Abstract
The paper is concerned with the effect of variable dispersal rates on Turing instability of a non-Lotka-Volterra reaction-diffusion system. In ecological applications, the dispersal rates of different species tends to oscillate in time. This oscillation is modeled by temporal variation in the diffusion coefficient with large as well as small periodicity. The case of large periodicity is analyzed using the theory of Floquet multipliers and that of the small periodicity by using Hill's equation. The effect of such variation on the resulting Turing space is studied. A comparative analysis of the Turing spaces with constant diffusivity and variable diffusivities is performed. Numerical simulations are carried out to support analytical findings.
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References
Auchmuty, J.F.G., Nicolis, G., 1975. Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equation and the steady state solutions. Bull. Math. Biol. 37, 323–365.
Benson, D.L., Sherratt, J., Maini, P.K., 1993a. Diffusion driven instability in an inhomogeneous domain. Bull. Math. Biol. 55, 365–384.
Benson, D.L., Maini, P.K., Sherratt, J., 1993b. Pattern formation in reaction diffusion models with spatially inhomogencous diffusion coefficients. Math. Comp. Model. 17, 29–34.
Bhattacharyya, R., Mukhopadhyay, B., Bandyopadhyay, M., 2003. Diffusive instability in a prey-predator system with time dependent diffusivity. Int. J. Math. Math. Sci. 66, 4195–4203.
Cantrell, R.S., Cosner, C., 1991. The effect of spatial heterogeneity in population dynamics. J. Math. Biol. 29, 315–338.
Levin, S.A., 1974. Dispersion and population interactions. Am. Nat. 108, 207–228.
Levin, S.A., 1977. A more functional response to predator-prey stability. Am. Nat. 111, 381–383.
Levin, S.A., Segel, L.A., 1976. Hypothesis for origin of planktonic patchiness. Nature 259, 659.
Levin, S.A., Segel, L.A., 1985. Pattern generation in space and aspect. SIAM Rev. 27, 45–67.
Maini, P.K., Benson, D.L., Sherratt, J.A., 1992. Pattern formation in reaction diffusion models with spatially inhomogeneous diffusion coefficients. IMA J. Math. Appl. Med. Biol. 9, 197–213.
Magnus, W., Winkler, S., 1979. Hill's Equation. Dover, New York.
May, R.M., 1973. Stability and Complexity in Model Ecosystems. Princeton University, Press Princeton, NJ.
Meinhardt, H., Gierer, A., 1974. Application of a theory of biological pattern formation based on lateral inhibition. J. Cell. Sci. 15, 321–346.
Mimura, M., Nishida, T., 1978. On a certain semilinear parabolic system related to Lotka-Volterra's ecological model. Publ. Res. Inst. Math. Sci. Kyoto Univ. 14, 269–282.
Mimura, M., Murray, J.D., 1979. On a planktonic prey-predator model which exhibits patchiness. J. Theor. Biol. 75, 249–262.
Murray, J.D., 1981. A pre-pattern formation mechanism for animal coat markings. J. Theor. Biol. 88, 161–199.
Murray, J.D., 2002. Mathematical Biology. Springer, Berlin.
Okubo, A., 1978. Horizontal dispersion and critical scales for phytoplankton patches. In: Steel, J.H. (Ed.), Spatial Pattern in Plankton Communities. Plenum, New York, pp. 21–42.
Okubo, A., 2001. Diffusion and Ecological Problems: Modern Perspective. Interdisciplinary Applied Mathematics, vol. 14, Springer, Berlin.
Pacala, S.W., Roughgarden, J., 1982. Spatial heterogeneity and interspecific competition. Theor. Popul. Biol. 21, 92–113.
Segel, L.A., Jackson, J.L., 1972. Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37, 545–559.
Sherratt, J.A., 1995a. Turing bifurcations with a temporally varying diffiusion coefficient. J. Math. Biol. 33, 295–308.
Sherratt, J.A., 1995b. Diffusion-driven instability in oscillating enviroments. Eur. J. Appl. Math. 6, 355–372.
Shigesada, N., Teramoto, E., 1978. A consideration on the theory of environmental density. Jpn. J. Ecol. 28, 1–8.
Shigesada, N., Kawasaki, K., Teramoto, E., 1979. Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99.
Shigesada, N., 1984. Spatial distribution of rapidly dispersing animals in heterogeneous environment. In: Levin, S.A., Hallam, T.G. (Eds.), Lecture Notes of Biomathematics, vol. 54. Springer-Verlag, Berlin, pp. 478–491.
Timm, U., Okubo, A., 1992. Diffusion-driven instability in a predator-prey system with time varying diffusivities. J. Math. Biol. 31, 307–320.
Turing, A.M., 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72.
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Mukhopadhyay, B., Bhattacharyya, R. Modeling the Role of Diffusion Coefficients on Turing Instability in a Reaction-diffusion Prey-predator System. Bull. Math. Biol. 68, 293–313 (2006). https://doi.org/10.1007/s11538-005-9007-2
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DOI: https://doi.org/10.1007/s11538-005-9007-2