Abstract
In this letter, we test a scalar stochastic nonlinear equation used to portray the growth of a population with Allee effects. We first testify that there is a unique dynamical bifurcation point Λ to the equation, and the sign of Λ determines the dynamical properties of the equation: if Λ is negative, then the equation has a unique invariant measure — the Dirac measure concentrated at zero; if Λ is positive, the equation has a unique invariant measure concentrated on \((0,+\infty )\), and the density function of the invariant measure can be expressed explicitly. Then we probe the lower-growth rate and the continuity of the solution. Finally, we apply the theoretical results to research the growth of African hunting dogs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
All data generated or analysed during this study are included in this published article.
References
Allee W (1949) Principles of animal ecology. Saunders Co, Philadelphia
Arnold L (1998) Random dynamical systems. Springer, New York
Boukal D, Berec L (2002) Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. J Theor Biol 218:375–394
Courchamp F, Clutton-Brock T, Grenfell B (1999) Inverse density dependence and the Allee effect. Trends Ecol Evol 14:405–410
Creel S, Mills M, McNutt J (2004) Demography and population dynamics of africanwild dogs in three critical populations in biology and conservations of wild canids. Oxford University Press, London
Davies-Mostert H, Mills M, Macdonald D (2015) The demography and dynamics of an expanding, managed African wild dog metapopulation. Afr J Wildl Res 45:258–273
Gard T (1988) Introduction to stochastic differential equations. Marcel Dekker, INC, New York
Huang Z, Yang Q, Cao J (2011) Stochastic stability and bifurcation for the chronic state in Marchuks model with noise. Appl Math Model 35:5842–5855
Ji W (2020) On a populationmodel with Allee effects and environmental perturbations. J Appl Math Comput 64:749–764
Ji W, Liu M (2020) Dynamics of a stochastic population model with Allee effects under regime switching. Adv Differ Equ 295:1–14
Karatzas J, Shreve S (1988) Brownian motion and stochastic calculus. Springer-Verlag, New York
Takeuchi Y (1996) Global dynamical properties of lotka-volterra systems. World Scientific Publishing Company, Singapore
Woodroffe R, Sillero-Zubiri C (2020) Lycaon pictus (amended version of 2012 assessment), The IUCN Red List of Threatened Species 2020 e.T12436A166502262. https://doi.org/10.2305/IUCN.UK.2020-1.RLTS.T12436A166502262.en
Xia J, Sun S, Liu G (2013) Evidence of a component Allee effect driven by predispersal seed predation in a plant (Pedicularis rex, Orobanchaceae). Biol Lett 9:20130387
Zou X, Zheng Y, Zhang L, Lv J (2020) Survivability and stochastic bifurcations for a stochastic Holling type II predator-prey model. Commun Nonlinear Sci Numer Simul 83:105136
Author information
Authors and Affiliations
Contributions
FMZ mainly finished the writing of the whole content of the paper. FMZ and GXH mainly finished the establishment of model and development. Both authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zheng, F., Hu, G. Dynamical Behaviors of a Stochastic Single-Species Model with Allee Effects. Methodol Comput Appl Probab 24, 1553–1563 (2022). https://doi.org/10.1007/s11009-021-09874-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-021-09874-6