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1. | Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413 |
2. | Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA 91711, United States |
References:
[1] |
L. A. Adamic and B. A. Huberman, Power-law distribution of the world wide web, Science, 287 (2000), 2115. |
[2] |
K. B. Athreya and J. Dai, Random logistic maps, J. Theoret. Probab., 13 (2000), 595-608.
doi: 10.1023/A:1007828804691. |
[3] |
K. B. Athreya and J. Dai, On the nonuniqueness of the invariant probability for i.i.d. random logistic maps, Ann. Probab., 30 (2002), 437-442.
doi: 10.1214/aop/1020107774. |
[4] |
R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, Fishery Invest., Ser. 2, 19 (1957), 533. |
[5] |
T. Bezandry, T. Diagana and S. Elaydi, On the stochastic Beverton-Holt equation with survival rates, J. Difference Equ. Appl., 14 (2008), 175-190.
doi: 10.1080/10236190701565610. |
[6] |
R. Bhattacharya and M. Majumdar, Random Dynamical Systems, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9780511618628. |
[7] |
R. N. Bhattacharya and B. V. Rao, Random iterations of two quadratic maps, in Stochastic Processes. A Festschrift in honor of Gopinath Kallianpur. (ed. S. Cambanis et al.), Springer Verlag, Berlin, Heidelberg, New York, 1993, 13-22. |
[8] |
O. Biham, O. Malcai, M. Levy and S. Solomon, Generic emergence of power law distributions and Lévy-Stable intermittent fluctuations in discrete logistic systems, Phys. Rev. E, 58 (1998), 1352-1358. |
[9] |
A. Blank and S. Solomon, Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components), Physica A, 287 (2000), 279-288.
doi: 10.1016/S0378-4371(00)00464-7. |
[10] |
M. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015.
doi: 10.1080/00036810701474140. |
[11] |
J. M. Cushing and S. M. Henson, Global dynamics of some periodically forced, monotone difference equations, J. Difference Equ. Appl., 7 (2001), 859-872.
doi: 10.1080/10236190108808308. |
[12] |
J. M. Cushing and S. M. Henson, A periodically forced Beverton-Holt equation, J. Difference Equ. Appl., 8 (2002), 1119-1120.
doi: 10.1080/1023619021000053980. |
[13] |
G. DaPrato, An Introduction to Infinite-Dimensional Analysis, Springer Verlag, Berlin, Heidelberg, New York, 2006.
doi: 10.1007/3-540-29021-4. |
[14] |
P. Diaconis and D. Freedman, Iterated random functions, SIAM Review., 41 (1999), 45-76.
doi: 10.1137/S0036144598338446. |
[15] |
P. Dubins and D. Freedman, Invariant probabilities for certain Markov processes, Ann. Math. Statist., 37 (1966), 837-848.
doi: 10.1214/aoms/1177699364. |
[16] |
C. Haskell and R. J. Sacker, The stochastic Beverton-Holt equation and the M. Neubert conjecture, J. Dynam. Diff. Eq., 17 (2005), 825-844.
doi: 10.1007/s10884-005-8273-x. |
[17] |
S. Kakutani, Random ergodic theorems and Markoff processes with a stable distribution, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, Los Angeles, 1951, 247-261. |
[18] |
Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, Boston, Basel, Stuttgart, 1986.
doi: 10.1007/978-1-4684-9175-3. |
[19] |
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, 2nd edition, Springer Verlag, Berlin, Heidelberg, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[20] |
Y. Louzoun, S. Solomon, H. Atlan and I. R. Cohen, Proliferation and competition in discrete biological systems, Bull. Math. Biol., 65 (2003), 375-396. |
[21] |
M. Mackey and J. G. Milton, A deterministic approach to survival statistics, J. Math. Biol., 28 (1990), 33-48.
doi: 10.1007/BF00171517. |
[22] |
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511626630. |
[23] |
E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, 1984.
doi: 10.1017/CBO9780511526237. |
[24] |
L. Pástor, E. Kisdi and G. Meszéna, Jensen's inequality and optimal life history strategies in stochastic environments, Trends. Ecol. Evol., 15 (2000), 117-118. |
[25] |
S. Solomon and P. Richmond, Power laws of wealth, market order volumes and market returns, Physica A, 299 (2001), 188-197. |
[26] |
S. Ulam and J. von Neumann, Random ergodic theorems (Abstract # 165), Bull. Amer. Math. Soc., 51 (1945), 660. |
show all references
References:
[1] |
L. A. Adamic and B. A. Huberman, Power-law distribution of the world wide web, Science, 287 (2000), 2115. |
[2] |
K. B. Athreya and J. Dai, Random logistic maps, J. Theoret. Probab., 13 (2000), 595-608.
doi: 10.1023/A:1007828804691. |
[3] |
K. B. Athreya and J. Dai, On the nonuniqueness of the invariant probability for i.i.d. random logistic maps, Ann. Probab., 30 (2002), 437-442.
doi: 10.1214/aop/1020107774. |
[4] |
R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, Fishery Invest., Ser. 2, 19 (1957), 533. |
[5] |
T. Bezandry, T. Diagana and S. Elaydi, On the stochastic Beverton-Holt equation with survival rates, J. Difference Equ. Appl., 14 (2008), 175-190.
doi: 10.1080/10236190701565610. |
[6] |
R. Bhattacharya and M. Majumdar, Random Dynamical Systems, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9780511618628. |
[7] |
R. N. Bhattacharya and B. V. Rao, Random iterations of two quadratic maps, in Stochastic Processes. A Festschrift in honor of Gopinath Kallianpur. (ed. S. Cambanis et al.), Springer Verlag, Berlin, Heidelberg, New York, 1993, 13-22. |
[8] |
O. Biham, O. Malcai, M. Levy and S. Solomon, Generic emergence of power law distributions and Lévy-Stable intermittent fluctuations in discrete logistic systems, Phys. Rev. E, 58 (1998), 1352-1358. |
[9] |
A. Blank and S. Solomon, Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components), Physica A, 287 (2000), 279-288.
doi: 10.1016/S0378-4371(00)00464-7. |
[10] |
M. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015.
doi: 10.1080/00036810701474140. |
[11] |
J. M. Cushing and S. M. Henson, Global dynamics of some periodically forced, monotone difference equations, J. Difference Equ. Appl., 7 (2001), 859-872.
doi: 10.1080/10236190108808308. |
[12] |
J. M. Cushing and S. M. Henson, A periodically forced Beverton-Holt equation, J. Difference Equ. Appl., 8 (2002), 1119-1120.
doi: 10.1080/1023619021000053980. |
[13] |
G. DaPrato, An Introduction to Infinite-Dimensional Analysis, Springer Verlag, Berlin, Heidelberg, New York, 2006.
doi: 10.1007/3-540-29021-4. |
[14] |
P. Diaconis and D. Freedman, Iterated random functions, SIAM Review., 41 (1999), 45-76.
doi: 10.1137/S0036144598338446. |
[15] |
P. Dubins and D. Freedman, Invariant probabilities for certain Markov processes, Ann. Math. Statist., 37 (1966), 837-848.
doi: 10.1214/aoms/1177699364. |
[16] |
C. Haskell and R. J. Sacker, The stochastic Beverton-Holt equation and the M. Neubert conjecture, J. Dynam. Diff. Eq., 17 (2005), 825-844.
doi: 10.1007/s10884-005-8273-x. |
[17] |
S. Kakutani, Random ergodic theorems and Markoff processes with a stable distribution, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, Los Angeles, 1951, 247-261. |
[18] |
Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, Boston, Basel, Stuttgart, 1986.
doi: 10.1007/978-1-4684-9175-3. |
[19] |
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, 2nd edition, Springer Verlag, Berlin, Heidelberg, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[20] |
Y. Louzoun, S. Solomon, H. Atlan and I. R. Cohen, Proliferation and competition in discrete biological systems, Bull. Math. Biol., 65 (2003), 375-396. |
[21] |
M. Mackey and J. G. Milton, A deterministic approach to survival statistics, J. Math. Biol., 28 (1990), 33-48.
doi: 10.1007/BF00171517. |
[22] |
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511626630. |
[23] |
E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, 1984.
doi: 10.1017/CBO9780511526237. |
[24] |
L. Pástor, E. Kisdi and G. Meszéna, Jensen's inequality and optimal life history strategies in stochastic environments, Trends. Ecol. Evol., 15 (2000), 117-118. |
[25] |
S. Solomon and P. Richmond, Power laws of wealth, market order volumes and market returns, Physica A, 299 (2001), 188-197. |
[26] |
S. Ulam and J. von Neumann, Random ergodic theorems (Abstract # 165), Bull. Amer. Math. Soc., 51 (1945), 660. |
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