Stochastic difference equations with the Allee effect

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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

Abstract
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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 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.

show all references

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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