Advertisement

Bulletin of Mathematical Biology

, Volume 65, Issue 1, pp 143–156 | Cite as

Interspecific influence on mobility and Turing instability

  • Yunxin HuangEmail author
  • Odo Diekmann
Article

Abstract

In this paper we formulate a multi-patch multi-species model in which the percapita emigration rate of one species depends on the density of some other species. We then focus on Turing instability to examine if and when this cross-emigration response has crucial effects. We find that the type of interaction matters greatly. In the case of competition a cross-emigration response promotes pattern formation by exercising a destabilizing influence; in particular, it may lead to diffusive instability provided that the response is sufficiently strong, which contrasts sharply with the well-known fact that the standard competition system does not exhibit Turing instability. In the case of prey-predator or activator-inhibitor interaction it acts against pattern formation by exerting a stabilizing effect; in particular, the diffusive instability, even though it may happen in a standard system, never occurs when the response is sufficiently strong. We conclude that the cross-emigration response is an important factor that should not be ignored when pattern formation is the issue.

Keywords

Pattern Formation Standard System Emigration Rate Turing Instability Diffusive Instability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allen, J. C. (1975). Mathematical models of species interactions in time and space. Am. Nat. 109, 319–342.CrossRefGoogle Scholar
  2. Almirantis, Y. and S. Papageorgiou (1991). Cross-diffusion effects on chemical and biological pattern formation. J. Theor. Biol. 151, 289–311.Google Scholar
  3. Amann, H. (1990). Dynamics of quasilinear parabolic equations II. Reaction-Diffusion Syst., Differ. Integral Equations 3, 13–75.zbMATHMathSciNetGoogle Scholar
  4. Amann, H. (1993). Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis, Teubnertexte Math. 133, H. Schmeisser and H. Triebel (Eds), Teubner: Stuttgart and Leipzig, pp. 9–126.Google Scholar
  5. Amann, H. (1995). Linear and Quasilinear Parabolic Problems, Vol. 1, Abstract Linear Theory, Basel, Boston, Berlin: Birkhauser Verlag.Google Scholar
  6. Aronson, D. G. (1985). The role of diffusion in mathematical population biology: skellam revisited, in Mathematics in Biology and Medicine, Lecture Notes in Biomathematics 57, V. Capasso (Ed.), pp. 2–6.Google Scholar
  7. Bernstein, C. (1984). Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis. Oecologia 61, 134–142.CrossRefGoogle Scholar
  8. Capasso, V. and A. Di Liddo (1994). Asymptotic behaviour of reaction-diffusion systems in population and epidemic models-the role of cross-diffusion. J. Math. Biol. 32, 453–463.MathSciNetCrossRefGoogle Scholar
  9. Chattopadhyay, J. and P. K. Tapaswi (1993). Order and disorder in biological systems through negative cross-diffusion of mitotic inhibitor—a mathematical model. Math. Comput. Modelling 17, 105–112.MathSciNetCrossRefGoogle Scholar
  10. Chattopadhyay, J. and P. K. Tapaswi (1997). Effect of cross-diffusion on pattern formation—a nonlinear analysis. Math. Comput. Modelling 48, 1–12.MathSciNetGoogle Scholar
  11. Crowley, P. H. (1981). Dispersal and stability of predator-prey interactions. Am. Nat. 118, 673–701.MathSciNetCrossRefGoogle Scholar
  12. Durrett, R. and S. Levin (1994). The importance of being discrete (and spatial). Theor. Pop. Biol. 46, 363–394.CrossRefGoogle Scholar
  13. Edelstein-Keshet, L. (1987). Mathematical Models in Biology, Birkhauser Mathematics Series, New York: McGraw-Hill Inc.Google Scholar
  14. Farkas, M. (1997). Two ways of modeling cross diffusion. Nonlinear Anal. TMA 30, 1225–1233.zbMATHMathSciNetCrossRefGoogle Scholar
  15. Grindrod, P. (1996). The Theory and Applications of Reaction-Diffusion Equations, Oxford: Clarendon Press.Google Scholar
  16. Hastings, A. (1990). Spatial heterogeneity and ecological models. Ecology 71, 426–428.CrossRefGoogle Scholar
  17. Holmes, E. E., M. A. Lewis, J. E. Banks and R. R. Veit (1994). Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75, 17–29.CrossRefGoogle Scholar
  18. Huang, Y. and O. Diekmann (2001). Predator migration in response to prey density: what are the consequences? J. Math. Biol. 43, 561–581.MathSciNetCrossRefGoogle Scholar
  19. Jansen, V. A. A. and A. L. Lloyd (2000). Local stability analysis of spatially homogeneous solutions of multi-patch systems. J. Math. Biol. 41, 232–252.MathSciNetCrossRefGoogle Scholar
  20. Jorńe, J. (1974). The effect of ionic migration on oscillations and pattern formation in chemical systems. J. Theor. Biol. 43, 375–380.Google Scholar
  21. Lou, Y. and W.-M. Ni (1996). Diffusion, self-diffusion and cross-diffusion. J. Differ. Equations 131, 79–131.MathSciNetCrossRefGoogle Scholar
  22. Marcus, M. and H. Minc (1964). A Survey of Matrix Theory and Matrix Inequalities, Boston: Allyn and Bacon Inc.Google Scholar
  23. Murray, J. D. (1993). Mathematical Biology, Berlin, Heidelberg, New York: Spinger.Google Scholar
  24. Ni, W.-M. (1998). Diffusion, cross-diffusion and their spike-layer steady states. Notices AMS 45, 9–18.zbMATHMathSciNetGoogle Scholar
  25. Okubo, A. (1980). Diffusion and Ecological Problems: Mathematical Models, Berlin: Springer.Google Scholar
  26. Okubo, A. and S. Levin (2000). Diffusion and Ecological Problems: Modern Perspectives, 2nd edn, Berlin: Spinger.Google Scholar
  27. Othmer, H. G. and L. E. Scriven (1971). Instability and dynamic patterns in cellular networks. J. Theor. Biol. 32, 507–537.CrossRefGoogle Scholar
  28. Pels, B. (2001). Evolutionary dynamics of dispersal in predatory mites, PhD thesis, Amsterdam University.Google Scholar
  29. Plahte, E. (2001). Pattern formation in discrete cell lattices. J. Math. Biol. 43, 411–445.zbMATHMathSciNetCrossRefGoogle Scholar
  30. Sabelis, M. W. (1981). Biological control of two-spotted spider mites using phytoseid predators. Part I: Modelling the Predator-prey Interactions at the Individual Level. Wageningen: Agric Res Rep Centre for Agricultural Publication and Documentation.Google Scholar
  31. Segel, L. A. (1984). Taxes in cellular ecology, in Mathematical Ecology, Lecture Notes in Biomathematics 54, S. A. Levin (Ed.), pp. 407–424.Google Scholar
  32. Segel, L. A. and J. L. Jackson (1972). Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 35, 545–559.CrossRefGoogle Scholar
  33. Shigesada, N., K. Kawasaki and E. Teramota (1979). Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99.CrossRefGoogle Scholar
  34. Takafuji, A. (1977). The effect of successful dispersal of a phytoseiid mite Phytoseiulus persimilis. Athias-Henriot (Acarina: phytoseiidae) on the persistence in the interactive system between the predator and its prey. Res. Popul. Ecol. 18, 210–222.Google Scholar
  35. Turing, A. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. B 237, 37–72.Google Scholar

Copyright information

© Society for Mathematical Biology 2003

Authors and Affiliations

  1. 1.Mathematical DepartmentUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations