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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 |
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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