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September  2016, 21(7): 2193-2210. doi: 10.3934/dcdsb.2016043

Ergodicity and loss of capacity for a random family of concave maps

Peter Hinow 1, and  Ami Radunskaya 2,

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

Received  July 2015 Revised  March 2016 Published  August 2016

Random fluctuations of an environment are common in ecological and economical settings. We consider a family of concave quadratic polynomials on the unit interval that model a self-limiting growth behavior. The maps are parametrized by an independent, identically distributed random parameter. We show the existence of a unique invariant ergodic measure of the resulting random dynamical system. Moreover, there is an attenuation of the mean of the state variable compared to the constant environment with the averaged parameter.
Keywords: Random dynamical systems, ergodic measure, self-limiting growth..
Mathematics Subject Classification: Primary: 37H10, 37A20; Secondary: 92D2.
Citation: Peter Hinow, Ami Radunskaya. Ergodicity and loss of capacity for a random family of concave maps. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2193-2210. doi: 10.3934/dcdsb.2016043
References:
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L. A. Adamic and B. A. Huberman, Power-law distribution of the world wide web, Science, 287 (2000), 2115. Google Scholar

[2]

K. B. Athreya and J. Dai, Random logistic maps, J. Theoret. Probab., 13 (2000), 595-608. doi: 10.1023/A:1007828804691.  Google Scholar

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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.  Google Scholar

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R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, Fishery Invest., Ser. 2, 19 (1957), 533. Google Scholar

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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.  Google Scholar

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R. Bhattacharya and M. Majumdar, Random Dynamical Systems, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618628.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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M. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015. doi: 10.1080/00036810701474140.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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G. DaPrato, An Introduction to Infinite-Dimensional Analysis, Springer Verlag, Berlin, Heidelberg, New York, 2006. doi: 10.1007/3-540-29021-4.  Google Scholar

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P. Diaconis and D. Freedman, Iterated random functions, SIAM Review., 41 (1999), 45-76. doi: 10.1137/S0036144598338446.  Google Scholar

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P. Dubins and D. Freedman, Invariant probabilities for certain Markov processes, Ann. Math. Statist., 37 (1966), 837-848. doi: 10.1214/aoms/1177699364.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[18]

Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, Boston, Basel, Stuttgart, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

M. Mackey and J. G. Milton, A deterministic approach to survival statistics, J. Math. Biol., 28 (1990), 33-48. doi: 10.1007/BF00171517.  Google Scholar

[22]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.  Google Scholar

[23]

E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, 1984. doi: 10.1017/CBO9780511526237.  Google Scholar

[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. Google Scholar

[25]

S. Solomon and P. Richmond, Power laws of wealth, market order volumes and market returns, Physica A, 299 (2001), 188-197. Google Scholar

[26]

S. Ulam and J. von Neumann, Random ergodic theorems (Abstract # 165), Bull. Amer. Math. Soc., 51 (1945), 660. Google Scholar

show all references

References:
[1]

L. A. Adamic and B. A. Huberman, Power-law distribution of the world wide web, Science, 287 (2000), 2115. Google Scholar

[2]

K. B. Athreya and J. Dai, Random logistic maps, J. Theoret. Probab., 13 (2000), 595-608. doi: 10.1023/A:1007828804691.  Google Scholar

[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.  Google Scholar

[4]

R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, Fishery Invest., Ser. 2, 19 (1957), 533. Google Scholar

[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.  Google Scholar

[6]

R. Bhattacharya and M. Majumdar, Random Dynamical Systems, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618628.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

M. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015. doi: 10.1080/00036810701474140.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

G. DaPrato, An Introduction to Infinite-Dimensional Analysis, Springer Verlag, Berlin, Heidelberg, New York, 2006. doi: 10.1007/3-540-29021-4.  Google Scholar

[14]

P. Diaconis and D. Freedman, Iterated random functions, SIAM Review., 41 (1999), 45-76. doi: 10.1137/S0036144598338446.  Google Scholar

[15]

P. Dubins and D. Freedman, Invariant probabilities for certain Markov processes, Ann. Math. Statist., 37 (1966), 837-848. doi: 10.1214/aoms/1177699364.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, Boston, Basel, Stuttgart, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

M. Mackey and J. G. Milton, A deterministic approach to survival statistics, J. Math. Biol., 28 (1990), 33-48. doi: 10.1007/BF00171517.  Google Scholar

[22]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.  Google Scholar

[23]

E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, 1984. doi: 10.1017/CBO9780511526237.  Google Scholar

[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. Google Scholar

[25]

S. Solomon and P. Richmond, Power laws of wealth, market order volumes and market returns, Physica A, 299 (2001), 188-197. Google Scholar

[26]

S. Ulam and J. von Neumann, Random ergodic theorems (Abstract # 165), Bull. Amer. Math. Soc., 51 (1945), 660. Google Scholar

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