Abstract
This paper is concerned with the spreading speeds and traveling wave solutions of discrete time recursion systems, which describe the spatial propagation mode of two competitive invaders. We first establish the existence of traveling wave solutions when the wave speed is larger than a given threshold. Furthermore, we prove that the threshold is the spreading speed of one species while the spreading speed of the other species is distinctly slower compared to the case when the interspecific competition disappears. Our results also show that the interspecific competition does affect the spread of both species so that the eventual population densities at the coexistence domain are lower than the case when the competition vanishes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Allen EJ, Allen LJS, Gilliam X (1996) Dispersal and competition models for plants. J Math Biol 34: 455–481
Aronson DG (1977) The asymptotic speed of propagation of a simple epidemic. In: Fitzgibbon WE, Walker HF (eds) Nonlinear diffusion. Pitman, London, pp 1–23
Aronson DG, Weinberger HF (1975) Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein JA (ed) Partial differential equations and related topics. Lecture notes in mathematics, vol 446. Springer, Berlin, pp 5–49
Aronson DG, Weinberger HF (1978) Multidimensional nonlinear diffusion arising in population dynamics. Adv Math 30: 33–76
Bengtsson J (1989) Interspecific competition increases local extinction rate in a metapopulation system. Nature 340: 713–715
Bleasdale JKA (1956) Interspecific competition in higher plants. Nature 178: 150–151
Britton N (1986) Reaction-diffusion equations and their applications to biology. Academic Press, San Diego
Carrillo C, Fife P (2005) Spatial effects in discrete generation population models. J Math Biol 50: 161–188
Cheng CP, Li W-T, Wang ZC (2008) Spreading speeds and traveling waves in a delayed population model with stage structure on a two-dimensional spatial lattice. IMA J Appl Math 73: 592–618
Cushing JM, Levarge S, Chitnis N, Henson SM (2004) Some discrete competition models and the competitive exclusion principle. J Differ Equ Appl 10: 1139–1151
Darlington PJ (1972) Competition, competitive repulsion, and coexistence. Proc Natl Acad Sci USA 69: 3151–3155
Diekmann O (1978) Thresholds and traveling waves for the geographical spread of infection. J Math Biol 6: 109–130
Diekmann O (1979) Run for your life. A note on the asymptotic speed of propagation of an epidemic. J Differ Equ 33: 58–73
Fife P (1979) Mathematical aspects of reacting and diffusing systems. Lecture notes in biomathematics, vol 28. Springer, Berlin
Hardin G (1960) The competitive exclusion principle. Science 131: 1292–1297
Hsu SB, Zhao XQ (2008) Spreading speeds and traveling waves for nonmonotone integrodifference equations. SIAM J Math Anal 40: 776–789
Kot M (1992) Discrete-time travelling waves: ecological examples. J Math Biol 30: 413–436
Leung AW (1989) Systems of nonlinear partial differential equations: applications to biology and engineering. Kluwer Academic Publishers, Dordrecht
Lewis MA (2000) Spread rate for a nonlinear stochastic invasion. J Math Biol 41: 430–454
Lewis MA, Li B, Weinberger HF (2002) Spreading speed and linear determinacy for two-species competiotion models. J Math Biol 45: 219–233
Li S (1992) Population ecology of freshwater fishes. Agricultural Press of China, Beijing
Li B (2009) Some remarks on traveling wave solutions in competition models. Discrete Contin Dyn Syst Ser B 12: 389–399
Li B, Weinberger HF, Lewis MA (2005) Spreading speeds as slowest wave speeds for cooperative systems. Math Biosci 196: 82–98
Li W-T, Lin G, Ruan S (2006) Existence of traveling wave solutions in delayed reaction diffusion systems with applications to diffusion-competition systems. Nonlinearity 19: 1253–1273
Liang X, Zhao XQ (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm Pure Appl Math 60: 1–40
Lin G, Li W-T, Ma M (2010) Travelling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete Contin Dyn Syst Ser B 19: 393–414
Lin G, Li W-T, Ruan S Asymptotic stability of monostable wavefronts in discrete-time integral recursions. Sci China Ser A (2010). doi:10.1007/s11425-009-0123-6
Lui R (1989a) Biological growth and spread modeled by systems of recursions. I. Mathematical theory. Math Biosci 93: 269–295
Lui R (1989b) Biological growth and spread modeled by systems of recursions. II. Biological theory. Math Biosci 93: 297–312
Ma S (2007) Traveling waves for non-local delayed diffusion equations via auxiliary equations. J Differ Equ 237: 259–277
Neubert MG, Caswell H (2000) Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations. Ecology 81: 1613–1628
Pan S (2009) Traveling wave solutions in delayed diffusion systems via a cross iteration scheme. Nonlinear Anal RWA 10: 2807–2818
Pao CV (1992) Nonlinear parabolic and elliptic equations. Plenum, New York
Pao CV (2005) Strongly coupled elliptic systems and applications to Lotka–Volterra models with cross-diffusion. Nonlinear Anal TMA 60: 1197–1217
Radcliffe J, Rass L (1983) Wave solutions for the deterministic nonreducible n-type epidemic. J Math Biol 17: 45–66
Radcliffe J, Rass L (1984) The uniqueness of wave solutions for the deterministic nonreducible n-type epidemic. J Math Biol 19: 303–308
Radcliffe J, Rass L (1986) The asymptotic spread of propagation of the deterministic non-reducible n-type epidemic. J Math Biol 23: 341–359
Smoller J (1994) Shock waves and reaction diffusion equations. Springer, New York
Thieme HR (1979a) Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J Reine Angew Math 306: 94–121
Thieme HR (1979b) Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J Math Biol 8: 173–187
Thieme HR, Zhao XQ (2003) Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models. J Differ Equ 195: 430–470
van den Bosch F, Metz JAJ, Diekmann O (1990) The velocity of spatial population expansion. J Math Biol 28:529–565
Volpert AI, Volpert VA, Volpert VA (1994) Traveling wave solutions of parabolic systems. Translations of mathematical monographs, vol 140. American Mathematical Society, Providence
Weinberger HF (1982) Long-time behavior of a class of biological model. SIAM J Math Anal 13: 353–396
Weinberger HF (2002) On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J Math Biol 45: 511–548
Weinberger HF, Lewis MA, Li B (2002) Analysis of linear determinacy for spread in cooperative models. J Math Biol 45: 183–218
Weinberger HF, Lewis MA, Li B (2007) Anomalous spreading speeds of cooperative recursion systems. J Math Biol 55: 207–222
Weng P, Huang H, Wu J (2003) Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J Appl Math 68: 409–439
Ye Q, Li Z (1990) Introduction to reaction diffusion equations. Science Press, Beijing
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lin, G., Li, WT. & Ruan, S. Spreading speeds and traveling waves in competitive recursion systems. J. Math. Biol. 62, 165–201 (2011). https://doi.org/10.1007/s00285-010-0334-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-010-0334-z