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

Stochastic difference equations with the Allee effect

Elena Braverman 1, and  Alexandra Rodkina 2,

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.
Keywords: a.s. persistence, Stochastic difference equations, a.s. low density, population dynamics., Allee effect.
Mathematics Subject Classification: Primary: 39A50, 37H10; Secondary: 93E10, 92D2.
Citation: Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. A. Burgman, S. Ferson and H. R. Akćakaya, Risk Assessment in Conservation Biology, Chapman & Hall, London, 1993. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, Cambridge, MA, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, 2011. doi: 10.1007/978-0-85729-685-6.  Google Scholar

[23]

A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar

show all references

References:
[1]

W. C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, 1931. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. A. Burgman, S. Ferson and H. R. Akćakaya, Risk Assessment in Conservation Biology, Chapman & Hall, London, 1993. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, Cambridge, MA, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, 2011. doi: 10.1007/978-0-85729-685-6.  Google Scholar

[23]

A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar

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