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

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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

Abstract
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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 and Continuous Dynamical Systems - B, 2016, 21 (7) : 2193-2210. doi: 10.3934/dcdsb.2016043

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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