for the publications made accessible in Discovery Research Portal are retained by the authors and/or othercopyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated withthese rights. • Users may download and print one copy of any publication from Discovery Research Portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain. • You may freely distribute the URL identifying the publication in the public portal.Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 18. Apr. 2023“mmnp-preface” — 2016/10/9 — 20:48 — page 1 — #1iiiiiiiiMath. Model. Nat. Phenom.Vol. , No. , 2016, pp. 1–3DOI: 10.1051/mmnp/20127102Bifurcations and Pattern Formation in BiologicalApplications. Preface.A. Morozov1, M. Ptashnyk2∗, V. Volpert1Department of Mathematics, University of Leicester, LE1 7RH Leicester, UK2Department of Mathematics, University of Dundee, DD1 4HN Dundee, UKAbstract. In the preface we present a short overview of articles included in the issue of Math-ematical Modelling of Natural Phenomena on ‘Bifurcations and pattern formation in biologicalapplications’.Keywords and phrases: pattern-formation, stochastic dynamics, continuation and bifurca-tion analysis, chaotic dynamics, evolutionary game modelsMathematics Subject Classification: 35B36, 92C15, 34C23, 34F10, 35B32, 37-xxAnalysis of pattern formations in biological and physical systems is a wide and active research area.Many different approaches (e.g. continuous, discrete, hybrid discrete-continuum) and techniques (e.g.linearised stability analysis and numerical continuations) are used to model and analyse the patternformation in biological systems. The reaction-diffusion theory for pattern formation was first proposedby A. Turing in his work in 1952 on morphogenesis [19]. In 1970s H. Meinhardt and A. Gierer developeda concept of activating and inhibiting chemicals to model and analyse the formation of patterns duringthe developmental processes [8, 14]. Since then the idea of diffusion-driven instability was applied toanalyse pattern formations in many different biological systems [12]. Mechano-chemical theory of patternformation developed by J.D. Murray and G. Oster [15] proposes mechanical interactions as a possiblemechanism responsible for pattern formation in biological systems.Following first analytical and numerical results, many different aspects of pattern formation in biolog-ical systems are currently being considered.It was identified that the growth of the spatial domain of interactions plays an important role in theformation of spatial patterns [5, 10]. In this issue the pattern formation for reaction-diffusion systemsdefined on evolving domains is presented by A. Madzvamuse and A.H. Chung [11]. A coupling betweenbulk and surface dynamics induces an interesting and important aspect of the analysis. For linearisedstability analysis Lagrangian transformation of the model equations to the equations defined on a fixeddomain is performed. The impact of cell death on the pattern formation in a growing bacterial biofilmsdescribed by a moving boundary model is considered by H.A. Wallace, L. Li, and F.A. Davidson in [22].Analysis indicates that the pattern formation in biofilms depends on the height (thickness) of the biofilms.Directed movement of cells or populations is also considered as one of the mechanisms of patternformation, e.g. pattern formation in tumour-invasion model [4] or in the chemotaxis system [16]. The∗Corresponding author. E-mail: m.ptashnyk@dundee.ac.ukc© EDP Sciences, 2016“mmnp-preface” — 2016/10/9 — 20:48 — page 2 — #2iiiiiiiiA. Morozov, M. Ptashnyk, V. Volpert Bifurcations and pattern formationtheory of spatial evolutionary game is used in the paper by A.E.F. Burgess, P.G. Schofield, S.F. Hubbard,M.A.J. Chaplain, and T. Lorenzi [3] to analyse the hybrid discrete-continuum model which includesrandom motion and chemotaxis. Numerical simulations of the model show that diffusion through spaceand chemotactic movement (i.e. up semiochemical gradients), can bring about self-generated patternswhich create favourable conditions for the coexistence of two populations in situations in which the twostrategies cannot coexist either in spatially homogeneous models or in cellular automaton models. Themodelling framework presented in this paper potentially covers a wide range of applications includingecology (e.g. predator-prey systems), epidemiology (e.g. the invasion of new host tissues by a pathogen)and oncology (e.g. the colonisation of a new niche by tumour cells following metastasis).The changes from regular patterns to chaos constitute an important and interesting aspect of thetemporal and spatio-temporal dynamics of biological systems [6, 16]. Complex behaviour and chaoticdynamics in the system of five ordinary differential equations modelling a tumour-immune-virus systemare studied by R. Eftimie, C.K. Macnamara, J. Dushoff, J.L. Bramson and D.J.D. Earn in [7].Analysis of mathematical models for ecosystems constitutes a vital part of the mathematical analysisof pattern formation in biological systems and is important due to limited opportunities for empiricalstudies of large ecosystems. In [17] applying mumerical bifurcation methods J.A. Sherratt analyses thepattern formation in the models for self-organisation observed in mussel beds and calculates the parameterregion in which patterns exist and the sub-region in which these patterns are stable. He also shows howthis results can be used to explain numerical observations of history-dependent wavelength selection asparameters are varied slowly. The impact of non-local interactions on the dynamics and formation ofpatterns in the spatio-temporal prey-predator model is studied by M. Banerjee and L. Zhang in [1]. It isobserved that the nonlocal interaction term in the prey growth has a stabilising effect on the dynamics ofsolutions of the model equations and can enhance the existence of stationary solutions for a wide rangeof parameter values.To better understand noise induced phenomena observed in biological systems it is important to anal-yse the influence of noise on the dynamics of populations and on pattern formation. Chemical MasterEquations [2], Stochastic ODEs and Stochastic PDEs [9] are used to describe the evolution of biologi-cal systems with external noise or intrinsic stochasticity. The influence of noise on the dynamics of anecosystem is considered in the work by D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio,B. Spagnolo, S. Mazzola, G. Basilone and A. Bonanno, presented in this issue in [20]. For Lotka–Volterraequations with a multiplicative noise the authors identify noise induced phenomena such as stochasticresonance, noise delayed extinction, and noise-induced pattern formation. The perturbation with multi-plicative white noise of a deterministic model is considered by I. Siekmann and H. Malchow to modelstochastic environmental fluctuations [18]. The authors show that different properties of the stochasticperturbations may not only lead to opposite predictions of survival and extinction but also considerablyaffect the patterns of the spatio-temporal dynamics. In [21] D. Valenti, G. Denaro, F. Giarratana, A. Giuf-frida, S. Mazzola, G. Basilone, S. Aronica, A. Bonanno and B. Spagnolo consider a system of stochasticODEs to model the dynamics of two bacterial populations responsible for food spoilage. The fluctuationsare modelled by multiplicative white Gaussian noise. It is observed that the numerical solutions of thestochastic model are in a better agreement with the experimental results than the numerical solutions ofthe corresponding deterministic model.References[1] M. Banerjee, L. Zhang, Stabilizing role of nonlocal interaction on spatial pattern formation, Math. Model. Nat. Phe-nom., 2016.[2] T. Biancalani, D. Fanelli, F. Di Patti, Stochastic Turing patterns in the Brusselator model., Phys. Rev. E 81, 046215,2010.[3] A.E.F. Burgess, P.G. Schofield, S.F. Hubbard, M.A.J. Chaplain, T. Lorenzi, Dynamical patterns of coexisting strategiesin a hybrid discrete-continuum spatial evolutionary game model, Math. Model. Nat. Phenom., 2016.[4] M.A.J. Chaplain, G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Netw. Het-erog. Media 1, 399–439, 2006.2“mmnp-preface” — 2016/10/9 — 20:48 — page 3 — #3iiiiiiiiA. Morozov, M. Ptashnyk, V. Volpert Bifurcations and pattern formation[5] E.J. Crampin, E.A. Gaffney, P.K. Maini, Reaction and diffusion on growing domains: scenarios for robust patternformation, Bull. Math. Biol. 61, 1093–1120, 1999.[6] H. Deng, A.V. Holden, L.F. Olsen (Editors), Chaos in Biological Systems, Springer 1987.[7] R. Eftimie, C.K. Macnamara, J. Dushoff, J.L. Bramson, D.J.D. Earn, Bifurcations and chaotic dynamics in a tumour-immune-virus system, Math. Model. Nat. Phenom., 2016.[8] A. Gierer, H. Meinhardt, Theory of biological pattern formation, Kybernetik, 12, 30-39, 1972.[9] J. Kelkel, C. Surulescu, On a stochastic reaction-diffusion system modeling pattern formation on seashells,J. Math. Biol. 60, 765–96, 2010.[10] S. Kondo, R. Asal, A reaction-diffusion wave on skin of the marine angelfish Pomacanthus, Nature 376, 765–768, 1995.[11] A. Madzvamuse, A.H. Chung, Analysis and simulations of coupled bulk-surface reaction-diffusion systems on expo-nentially evolving volumes, Math. Model. Nat. Phenom., 2016.[12] P.K. Maini, T.E. Woolley, R.E. Baker, E.A. Gaffney, S.S. Lee, Turing’s model for biological pattern formation and therobustness problem, Interface Focus, 2, 487–496, 2012[13] A.J. McKane, T. Biancalani, T. Rogers, Stochastic pattern formation and spontaneous polarisation: the linear noiseapproximation and beyond, Bull. Math. Biol. 76, 895–921, 2014.[14] H. Meinhardt, Modelling of biological pattern formation, Academic Press, London, 1982.[15] J.D. Murray, G. Oster, Generation of biological pattern and form, IMA J. Math. Med. Biol. 1, 1-25, 1984.[16] K.J. Painter, T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240, 363–375, 2011.[17] J.A. Sherratt, Using numerical bifurcation analysis to study pattern formation in mussel beds, Math. Model. Nat. Phe-nom., 2016.[18] I. Siekmann, H. Malchow, Fighting enemies and noise: Competition of residents and invaders in a stochasticallyfluctuating environment, Math. Model. Nat. Phenom., 2016.[19] A.M. Turing, Chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond., B 237, 37-72, 1952.[20] D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio, B. Spagnolo, S. Mazzola, G. Basilone, A. Bonanno, Noiseinduced phenomena in population dynamics, Math. Model. Nat. Phenom., 2016.[21] D. Valenti, G. Denaro, F. Giarratana, A. Giuffrida, S. Mazzola, G. Basilone, S. Aronica, A. Bonanno, B. Spagnolo,Modeling of sensory characteristics based on the growth of food spoilage bacteria, Math. Model. Nat. Phenom., 2016.[22] H.A. Wallace, L. Li, F.A. Davidson, Instability in a moving boundary: heterogeneous growth of bacterial biofilms,Math. Model. Nat. Phenom., 2016.3

English

Preface. Bifurcations and Pattern Formation in Biological Applications

Heriot Watt Pure

                                                              University of DundeePreface. Bifurcations and Pattern Formation in Biological ApplicationsMorozov, A.; Ptashnyk, Mariya; Volpert, V.Published in:Mathematical Modelling of Natural PhenomenaDOI:10.1051/mmnp/201611501Publication date:2016Document VersionAccepted author manuscriptLink to publication in Discovery Research PortalCitation for published version (APA):Morozov, A., Ptashnyk, M., & Volpert, V. (2016). Preface. Bifurcations and Pattern Formation in BiologicalApplications. Mathematical Modelling of Natural Phenomena, 11(5), 1-3. DOI: 10.1051/mmnp/201611501General rightsCopyright and moral rights for the publications made accessible in Discovery Research Portal are retained by the authors and/or othercopyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated withthese rights. • Users may download and print one copy of any publication from Discovery Research Portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain. • You may freely distribute the URL identifying the publication in the public portal.Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 01. Jul. 2017“mmnp-preface” — 2016/10/9 — 20:48 — page 1 — #1iiiiiiiiMath. Model. Nat. Phenom.Vol. , No. , 2016, pp. 1–3DOI: 10.1051/mmnp/20127102Bifurcations and Pattern Formation in BiologicalApplications. Preface.A. Morozov1, M. Ptashnyk2∗, V. Volpert1Department of Mathematics, University of Leicester, LE1 7RH Leicester, UK2Department of Mathematics, University of Dundee, DD1 4HN Dundee, UKAbstract. In the preface we present a short overview of articles included in the issue of Math-ematical Modelling of Natural Phenomena on ‘Bifurcations and pattern formation in biologicalapplications’.Keywords and phrases: pattern-formation, stochastic dynamics, continuation and bifurca-tion analysis, chaotic dynamics, evolutionary game modelsMathematics Subject Classification: 35B36, 92C15, 34C23, 34F10, 35B32, 37-xxAnalysis of pattern formations in biological and physical systems is a wide and active research area.Many different approaches (e.g. continuous, discrete, hybrid discrete-continuum) and techniques (e.g.linearised stability analysis and numerical continuations) are used to model and analyse the patternformation in biological systems. The reaction-diffusion theory for pattern formation was first proposedby A. Turing in his work in 1952 on morphogenesis [19]. In 1970s H. Meinhardt and A. Gierer developeda concept of activating and inhibiting chemicals to model and analyse the formation of patterns duringthe developmental processes [8, 14]. Since then the idea of diffusion-driven instability was applied toanalyse pattern formations in many different biological systems [12]. Mechano-chemical theory of patternformation developed by J.D. Murray and G. Oster [15] proposes mechanical interactions as a possiblemechanism responsible for pattern formation in biological systems.Following first analytical and numerical results, many different aspects of pattern formation in biolog-ical systems are currently being considered.It was identified that the growth of the spatial domain of interactions plays an important role in theformation of spatial patterns [5, 10]. In this issue the pattern formation for reaction-diffusion systemsdefined on evolving domains is presented by A. Madzvamuse and A.H. Chung [11]. A coupling betweenbulk and surface dynamics induces an interesting and important aspect of the analysis. For linearisedstability analysis Lagrangian transformation of the model equations to the equations defined on a fixeddomain is performed. The impact of cell death on the pattern formation in a growing bacterial biofilmsdescribed by a moving boundary model is considered by H.A. Wallace, L. Li, and F.A. Davidson in [22].Analysis indicates that the pattern formation in biofilms depends on the height (thickness) of the biofilms.Directed movement of cells or populations is also considered as one of the mechanisms of patternformation, e.g. pattern formation in tumour-invasion model [4] or in the chemotaxis system [16]. The∗Corresponding author. E-mail: m.ptashnyk@dundee.ac.ukc© EDP Sciences, 2016“mmnp-preface” — 2016/10/9 — 20:48 — page 2 — #2iiiiiiiiA. Morozov, M. Ptashnyk, V. Volpert Bifurcations and pattern formationtheory of spatial evolutionary game is used in the paper by A.E.F. Burgess, P.G. Schofield, S.F. Hubbard,M.A.J. Chaplain, and T. Lorenzi [3] to analyse the hybrid discrete-continuum model which includesrandom motion and chemotaxis. Numerical simulations of the model show that diffusion through spaceand chemotactic movement (i.e. up semiochemical gradients), can bring about self-generated patternswhich create favourable conditions for the coexistence of two populations in situations in which the twostrategies cannot coexist either in spatially homogeneous models or in cellular automaton models. Themodelling framework presented in this paper potentially covers a wide range of applications includingecology (e.g. predator-prey systems), epidemiology (e.g. the invasion of new host tissues by a pathogen)and oncology (e.g. the colonisation of a new niche by tumour cells following metastasis).The changes from regular patterns to chaos constitute an important and interesting aspect of thetemporal and spatio-temporal dynamics of biological systems [6, 16]. Complex behaviour and chaoticdynamics in the system of five ordinary differential equations modelling a tumour-immune-virus systemare studied by R. Eftimie, C.K. Macnamara, J. Dushoff, J.L. Bramson and D.J.D. Earn in [7].Analysis of mathematical models for ecosystems constitutes a vital part of the mathematical analysisof pattern formation in biological systems and is important due to limited opportunities for empiricalstudies of large ecosystems. In [17] applying mumerical bifurcation methods J.A. Sherratt analyses thepattern formation in the models for self-organisation observed in mussel beds and calculates the parameterregion in which patterns exist and the sub-region in which these patterns are stable. He also shows howthis results can be used to explain numerical observations of history-dependent wavelength selection asparameters are varied slowly. The impact of non-local interactions on the dynamics and formation ofpatterns in the spatio-temporal prey-predator model is studied by M. Banerjee and L. Zhang in [1]. It isobserved that the nonlocal interaction term in the prey growth has a stabilising effect on the dynamics ofsolutions of the model equations and can enhance the existence of stationary solutions for a wide rangeof parameter values.To better understand noise induced phenomena observed in biological systems it is important to anal-yse the influence of noise on the dynamics of populations and on pattern formation. Chemical MasterEquations [2], Stochastic ODEs and Stochastic PDEs [9] are used to describe the evolution of biologi-cal systems with external noise or intrinsic stochasticity. The influence of noise on the dynamics of anecosystem is considered in the work by D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio,B. Spagnolo, S. Mazzola, G. Basilone and A. Bonanno, presented in this issue in [20]. For Lotka–Volterraequations with a multiplicative noise the authors identify noise induced phenomena such as stochasticresonance, noise delayed extinction, and noise-induced pattern formation. The perturbation with multi-plicative white noise of a deterministic model is considered by I. Siekmann and H. Malchow to modelstochastic environmental fluctuations [18]. The authors show that different properties of the stochasticperturbations may not only lead to opposite predictions of survival and extinction but also considerablyaffect the patterns of the spatio-temporal dynamics. In [21] D. Valenti, G. Denaro, F. Giarratana, A. Giuf-frida, S. Mazzola, G. Basilone, S. Aronica, A. Bonanno and B. Spagnolo consider a system of stochasticODEs to model the dynamics of two bacterial populations responsible for food spoilage. The fluctuationsare modelled by multiplicative white Gaussian noise. It is observed that the numerical solutions of thestochastic model are in a better agreement with the experimental results than the numerical solutions ofthe corresponding deterministic model.References[1] M. Banerjee, L. Zhang, Stabilizing role of nonlocal interaction on spatial pattern formation, Math. Model. Nat. Phe-nom., 2016.[2] T. Biancalani, D. Fanelli, F. Di Patti, Stochastic Turing patterns in the Brusselator model., Phys. Rev. E 81, 046215,2010.[3] A.E.F. Burgess, P.G. Schofield, S.F. Hubbard, M.A.J. Chaplain, T. Lorenzi, Dynamical patterns of coexisting strategiesin a hybrid discrete-continuum spatial evolutionary game model, Math. Model. Nat. Phenom., 2016.[4] M.A.J. Chaplain, G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Netw. Het-erog. Media 1, 399–439, 2006.2“mmnp-preface” — 2016/10/9 — 20:48 — page 3 — #3iiiiiiiiA. Morozov, M. Ptashnyk, V. Volpert Bifurcations and pattern formation[5] E.J. Crampin, E.A. Gaffney, P.K. Maini, Reaction and diffusion on growing domains: scenarios for robust patternformation, Bull. Math. Biol. 61, 1093–1120, 1999.[6] H. Deng, A.V. Holden, L.F. Olsen (Editors), Chaos in Biological Systems, Springer 1987.[7] R. Eftimie, C.K. Macnamara, J. Dushoff, J.L. Bramson, D.J.D. Earn, Bifurcations and chaotic dynamics in a tumour-immune-virus system, Math. Model. Nat. Phenom., 2016.[8] A. Gierer, H. Meinhardt, Theory of biological pattern formation, Kybernetik, 12, 30-39, 1972.[9] J. Kelkel, C. Surulescu, On a stochastic reaction-diffusion system modeling pattern formation on seashells,J. Math. Biol. 60, 765–96, 2010.[10] S. Kondo, R. Asal, A reaction-diffusion wave on skin of the marine angelfish Pomacanthus, Nature 376, 765–768, 1995.[11] A. Madzvamuse, A.H. Chung, Analysis and simulations of coupled bulk-surface reaction-diffusion systems on expo-nentially evolving volumes, Math. Model. Nat. Phenom., 2016.[12] P.K. Maini, T.E. Woolley, R.E. Baker, E.A. Gaffney, S.S. Lee, Turing’s model for biological pattern formation and therobustness problem, Interface Focus, 2, 487–496, 2012[13] A.J. McKane, T. Biancalani, T. Rogers, Stochastic pattern formation and spontaneous polarisation: the linear noiseapproximation and beyond, Bull. Math. Biol. 76, 895–921, 2014.[14] H. Meinhardt, Modelling of biological pattern formation, Academic Press, London, 1982.[15] J.D. Murray, G. Oster, Generation of biological pattern and form, IMA J. Math. Med. Biol. 1, 1-25, 1984.[16] K.J. Painter, T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240, 363–375, 2011.[17] J.A. Sherratt, Using numerical bifurcation analysis to study pattern formation in mussel beds, Math. Model. Nat. Phe-nom., 2016.[18] I. Siekmann, H. Malchow, Fighting enemies and noise: Competition of residents and invaders in a stochasticallyfluctuating environment, Math. Model. Nat. Phenom., 2016.[19] A.M. Turing, Chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond., B 237, 37-72, 1952.[20] D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio, B. Spagnolo, S. Mazzola, G. Basilone, A. Bonanno, Noiseinduced phenomena in population dynamics, Math. Model. Nat. Phenom., 2016.[21] D. Valenti, G. Denaro, F. Giarratana, A. Giuffrida, S. Mazzola, G. Basilone, S. Aronica, A. Bonanno, B. Spagnolo,Modeling of sensory characteristics based on the growth of food spoilage bacteria, Math. Model. Nat. Phenom., 2016.[22] H.A. Wallace, L. Li, F.A. Davidson, Instability in a moving boundary: heterogeneous growth of bacterial biofilms,Math. Model. Nat. Phenom., 2016.3

In the preface we present a short overview of articles included in the issue “Bifurcations and pattern formation in biological applications” of the journal Mathematical Modelling of Natural Phenomena

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Mathematical Modelling of Natural Phenomena

A. Morozov (7746905)

M. Ptashnyk (7784525)

V. Volpert (7781363)

Leicester Research Archive

                                                                    University of DundeePreface. Bifurcations and Pattern Formation in Biological ApplicationsMorozov, A.; Ptashnyk, M.; Volpert, V.Published in:Mathematical Modelling of Natural PhenomenaDOI:10.1051/mmnp/201611501Publication date:2016Document VersionPeer reviewed versionLink to publication in Discovery Research PortalCitation for published version (APA):Morozov, A., Ptashnyk, M., & Volpert, V. (2016). Preface. Bifurcations and Pattern Formation in BiologicalApplications. Mathematical Modelling of Natural Phenomena, 11(5), 1-3.https://doi.org/10.1051/mmnp/201611501General rightsCopyright and moral rights for the publications made accessible in Discovery Research Portal are retained by the authors and/or othercopyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated withthese rights.Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 30. Oct. 2025“mmnp-preface” — 2016/10/9 — 20:48 — page 1 — #1iiiiiiiiMath. Model. Nat. Phenom.Vol. , No. , 2016, pp. 1–3DOI: 10.1051/mmnp/20127102Bifurcations and Pattern Formation in BiologicalApplications. Preface.A. Morozov1, M. Ptashnyk2∗, V. Volpert1Department of Mathematics, University of Leicester, LE1 7RH Leicester, UK2Department of Mathematics, University of Dundee, DD1 4HN Dundee, UKAbstract. In the preface we present a short overview of articles included in the issue of Math-ematical Modelling of Natural Phenomena on ‘Bifurcations and pattern formation in biologicalapplications’.Keywords and phrases: pattern-formation, stochastic dynamics, continuation and bifurca-tion analysis, chaotic dynamics, evolutionary game modelsMathematics Subject Classification: 35B36, 92C15, 34C23, 34F10, 35B32, 37-xxAnalysis of pattern formations in biological and physical systems is a wide and active research area.Many different approaches (e.g. continuous, discrete, hybrid discrete-continuum) and techniques (e.g.linearised stability analysis and numerical continuations) are used to model and analyse the patternformation in biological systems. The reaction-diffusion theory for pattern formation was first proposedby A. Turing in his work in 1952 on morphogenesis [19]. In 1970s H. Meinhardt and A. Gierer developeda concept of activating and inhibiting chemicals to model and analyse the formation of patterns duringthe developmental processes [8, 14]. Since then the idea of diffusion-driven instability was applied toanalyse pattern formations in many different biological systems [12]. Mechano-chemical theory of patternformation developed by J.D. Murray and G. Oster [15] proposes mechanical interactions as a possiblemechanism responsible for pattern formation in biological systems.Following first analytical and numerical results, many different aspects of pattern formation in biolog-ical systems are currently being considered.It was identified that the growth of the spatial domain of interactions plays an important role in theformation of spatial patterns [5, 10]. In this issue the pattern formation for reaction-diffusion systemsdefined on evolving domains is presented by A. Madzvamuse and A.H. Chung [11]. A coupling betweenbulk and surface dynamics induces an interesting and important aspect of the analysis. For linearisedstability analysis Lagrangian transformation of the model equations to the equations defined on a fixeddomain is performed. The impact of cell death on the pattern formation in a growing bacterial biofilmsdescribed by a moving boundary model is considered by H.A. Wallace, L. Li, and F.A. Davidson in [22].Analysis indicates that the pattern formation in biofilms depends on the height (thickness) of the biofilms.Directed movement of cells or populations is also considered as one of the mechanisms of patternformation, e.g. pattern formation in tumour-invasion model [4] or in the chemotaxis system [16]. The∗Corresponding author. E-mail: m.ptashnyk@dundee.ac.ukc© EDP Sciences, 2016“mmnp-preface” — 2016/10/9 — 20:48 — page 2 — #2iiiiiiiiA. Morozov, M. Ptashnyk, V. Volpert Bifurcations and pattern formationtheory of spatial evolutionary game is used in the paper by A.E.F. Burgess, P.G. Schofield, S.F. Hubbard,M.A.J. Chaplain, and T. Lorenzi [3] to analyse the hybrid discrete-continuum model which includesrandom motion and chemotaxis. Numerical simulations of the model show that diffusion through spaceand chemotactic movement (i.e. up semiochemical gradients), can bring about self-generated patternswhich create favourable conditions for the coexistence of two populations in situations in which the twostrategies cannot coexist either in spatially homogeneous models or in cellular automaton models. Themodelling framework presented in this paper potentially covers a wide range of applications includingecology (e.g. predator-prey systems), epidemiology (e.g. the invasion of new host tissues by a pathogen)and oncology (e.g. the colonisation of a new niche by tumour cells following metastasis).The changes from regular patterns to chaos constitute an important and interesting aspect of thetemporal and spatio-temporal dynamics of biological systems [6, 16]. Complex behaviour and chaoticdynamics in the system of five ordinary differential equations modelling a tumour-immune-virus systemare studied by R. Eftimie, C.K. Macnamara, J. Dushoff, J.L. Bramson and D.J.D. Earn in [7].Analysis of mathematical models for ecosystems constitutes a vital part of the mathematical analysisof pattern formation in biological systems and is important due to limited opportunities for empiricalstudies of large ecosystems. In [17] applying mumerical bifurcation methods J.A. Sherratt analyses thepattern formation in the models for self-organisation observed in mussel beds and calculates the parameterregion in which patterns exist and the sub-region in which these patterns are stable. He also shows howthis results can be used to explain numerical observations of history-dependent wavelength selection asparameters are varied slowly. The impact of non-local interactions on the dynamics and formation ofpatterns in the spatio-temporal prey-predator model is studied by M. Banerjee and L. Zhang in [1]. It isobserved that the nonlocal interaction term in the prey growth has a stabilising effect on the dynamics ofsolutions of the model equations and can enhance the existence of stationary solutions for a wide rangeof parameter values.To better understand noise induced phenomena observed in biological systems it is important to anal-yse the influence of noise on the dynamics of populations and on pattern formation. Chemical MasterEquations [2], Stochastic ODEs and Stochastic PDEs [9] are used to describe the evolution of biologi-cal systems with external noise or intrinsic stochasticity. The influence of noise on the dynamics of anecosystem is considered in the work by D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio,B. Spagnolo, S. Mazzola, G. Basilone and A. Bonanno, presented in this issue in [20]. For Lotka–Volterraequations with a multiplicative noise the authors identify noise induced phenomena such as stochasticresonance, noise delayed extinction, and noise-induced pattern formation. The perturbation with multi-plicative white noise of a deterministic model is considered by I. Siekmann and H. Malchow to modelstochastic environmental fluctuations [18]. The authors show that different properties of the stochasticperturbations may not only lead to opposite predictions of survival and extinction but also considerablyaffect the patterns of the spatio-temporal dynamics. In [21] D. Valenti, G. Denaro, F. Giarratana, A. Giuf-frida, S. Mazzola, G. Basilone, S. Aronica, A. Bonanno and B. Spagnolo consider a system of stochasticODEs to model the dynamics of two bacterial populations responsible for food spoilage. The fluctuationsare modelled by multiplicative white Gaussian noise. It is observed that the numerical solutions of thestochastic model are in a better agreement with the experimental results than the numerical solutions ofthe corresponding deterministic model.References[1] M. Banerjee, L. Zhang, Stabilizing role of nonlocal interaction on spatial pattern formation, Math. Model. Nat. Phe-nom., 2016.[2] T. Biancalani, D. Fanelli, F. Di Patti, Stochastic Turing patterns in the Brusselator model., Phys. Rev. E 81, 046215,2010.[3] A.E.F. Burgess, P.G. Schofield, S.F. Hubbard, M.A.J. Chaplain, T. Lorenzi, Dynamical patterns of coexisting strategiesin a hybrid discrete-continuum spatial evolutionary game model, Math. Model. Nat. Phenom., 2016.[4] M.A.J. Chaplain, G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Netw. Het-erog. Media 1, 399–439, 2006.2“mmnp-preface” — 2016/10/9 — 20:48 — page 3 — #3iiiiiiiiA. Morozov, M. Ptashnyk, V. Volpert Bifurcations and pattern formation[5] E.J. Crampin, E.A. Gaffney, P.K. Maini, Reaction and diffusion on growing domains: scenarios for robust patternformation, Bull. Math. Biol. 61, 1093–1120, 1999.[6] H. Deng, A.V. Holden, L.F. Olsen (Editors), Chaos in Biological Systems, Springer 1987.[7] R. Eftimie, C.K. Macnamara, J. Dushoff, J.L. Bramson, D.J.D. Earn, Bifurcations and chaotic dynamics in a tumour-immune-virus system, Math. Model. Nat. Phenom., 2016.[8] A. Gierer, H. Meinhardt, Theory of biological pattern formation, Kybernetik, 12, 30-39, 1972.[9] J. Kelkel, C. Surulescu, On a stochastic reaction-diffusion system modeling pattern formation on seashells,J. Math. Biol. 60, 765–96, 2010.[10] S. Kondo, R. Asal, A reaction-diffusion wave on skin of the marine angelfish Pomacanthus, Nature 376, 765–768, 1995.[11] A. Madzvamuse, A.H. Chung, Analysis and simulations of coupled bulk-surface reaction-diffusion systems on expo-nentially evolving volumes, Math. Model. Nat. Phenom., 2016.[12] P.K. Maini, T.E. Woolley, R.E. Baker, E.A. Gaffney, S.S. Lee, Turing’s model for biological pattern formation and therobustness problem, Interface Focus, 2, 487–496, 2012[13] A.J. McKane, T. Biancalani, T. Rogers, Stochastic pattern formation and spontaneous polarisation: the linear noiseapproximation and beyond, Bull. Math. Biol. 76, 895–921, 2014.[14] H. Meinhardt, Modelling of biological pattern formation, Academic Press, London, 1982.[15] J.D. Murray, G. Oster, Generation of biological pattern and form, IMA J. Math. Med. Biol. 1, 1-25, 1984.[16] K.J. Painter, T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240, 363–375, 2011.[17] J.A. Sherratt, Using numerical bifurcation analysis to study pattern formation in mussel beds, Math. Model. Nat. Phe-nom., 2016.[18] I. Siekmann, H. Malchow, Fighting enemies and noise: Competition of residents and invaders in a stochasticallyfluctuating environment, Math. Model. Nat. Phenom., 2016.[19] A.M. Turing, Chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond., B 237, 37-72, 1952.[20] D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio, B. Spagnolo, S. Mazzola, G. Basilone, A. Bonanno, Noiseinduced phenomena in population dynamics, Math. Model. Nat. Phenom., 2016.[21] D. Valenti, G. Denaro, F. Giarratana, A. Giuffrida, S. Mazzola, G. Basilone, S. Aronica, A. Bonanno, B. Spagnolo,Modeling of sensory characteristics based on the growth of food spoilage bacteria, Math. Model. Nat. Phenom., 2016.[22] H.A. Wallace, L. Li, F.A. Davidson, Instability in a moving boundary: heterogeneous growth of bacterial biofilms,Math. Model. Nat. Phenom., 2016.3

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Preface. Bifurcations and Pattern Formation in Biological Applications

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