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Asymptotic spreading of competition diffusion systems: The role of interspecific competitions

  • GUO LIN (a1) and WAN-TONG LI (a1)


This paper is concerned with the asymptotic spreading of competition diffusion systems, with the purpose of formulating the propagation modes of a co-invasion–coexistence process of two competitors. Using the comparison principle for competitive systems, some results on asymptotic spreading are obtained. Our conclusions imply that the interspecific competitions slow the invasion of one species and decrease the population densities in the coexistence domain. Therefore, the interspecific competitions play a negative role in the evolution of competitive communities.



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[1]Ahmad, S., Lazer, A. C. & Tineo, A. (2008) Traveling waves for a system of equations. Nonlinear Anal. 68 (12), 39093912.
[2]Aronson, D. G. (1977) The asymptotic speed of propagation of a simple epidemic. In: Fitzgibbon, W.E. & Walker, H. F. (editors), Nonlinear Diffusion, Pitman, London, pp. 123.
[3]Aronson, D. G. & Weinberger, H. F. (1975) Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J. A. (editor), Partial Differential Equations and Related Topics), Lecture Notes in Mathematics, Vol. 446, Springer, Berlin, Germany, pp. 549.
[4]Aronson, D. G. & Weinberger, H. F. (1978) Multidimensional nonlinear diffusion arising in population dynamics. Adv. Math. 30 (1), 3376.
[5]Bengtsson, J. (1989) Interspecific competition increases local extinction rate in a metapopulation system. Nature 340, 713715.
[6]Bleasdale, J. K. A. (1956) Interspecific competition in higher plants. Nature 178, 150151.
[7]Chesson, P. (2000) General theory of competitive coexistence in spatially-varying environments. Theor. Popul. Biol. 58, 211237.
[8]Conley, C. & Gardner, R. (1984) An application of the generalized Morse index to travelling wave solutions of a competitive reaction diffusion model. Indiana Univ. Math. J. 33 (3), 319343.
[9]Davis, M. B. (1981) Quaternary history and stability of forest communities. In: West, D.C., Shugart, H. H. & Botkin, D. B. (editors), Forest Succession: Concepts and Applications, Springer-Verlag, New York, pp. 132153.
[10]Diekmann, O. (1979) Run for your life. A note on the asymptotic speed of propagation of an epidemic. J. Differ. Equ. 33 (1), 5873.
[11]Fife, P. C. & Tang, M. (1981) Comparison principles for reaction–diffusion systems: Irregular comparison functions and applications to questions of stability and speed of propagation of disturbances. J. Differ. Equ. 40 (2), 168185.
[12]Fusco, G., Hale, J. K. & Xun, J. (1996) Traveling waves as limits of solutions on bounded domain. SIAM J. Math. Anal. 27 (6), 15441558.
[13]Gardner, S. A. (1982) Existence and stability of traveling wave solutions of competition model: A degree theoretical approach. J. Differ. Equ. 44 (3), 343364.
[14]Gilpin, M. & Ayala, F. (1973) Global models of growth and competition. Proc. Nat. Acad. Sci. USA 70, 35903593.
[15]Goel, N. S., Maitra, S. C. & Montrol, E. W. (1971) On the Volterra and other nonlinear models of interacting populations. Revs. Mod. Phys. 43, 231276.
[16]Hardin, G. (1960) The competitive exclusion principle. Science 131, 12921297.
[17]Hosono, Y. (1995) Travelling waves for a diffusive Lotka-Volterra competition model II. A geometric approach. Forma 10 (3), 235257.
[18]Hosono, Y. (2003) Traveling waves for a diffusive Lotka-Volterra competition model I. Singular perturbations. Discrete Contin. Dyn. Syst. Ser. B 3 (1), 7995.
[19]Huang, W. & Han, M. (2011) Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model. J. Differ. Equ. 251 (6), 15491561.
[20]Huston, M. A. & DeAngelis, D. L. (1994) Competition and coexistence: The effects of resource transport and supply rates. Am. Nat. 144, 954977.
[21]Kanel, J. I. & Zhou, L. (1996) Existence of wave front solutions and estimates of wave speed for a competition-diffusion system. Nonlinear Anal. 27 (5), 579587.
[22]Kan-on, Y. (1995) Parameter dependence of propagation speed of traveling waves for competition-diffusion equations. SIAM J. Math. Anal. 26 (2), 340363.
[23]Kan-on, Y. & Fang, Q. (1996) Stability of monotone travelling waves for competition-diffusion equations. Japan J. Indust. Appl. Math. 13 (2), 343349.
[24]Lewis, M. A., Li, B. & Weinberger, H. F. (2002) Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45 (3), 219233.
[25]Li, W.-T., Lin, G. & Ruan, S. (2006) Existence of travelling wave solutions in delayed reaction diffusion systems with applications to diffusion-competition systems. Nonlinearity 19 (6), 12531273.
[26]Liang, X. & Zhao, X. (2007) Asymptotic speeds of spread and traveling waves for monotone semi-flows with applications. Comm. Pure Appl. Math. 60 (1), 140.
[27]Lin, G., Li, W.-T. & Ma, M. (2010) Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete Contin. Dyn. Syst. Ser. B 13 (3), 393414.
[28]Lin, G., Li, W.-T. & Ruan, S. (2011) Spreading speeds and traveling waves in competitive recursion systems. J. Math. Biol. 62 (2), 165201.
[29]Martin, R.-H. & Smith, H. L. (1990) Abstract functional differential equations and reaction–diffusion systems. Trans. Am. Math. Soc. 321 (1), 144.
[30]Martin, R. H. & Smith, H. L. (1991) Reaction–diffusion systems with the time delay: Monotonicity, invariance, comparison and convergence. J. Reine. Angew. Math. 413, 135.
[31]May, R. M. (1973) Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ.
[32]Murray, J. D. (1993) Mathematical Biology, Springer, New York, xiv+767 pp.
[33]Okubo, A., Maini, P. K., Williamson, M. H. & Murray, J. D. (1989) On the spatial spread of the grey squirrel in Britain. Proc. R. Soc. Lond. B 238, 113125.
[34]Pan, S. (2009) Traveling wave solutions in delayed diffusion systems via a cross iteration scheme. Nonlinear Anal. Real World Appl. 10 (5), 28072818.
[35]Pao, C. V. (1992) Nonlinear Parabolic and Elliptic Equations, Plenum, New York, xvi+777 pp.
[36]Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, viii+279 pp.
[37]Ruan, S. & Wu, J. (1994) Reaction-diffusion systems with infinite delay. Canad. Appl. Math. Quart. 2, 485550.
[38]Ruan, S. & Zhao, X. (1999) Persistence and extinction in two species reaction-diffusion systems with delays. J. Differ. Equ. 156 (1), 7192.
[39]Shigesada, N. & Kawasaki, K. (1997) Biological Invasions: Theory and Practice, Oxford University Press, Oxford, UK, xiii+205 pp.
[40]Smith, H. L. (1995) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, RI, x+174 pp.
[41]Smith, H. L. & Zhao, X. (2000) Global asymptotic stability of travelling waves in delayed reaction-diffusion equations. SIAM J. Math. Anal. 31 (3), 514534.
[42]Smoller, J. (1994) Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York, xxi+581 pp.
[43]Tang, M. M. & Fife, P. (1980) Propagating fronts for competing species equations with diffusion. Arch. Ration. Mech. Anal. 73 (1), 6977.
[44]Thieme, H. R. & Zhao, X. (2003) Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models. J. Differ. Equ. 195 (2), 430470.
[45]Travis, C. C. & Webb, G. F. (1974) Existence and stability for partial functional differential equations. Trans. Am. Math. Soc. 200, 395418.
[46]Volpert, A. I., Volpert, V. A. & Volpert, V. A. (1994) Traveling Wave Solutions of Parabolic Systems (Translations of Mathematical Monographs, 140), AMS, Providence, RI, xii+448 pp.
[47]Wang, Z.-C., Li, W.-T. & Ruan, S. (2008) Travelling fronts in monostable equations with nonlocal delayed effects. J. Dyn. Differ. Equ. 20 (3), 573603.
[48]Weinberger, H. F., Lewis, M. A. & Li, B. (2002) Analysis of linear determinacy for spread in cooperative models.. J. Math. Biol. 45 (3), 183218.
[49]Wu, J. (1996) Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, x+429 pp.
[50]Ye, Q., Li, Z., Wang, M. & Wu, Y. (2011) Introduction to Reaction Diffusion Equations, Science Press, Beijing, China, xvii+450 pp.



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