Journal of Mathematical Biology

, Volume 55, Issue 2, pp 207–222 | Cite as

Anomalous spreading speeds of cooperative recursion systems

  • Hans F. Weinberger
  • Mark A. LewisEmail author
  • Bingtuan Li


This work presents an example of a cooperative system of truncated linear recursions in which the interaction between species causes one of the species to have an anomalous spreading speed. By this we mean that this species spreads at a speed which is strictly greater than its spreading speed in isolation from the other species and the speeds at which all the other species actually spread. An ecological implication of this example is discussed in Sect. 5. Our example shows that the formula for the fastest spreading speed given in Lemma 2.3 of our paper (Weinberger et al. in J Math Biol 45:183–218, 2002) is incorrect. However, we find an extra hypothesis under which the formula for the faster spreading speed given in (Weinberger et al. in J Math Biol 45:183–218, 2002) is valid. We also show that the hypotheses of all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) whose proofs rely on Lemma 2.3 imply this extra hypothesis, so that all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) and all the examples given there are valid as they stand.

Mathematics Subject Classification (2000)

92D40 92D25 35K55 35K57 


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  1. 1.
    Lewis M.A., Li B. and Weinberger H.F. (2002). Spreading speeds and linear determinacy for two-species competition models. J. Math. Biol. 45: 219–233 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Li B., Weinberger H.F. and Lewis M.A. (2005). Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosc. 196: 82–98 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Lui R. (1989). Biological growth and spread modeled by systems of recursions. I Mathematical theory. Math. Biosc. 93: 269–295 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Weinberger H.F., Lewis M.A. and Li B. (2002). Analysis of the linear conjecture for spread in cooperative models. J. Math. Biol. 45: 183–218 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Hans F. Weinberger
    • 1
  • Mark A. Lewis
    • 2
    Email author
  • Bingtuan Li
    • 3
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Mathematical and Statistical Sciences, Department of Biological SciencesCAB 632 University of AlbertaEdmontonCanada
  3. 3.Department of MathematicsUniversity of LouisvilleLouisvilleUSA

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