Existence and Stability of a Spike in the Central Component for a Consumer Chain Model

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• 作者：Juncheng Wei ; Matthias Winter
• 关键词：Pattern formation ; Consumer chain model ; Predator–prey model ; Autocatalytic reaction ; Reaction–diffusion systems ; Spiky solutions ; Stability ; Primary 35B35 ; 92C40 ; Secondary 35B40
• 刊名：Journal of Dynamics and Differential Equations
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：27
• 期：3-4
• 页码：1141-1171
• 全文大小：686 KB
• 参考文献：1.Benson, D.L., Maini, P.K., Sherratt, J.A.: Unravelling the Turing bifurcation using spatially varying diffusion coefficients. J. Math. Biol. 37, 381-17 (1998)CrossRef MathSciNet
  2.Bettencourt, L., West, G.: A unified theory of urban living. Nature 467, 912-13 (2010)CrossRef
  3.Dancer, E.N.: On stability and Hopf bifurcations for chemotaxis systems. Methods Appl. Anal. 8, 245-56 (2001)MathSciNet
  4.Doelman, A., Gardner, A., Kaper, T.J.: Stability analysis of singular patterns in the 1-D Gray–Scott model: a matched asymptotic approach. Phys. D 122, 1-6 (1998)CrossRef MathSciNet
  5.Doelman, A., Gardner, R.A., Kaper, T.J.: A stability index analysis of 1-D patterns of the Gray—Scott model. Mem. Amer. Math. Soc. 155(737), xii+64 (2002)
  6.Doelman, A., Kaper, T., Zegeling, P.A.: Pattern formation in the one-dimensional Gray–Scott model. Nonlinearity 10, 523-63 (1997)CrossRef MathSciNet
  7.Doelman, A., Gardner, R., Kaper, T.J.: Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50, 443-07 (2001)CrossRef MathSciNet
  8.Eigen, M., Schuster, P.: The hypercycle. A principle of natural self organisation. Part A: emergence of the hypercycle. Naturwissenschaften 64, 541-65 (1977)CrossRef
  9.Eigen, M., Schuster, P.: The hypercycle. A principle of natural self organisation. Part B: the abstract hypercycle. Naturwissenschaften 65, 7-1 (1978)CrossRef
  10.Eigen, M., Schuster, P.: The hypercycle. A principle of natural self organisation. Part C: the realistic hypercycle. Naturwissenschaften 65, 341-69 (1978)CrossRef
  11.Eigen, M., Schuster, P.: The Hypercycle: A Principle of Natural Selforganisation. Springer, Berlin (1979)CrossRef
  12.Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik (Berlin) 12, 30-9 (1972)
  13.Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38, 29-3 (1983)CrossRef
  14.Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system \(A+ 2B \rightarrow 3B, B \rightarrow C\) . Chem. Eng. Sci. 39, 1087-097 (1984)CrossRef
  15.Iron, D., Wei, J., Winter, M.: Stability analysis of Turing patterns generated by the Schnakenberg model. J. Math. Biol. 49, 358-90 (2004)CrossRef MathSciNet
  16.Kolokolnikov, T., Ward, M.J., Wei, J.: The existence and stability of spike equilibria in the one-dimensional Gray–Scott model: the low feed-rate regime. Stud. Appl. Math. 115, 21-1 (2005)CrossRef MathSciNet
  17.Kolokolnikov, T., Ward, M.J., Wei, J.: The existence and stability of spike equilibria in the one-dimensional Gray–Scott model: the pulse-splitting regime. Phys. D 202, 258-93 (2005)CrossRef MathSciNet
  18.Liu, J., Ou, C.: How many consumer levels can survive in a chemotactic food chain? Front. Math. China 4, 495-21 (2009)CrossRef MathSciNet
  19.Meinhardt, H.: Models of Biological Pattern Formation. Academic Press, London (1982)
  20.Meinhardt, H., Gierer, A.: Generation and regeneration of sequences of structures during morphogenesis. J. Theor. Biol. 85, 429-50 (1980)CrossRef MathSciNet
  21.Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics. Springer, New York (2003)
  22.Ni, W.-M., Takagi, I.: On the shape of least energy solution to a semilinear Neumann problem. Comm. Pure Appl. Math. 41, 819-51 (1991)CrossRef MathSciNet
  23.Ni, W.-M., Takagi, I.: Locating the peaks of least energy solutions to a semilinear Neumann problem. Duke Math. J. 70, 247-81 (1993)CrossRef MathSciNet
  24.Ni, W.-M., Takagi, I.: Point-condensation generated by a reaction-diffusion system in axially symmetric domains. Jpn. J. Ind. Appl. Math. 12, 327-65 (1995)CrossRef MathSciNet
  25.Nishiura, Y., Ueyama, D.: A skeleton structure of self-replicating dynamics. Phys. D 130, 73-04 (1999)CrossRef
  26.Nishiura, Y., Teramoto, T., Ueyama, D.: Scattering and separators in dissipative systems. Phys. Rev. E 67, 056210 (2003)CrossRef
  27.Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theoret. Biol. 81, 389-00 (1979)CrossRef MathSciNet
  28.Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B 237, 37-2 (1952)CrossRef
  29.Volpert, A.I., Volpert, Vitaly A., Volpert, Vladimir A.: Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs. American Mathematical Society, Providence (1994)
  30.Ward, M.J., Wei, J.: Asymmetric spike patterns for the one-dimensional Gierer–Meinhardt model: equilibria and stability. Eur. J. Appl. Math. 13, 283-20 (2002)CrossRef MathSciNet
  31.Wei, J.: On single interior spike solutions of Gierer–Meinhardt system: uniqueness, spectrum estimates and stability analysis. Eur. J. Appl. Math. 10, 353-78 (1999)CrossRef
  32.Wei, J
• 作者单位：Juncheng Wei (1)
  Matthias Winter (2)
  
  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
  2. Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, UK
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Ordinary Differential Equations
  Partial Differential Equations
  Applications of Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9222

文摘

We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in Gierer and Meinhardt (Kybernetik 12:30-9, 1972) and Schnakenberg (J Theoret Biol 81:389-00, 1979) for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction–diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. Keywords Pattern formation Consumer chain model Predator–prey model Autocatalytic reaction Reaction–diffusion systems Spiky solutions Stability