Skip to main content
SpringerLink
Log in

On a population model with density dependence and Allee effect

  • Original Article
  • Published:
Theory in Biosciences Aims and scope Submit manuscript

Abstract

We study the dynamics of a discrete model with two different stages of the population, the pre-adult stage governed by a Beverton–Holt-type map and the adult stage by a \(\gamma\)-Ricker map. The composition of both maps gives the dynamics. The existence of the Allee effect is easily observed. We check that the model can evolve from a sure extinction to complicated dynamics. The presence of an almost sure extinction is proved to exist when the dynamical complexity is the highest possible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

Notes

  1. In fact, the equation provides two solutions, but one of them makes no sense for this model.

References

  • Aiello WG, Freedman HI (1990) A time-delay model of single-species growth with stage structure. Math Biosci 101:139–153

    CAS  PubMed  Google Scholar 

  • Adler RL, Konheim AG, McAndrew MH (1965) Topological entropy. Trans Am Math Soc 114:309–319

    Google Scholar 

  • Allee WC (1938) The social life of animals. William Heinemann, London

    Google Scholar 

  • Alsedá L, Llibre J, Misiurewicz M (1993) Combinatorial dynamics and entropy in dimension one. World Scientific Publishing, Singapore

    Google Scholar 

  • Avilés L (1999) Cooperation and non-linear dynamics: an ecological perspective on the evolution of sociality. Evol Ecol Res 1:459–477

    Google Scholar 

  • Ayala FJ, Gilpin MEJ, Ehrenfeld JG (1973) Competition between species: theoretical models and experimental tests. Theor Popul Biol 4:331–356

    CAS  PubMed  Google Scholar 

  • Balibrea F, Jiménez-López V (1999) The measure of scrambled sets: a survey. Acta Univ M Belii Ser Math 7:3–11

    Google Scholar 

  • Bellows TS (1982) Analytical models for laboratory populations of Callosobruchus chinensis and C. Maculatus (Coleoptera, Bruchidae). J Anim Ecol 51:263–287

    Google Scholar 

  • Bellows TS (1981) The Descriptive Properties of Some Models for Density Dependence. J Anim Ecol 50:139–156

    Google Scholar 

  • Bellows TS, Hassell MP (1988) The dynamics of age-structured host-parasitoid interactions. J Anim Ecol 57:259–268

    Google Scholar 

  • Block LS, Coppel WA (1992) Dynamics on one dimension, Lecture Notes in mathematics, 1513. Springer

  • Block LS, Keesling J, Li SH, Peterson K (1989) An improved algorithm for computing topological entropy. J Stat Phys 55:929–939

    Google Scholar 

  • Bulmer MG (1977) Periodical insects. Am Nat 111:1099–1117

    Google Scholar 

  • Courchamp F, Berec L, Gascoigne J (2008) Allee effects. Oxford University Press, Oxford

    Google Scholar 

  • Cánovas JS, Muñoz-Guillermo M (2017) On the complexity of economic dynamics: an approach through topological entropy. Chaos Solitons Fractals 103:163–176

    Google Scholar 

  • Cánovas JS, Muñoz-Guillermo M (2022) On the dynamics of a hyperbolic-exponential model of growth with density dependence. Commun Nonlinear Sci Numer Simul 104:106050

    Google Scholar 

  • de Melo W, van Strien S (1993) One dimensional dynamics. Springer Verlag, Heidelberg

    Google Scholar 

  • Elaydi SN, Sacker RJ (2010) Population models with Allee effect: a new model. J Biol Dyn 4:397–408

    PubMed  Google Scholar 

  • Fisher ME, Goh BS (1984) Stability results for delayed-recruitment models in population dynamics. J Math Biol 19:147–156

    Google Scholar 

  • Goh BS (1977) Stability in a stock-recruitment model of an exploited fishery. Math Biosci 33:359–372

    Google Scholar 

  • Guckenheimer J, Oster G, Ipaktchi A (1977) The dynamics of density dependent population models. J Math Biol 4:8–147

    CAS  PubMed  Google Scholar 

  • Graczyk J, Sands D, Światek G (2004) Metric attractors for smooth unimodal maps. Ann Math 159:725–740

    Google Scholar 

  • Hassell MP (1975) Density-dependence in single-species populations. J Anim Ecol 44:283–295

    Google Scholar 

  • Hassell MP, Comins HN (1976) Discrete time models for two-species competition. Theor Popul Biol 9:202–221

    CAS  PubMed  Google Scholar 

  • Iles T (1994) A review of stock-recruitment relationships with reference to flatfish populations. Neth J Sea Res 32:399–420

    Google Scholar 

  • Keller G (1991) Lyapunov exponents and complexity for interval maps. Lyapunov Exponents. Lecture notes in mathematics, vol. 1486. Springer, Berlin, Heidelberg

  • Kot M (2001) Elements of mathematical ecology. Cambridge University Press, Cambridge

    Google Scholar 

  • Lasota A (1977) Ergodic problems in biology. Astérisque 50:239–250

    Google Scholar 

  • Lehtonen J (2016) The Lambert W function in ecological and evolutionary models. Methods Ecol Evol 7:1110–1118

    Google Scholar 

  • Levin SA (2013) Encyclopedia of biodiversity. Elsevier, Amsterdam

    Google Scholar 

  • Leonel Rocha J, Taha AK (2020) Bifurcation analysis of the \(\gamma\)- ricker population model using the Lambert W function. Int J Bifurcat Chaos 30:07

    Google Scholar 

  • Li J (2004) Simple mathematical models for interacting wild and transgenic mosquito populations. Math Biosci 189:39–59

    PubMed  Google Scholar 

  • Li TY, Yorke JA (1975) Period three implies chaos. Am Math Month 82:985–992

    Google Scholar 

  • Liz E (2018) A global picture of the gamma-ricker map: a flexible discrete-time model with factors of positive and negative density dependence. Bull Math Biol 80:417–434

    CAS  PubMed  Google Scholar 

  • Lynch S (2005) Analysis of a blood cell population model. Int J Bifurcat Chaos 15:2311–2316

    Google Scholar 

  • May RM (1974) Biological populations with non-overlapping generations: stable points, stable cycles and chaos. Science 186:645–647

    CAS  PubMed  Google Scholar 

  • May R (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–467

    CAS  PubMed  Google Scholar 

  • May RM (2001) Stability and complexity in model ecosystems. Princeton Landmarks in Biology, Princeton

    Google Scholar 

  • May RM, Conway GR, Hassell MP, Southwood TRE (1974) Time delays, density-dependence and single-species oscillations. J Anim Ecol 43:747–770

    Google Scholar 

  • May RM, Oster GF (1976) Bifurcations and dynamic complexity in simple ecological models. Am Nat 110:573–599

    Google Scholar 

  • Milnor J (1985) On the concept of attractor. Comm Math Phys 99:177–95

    Google Scholar 

  • Misiurewicz M, Szlenk W (1980) Entropy of piecewise monotone mappings. Studia Math 67:45–63

    Google Scholar 

  • Nagoshi M, Kanda T, Horiya K (1978) Relation between exploitation and production in experimental populations of Poecilia reticulata. Popul Ecol 20:33–49

    Google Scholar 

  • Prout T, McChesney F (1985) Competition among immatures affects their adult fertility: population dynamics. Am Nat 126:521–558

    Google Scholar 

  • Quinn TJ, Derison RB (1999) Quantitative fish dynamics. Oxford University Press, Oxford

    Google Scholar 

  • Reish R, Deriso R, Ruppert D, Carroll R (1985) An investigation of the population dynamics of Atlantic Menhaden (Brevoortia tyrannus). Can J Fish Aquatic Sci 42:s147–s157

    Google Scholar 

  • Ricker WE (1954) Stock and Recruitment. J Fish Board Can 11:559–623

    Google Scholar 

  • Rocha JL, Taha AK (2019) Allee’s effect bifurcation in generalized logistic maps. Int J Bifurcat nd Chaos 29:1950039

    Google Scholar 

  • Rockwood LL (2015) Introduction to population ecology. Wiley, Hoboken

    Google Scholar 

  • Rodríguez DJ (1988) Models of growth with density regulation in more than one life stage. Theor Popul Biol 34:93–117

    PubMed  Google Scholar 

  • Singer D (1978) Stable orbits and bifurcations of maps on the interval. SIAM J Appl Math 35:260–267

    Google Scholar 

  • Smítal J (1986) Chaotic functions with zero topological entropy. Trans Am Math Soc 297:269–282

    Google Scholar 

  • Stephens PA, Sutherland WJ (2000) Vertebrate mating systems, Allee effects and conservation. In: Apollonio M, Festa-Bianchet M, Mainardi D (eds) Vertebrate mating systems. World Scientific Publishing, Singapore, pp 168–213

    Google Scholar 

  • Stephens PA, Sutherland WJ, Freckleton P (1999) What is the Allee effect? Oikos 87:185–190

    Google Scholar 

  • Varley GC, Gradwell GR (1970) Recent advances in insect population dynamics. Annu Rev Entomol 15:1–24

    Google Scholar 

  • Verhulst PF (1845) Recherches mathematiques sur la loi d’accroissement de la population. Nouv Mem l’Acad R Belles-Lett Brux 18:14–54

    Google Scholar 

  • Zheng J, Kruse GH (2003) Stock-recruitment relationships for three major Alaskan crab stocks. Fish Res 65:103–121

    Google Scholar 

  • Yang Y, Sacker RJ (2014) Periodic unimodal Allee maps, the semigroup property and the \(\lambda\)-Ricker map with Allee effect. Discr Contin Dyn Syst Ser B 19:589–606

    Google Scholar 

Download references

Funding

This work has been supported by the grant MTM2017-84079-P funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe,” by the “European Union.” We would like to thank the anonymous reviewer for his/her comments toward improving our manuscript. His/her suggestions made us to think in this paper with different views, and so we are grateful for that.

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, C/ Doctor Fleming sn, 30202, Cartagena, Spain

    J. S. Cánovas & M. Muñoz-Guillermo

Authors
  1. J. S. Cánovas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Muñoz-Guillermo
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Jose S. Cánovas and Marìa Muñoz Guillermo contributed equally to the paper’s conceptualization, methodology, formal analysis and writing of the original draft.

Corresponding author

Correspondence to J. S. Cánovas.

Ethics declarations

Conflict of interest

All authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cánovas, J.S., Muñoz-Guillermo, M. On a population model with density dependence and Allee effect. Theory Biosci. 142, 423–441 (2023). https://doi.org/10.1007/s12064-023-00407-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12064-023-00407-y

Keywords

Search

Navigation