Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On Allee effects in structured populations

Author: Sebastian J. Schreiber
Journal: Proc. Amer. Math. Soc. 132 (2004), 3047-3053
MSC (2000): Primary 37N25, 92D25, 37C65
Published electronically: May 12, 2004
MathSciNet review: 2063126
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Maps $f(x)=A(x)x$ of the nonnegative cone $C$ of ${\mathbf R}^k$into itself are considered where $A(x)$ are nonnegative, primitive matrices with nondecreasing entries and at least one increasing entry. Let $\lambda(x)$ denote the dominant eigenvalue of $A(x)$ and $\lambda(\infty)=\sup_{x\in C} \lambda(x)$. These maps are shown to exhibit a dynamical trichotomy. First, if $\lambda(0)\ge 1$, then $\lim_{n\to\infty} \Vert f^n(x)\Vert=\infty$ for all nonzero $x\in C$. Second, if $\lambda(\infty)\le 1$, then $\lim_{n\to\infty}f^n(x)=0$ for all $x\in C$. Finally, if $\lambda(0)<1$ and $\lambda(\infty)>1$, then there exists a compact invariant hypersurface $\Gamma$ separating $C$. For $x$ below $\Gamma$, $\lim_{n\to\infty}f^n(x)=0$, while for $x$ above, $\lim_{n\to\infty}\Vert f^n(x)\Vert=\infty$. An application to nonlinear Leslie matrices is given.

References [Enhancements On Off] (What's this?)

  • 1. Michel Benaïm, On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 137 (1997), no. 2, 302–319. MR 1456599,
  • 2. H. Caswell, Matrix population models, Sinauer, Sunderland, Massachuesetts, 2001.
  • 3. F. Courchamp, T. Clutton-Brock, and B. Grenfell, Inverse density dependence and the Allee effect, TREE 14 (1999), 405-410.
  • 4. Michael P. Hassell, The dynamics of arthropod predator-prey systems, Monographs in Population Biology, vol. 13, Princeton University Press, Princeton, N.J., 1978. MR 508052
  • 5. D. Ruelle, Analycity properties of the characteristic exponents of random matrix products, Adv. in Math. 32 (1979), no. 1, 68–80. MR 534172,
  • 6. Sebastian J. Schreiber, On growth rates of subadditive functions for semiflows, J. Differential Equations 148 (1998), no. 2, 334–350. MR 1643183,
  • 7. E. Seneta, Nonnegative matrices and Markov chains, 2nd ed., Springer Series in Statistics, Springer-Verlag, New York, 1981. MR 719544
  • 8. P. A. Stephens and W. J. Sutherland, Conseqeuences of the Allee effect for behavior, ecology, and conservation, TREE 14 (1999), 401-405.
  • 9. P. A. Stephens, W. J. Sutherland, and R. P. Freckleton, What is the Allee effect?, Oikos 87 (1999), 185-190.
  • 10. Peter Takáč, Domains of attraction of generic 𝜔-limit sets for strongly monotone discrete-time semigroups, J. Reine Angew. Math. 423 (1992), 101–173. MR 1142485,
  • 11. I. Terescák, Dynamics of $C^1$ smooth strongly monotone discrete-time dynamical systems, preprint (1996).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37N25, 92D25, 37C65

Retrieve articles in all journals with MSC (2000): 37N25, 92D25, 37C65

Additional Information

Sebastian J. Schreiber
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795

Received by editor(s): March 17, 2003
Received by editor(s) in revised form: May 20, 2003
Published electronically: May 12, 2004
Additional Notes: This research was supported in part by National Science Foundation Grant DMS-0077986
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society