# American Institute of Mathematical Sciences

November  2016, 36(11): 5929-5949. doi: 10.3934/dcds.2016060

## Stochastic difference equations with the Allee effect

 1 Dept. of Math. and Stats., University of Calgary, 2500 University Drive N.W., Calgary, AB, Canada T2N 1N4, Canada 2 Department of Mathematics and Computer Science, The University of the West Indies, Mona, Kingston 7

Received  March 2015 Revised  May 2016 Published  August 2016

For a stochastically perturbed equation $x_{n+1} =\max\{f(x_n)+l\chi_{n+1}, 0 \}$ with $f(x) < x$ on $(0,m)$, which corresponds to the Allee effect, we observe that for very small perturbation amplitude $l$, the eventual behavior is similar to a non-perturbed case: there is extinction for small initial values in $(0,m-\varepsilon)$ and persistence for $x_0 \in (m + \delta, H]$ for some $H$ satisfying $H > f(H)> m$. As the amplitude grows, an interval $(m-\varepsilon, m + \delta)$ of initial values arises and expands, such that with a certain probability, $x_n$ sustains in $[m, H]$, and possibly eventually gets into the interval $(0,m-\varepsilon)$, with a positive probability. Lower estimates for these probabilities are presented. If $H$ is large enough, as the amplitude of perturbations grows, the Allee effect disappears: a solution persists for any positive initial value.
Citation: Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060
##### References:
 [1] W. C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, 1931. [2] J. A. D. Appleby, G. Berkolaiko and A. Rodkina, On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations, J. Difference Equ. Appl., 14 (2008), 923-951. doi: 10.1080/10236190701871786. [3] J. A. D. Appleby, G. Berkolaiko and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise, Stochastics, 81 (2009), 99-127. doi: 10.1080/17442500802088541. [4] J. A. D. Appleby, X. Mao and A. Rodkina, A., On stochastic stabilization of difference equations, Dynamics of Continuous and Discrete System, 15 (2006), 843-857. doi: 10.3934/dcds.2006.15.843. [5] G. Berkolaiko and A. Rodkina, Almost sure convergence of solutions to nonhomogeneous stochastic difference equation, J. Difference Equ. Appl., 12 (2006), 535-553. doi: 10.1080/10236190600574093. [6] D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theor. Biol., 218 (2002), 375-394. doi: 10.1006/jtbi.2002.3084. [7] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4757-3516-1. [8] E. Braverman, Random perturbations of difference equations with Allee effect: Switch of stability properties, Proceedings of the Workshop Future Directions in Difference Equations, 51-60, Colecc. Congr., 69, Univ. Vigo, Serv. Publ., Vigo, 2011. [9] E. Braverman and J. J. Haroutunian, Chaotic and stable perturbed maps: 2-cycles and spatial models, Chaos, 20 (2010), 023114. doi: 10.1063/1.3404774. [10] E. Braverman and A. Rodkina, Stabilization of two-cycles of difference equations with stochastic perturbations, J. Difference Equ. Appl., 19 (2013), 1192-1212. doi: 10.1080/10236198.2012.726989. [11] E. Braverman and A. Rodkina, Difference equations of Ricker and logistic types under bounded stochastic perturbations with positive mean, Comput. Math. Appl., 66 (2013), 2281-2294. doi: 10.1016/j.camwa.2013.06.014. [12] M. A. Burgman, S. Ferson and H. R. Akćakaya, Risk Assessment in Conservation Biology, Chapman & Hall, London, 1993. [13] S. N. Cohen and R. J. Elliott, Backward stochastic difference equations and nearly time-consistent nonlinear expectations, SIAM J. Control Optim., 49 (2011), 125-139. doi: 10.1137/090763688. [14] N. Dokuchaev and A. Rodkina, Instability and stability of solutions of systems of nonlinear stochastic difference equations with diagonal noise, J. Difference Equ. Appl., 14 (2014), 744-764. doi: 10.1080/10236198.2013.815748. [15] F. C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, Cambridge, MA, 1982. [16] J. Jacobs, Cooperation, optimal density and low density thresholds: Yet another modification of the logistic model, Oecologia, 64 (1984), 389-395. doi: 10.1007/BF00379138. [17] C. Kelly and A. Rodkina, Constrained stability and instability of polynomial difference equations with state-dependent noise, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 913-933. doi: 10.3934/dcdsb.2009.11.913. [18] V. Kolmanovskii and L. Shaikhet, Some conditions for boundedness of solutions of difference Volterra equations, Appl. Math. Lett., 16 (2003), 857-862. doi: 10.1016/S0893-9659(03)90008-5. [19] A. Rodkina and M. Basin, On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term, Syst. Control Lett., 56 (2007), 423-430. doi: 10.1016/j.sysconle.2006.11.001. [20] S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8. [21] S. J. Schreiber, Persistence for stochastic difference equations: A mini-review, J. Difference Equ. Appl., 18 (2012), 1381-1403. doi: 10.1080/10236198.2011.628662. [22] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, 2011. doi: 10.1007/978-0-85729-685-6. [23] A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1.

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##### References:
 [1] W. C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, 1931. [2] J. A. D. Appleby, G. Berkolaiko and A. Rodkina, On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations, J. Difference Equ. Appl., 14 (2008), 923-951. doi: 10.1080/10236190701871786. [3] J. A. D. Appleby, G. Berkolaiko and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise, Stochastics, 81 (2009), 99-127. doi: 10.1080/17442500802088541. [4] J. A. D. Appleby, X. Mao and A. Rodkina, A., On stochastic stabilization of difference equations, Dynamics of Continuous and Discrete System, 15 (2006), 843-857. doi: 10.3934/dcds.2006.15.843. [5] G. Berkolaiko and A. Rodkina, Almost sure convergence of solutions to nonhomogeneous stochastic difference equation, J. Difference Equ. Appl., 12 (2006), 535-553. doi: 10.1080/10236190600574093. [6] D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theor. Biol., 218 (2002), 375-394. doi: 10.1006/jtbi.2002.3084. [7] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4757-3516-1. [8] E. Braverman, Random perturbations of difference equations with Allee effect: Switch of stability properties, Proceedings of the Workshop Future Directions in Difference Equations, 51-60, Colecc. Congr., 69, Univ. Vigo, Serv. Publ., Vigo, 2011. [9] E. Braverman and J. J. Haroutunian, Chaotic and stable perturbed maps: 2-cycles and spatial models, Chaos, 20 (2010), 023114. doi: 10.1063/1.3404774. [10] E. Braverman and A. Rodkina, Stabilization of two-cycles of difference equations with stochastic perturbations, J. Difference Equ. Appl., 19 (2013), 1192-1212. doi: 10.1080/10236198.2012.726989. [11] E. Braverman and A. Rodkina, Difference equations of Ricker and logistic types under bounded stochastic perturbations with positive mean, Comput. Math. Appl., 66 (2013), 2281-2294. doi: 10.1016/j.camwa.2013.06.014. [12] M. A. Burgman, S. Ferson and H. R. Akćakaya, Risk Assessment in Conservation Biology, Chapman & Hall, London, 1993. [13] S. N. Cohen and R. J. Elliott, Backward stochastic difference equations and nearly time-consistent nonlinear expectations, SIAM J. Control Optim., 49 (2011), 125-139. doi: 10.1137/090763688. [14] N. Dokuchaev and A. Rodkina, Instability and stability of solutions of systems of nonlinear stochastic difference equations with diagonal noise, J. Difference Equ. Appl., 14 (2014), 744-764. doi: 10.1080/10236198.2013.815748. [15] F. C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, Cambridge, MA, 1982. [16] J. Jacobs, Cooperation, optimal density and low density thresholds: Yet another modification of the logistic model, Oecologia, 64 (1984), 389-395. doi: 10.1007/BF00379138. [17] C. Kelly and A. Rodkina, Constrained stability and instability of polynomial difference equations with state-dependent noise, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 913-933. doi: 10.3934/dcdsb.2009.11.913. [18] V. Kolmanovskii and L. Shaikhet, Some conditions for boundedness of solutions of difference Volterra equations, Appl. Math. Lett., 16 (2003), 857-862. doi: 10.1016/S0893-9659(03)90008-5. [19] A. Rodkina and M. Basin, On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term, Syst. Control Lett., 56 (2007), 423-430. doi: 10.1016/j.sysconle.2006.11.001. [20] S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8. [21] S. J. Schreiber, Persistence for stochastic difference equations: A mini-review, J. Difference Equ. Appl., 18 (2012), 1381-1403. doi: 10.1080/10236198.2011.628662. [22] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, 2011. doi: 10.1007/978-0-85729-685-6. [23] A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1.
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