Uncertainty in a Measurement of Density Dependence on Population Fluctuations

Author(s)
Hiro-Sato Niwa

ABSTRACT

This article discusses the question of how elasticity of the system is intertwined with external stochastic disturbances. The speed at which a displaced system returns to its equilibrium is a measure of density dependence in population dynamics. Population dynamics in random environments, linearized around the equilibrium point, can be represented by a Langevin equation, where populations fluctuate under locally stable (not periodic or chaotic) dynamics. I consider a Langevin model in discrete time, driven by time-correlated random forces, and examine uncertainty in locating the population equilibrium. There exists a time scale such that for times shorter than this scale the dynamics can be approximately described by a random walk; it is difficult to know whether the system is heading toward the equilibrium point. Density dependence is a concept that emerges from a proper coarse-graining procedure applied for time-series analysis of population data. The analysis is illustrated using time-series data from fisheries in the North Atlantic, where fish populations are buffeted by stochastic harvesting in a random environment.

KEYWORDS

Population Dynamics, Stochastic Difference Equation, Noise Color, Coarse Graining, Ecological Time-Series

Population Dynamics, Stochastic Difference Equation, Noise Color, Coarse Graining, Ecological Time-Series

Cite this paper

Niwa, H. (2014) Uncertainty in a Measurement of Density Dependence on Population Fluctuations.*Applied Mathematics*, **5**, 1108-1119. doi: 10.4236/am.2014.58104.

Niwa, H. (2014) Uncertainty in a Measurement of Density Dependence on Population Fluctuations.

References

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http://dx.doi.org/10.1098/rstb.2002.1117

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http://dx.doi.org/10.1086/338988

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http://dx.doi.org/10.1126/science.283.5407.1528

[21] Niwa, H.S. (2006) Recruitment Variability in Exploited Aquatic Populations. Aquatic Living Resources, 19, 195-206.

http://dx.doi.org/10.1051/alr:2006020

[22] von Neumann, J. (1941) Distribution of the Ratio of the Mean Square Successive Difference to the Variance. Annals of Mathematical Statistics, 12, 367-395.

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http://dx.doi.org/10.2307/2529815

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http://dx.doi.org/10.1007/978-3-642-58244-8

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http://dx.doi.org/10.1093/icesjms/fsm004

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http://dx.doi.org/10.1093/icesjms/fsm038

[27] ICES (2008) Report of the ICES Advisory Committee, ICES Advice 2008, Books 1-10.

http://www.ices.dk/publications/our-publications/Pages/ICES-Advice.aspx

[1] Sibly, R.M. and Hone, J. (2002) Population Growth Rate and its Determinants: An Overview. Philosophical Transactions of the Royal Society of London B, 357, 1153-1170.

http://dx.doi.org/10.1098/rstb.2002.1117

[2] Hassell, M.P., Latto, J. and May, R.M. (1989) Seeing the Wood for the Trees: Detecting Density Dependence from Existing Life-Table Studies. Journal of Animal Ecology, 58, 883-892.

http://dx.doi.org/10.2307/5130

[3] Brook, B.W. and Bradshaw, C.J.A. (2006) Strength of Evidence for Density Dependence in Abundance Time Series of 1198 Species. Ecology, 87, 1445-1451.

http://dx.doi.org/10.1890/0012-9658(2006)87[1445:SOEFDD]2.0.CO;2

[4] Connell, J.H. and Sousa, W.P. (1983) On the Evidence Needed to Judge Ecological Stability or Persistence. American Naturalist, 121, 789-824.

http://dx.doi.org/10.1086/284105

[5] Hixon, M.A., Pacala, S.W. and Sandin, S.A. (2002) Population Regulation: Historical Context and Contemporary Challenges of Open vs. Closed Systems. Ecology, 83, 1490-1508.

http://dx.doi.org/10.1890/0012-9658(2002)083[1490:PRHCAC]2.0.CO;2

[6] Solow, A.R. (1990) Testing for Density Dependence: A Cautionary Note. Oecologia, 83, 47-49.

http://dx.doi.org/10.1007/BF00324632

[7] Beddington, J.R. and May, R.M. (1977) Harvesting Natural Populations in a Randomly Fluctuating Environment. Science, 197, 463-465.

http://dx.doi.org/10.1126/science.197.4302.463

[8] Shepherd, J.G. and Horwood, J.W. (1979) The Sensitivity of Exploited Populations to Environmental “Noise”, and the Implications for Management. Journal du Conseil International pour l’Exploration de la Mer, 38, 318-323.

http://dx.doi.org/10.1093/icesjms/38.3.318

[9] Ives, A.R., Dennis, B., Cottingham, K.L. and Carpenter, S.R. (2003) Estimating Community Stability and Ecological Interactions from Time-Series Data. Ecological Monographs, 73, 301-330.

http://dx.doi.org/10.1890/0012-9615(2003)073[0301:ECSAEI]2.0.CO;2

[10] Ives, A.R., Abbott, K.C. and Ziebarth, N.L. (2010) Analysis of Ecological Time Series with ARMA(p, q) Models. Ecology, 91, 858-871.

http://dx.doi.org/10.1890/09-0442.1

[11] Roughgarden, J. (1975) A Simple Model for Population Dynamics in Stochastic Environments. American Naturalist, 109, 713-736.

http://dx.doi.org/10.1086/283039

[12] Royama, T. (1981) Fundamental Concepts and Methodology for the Analysis of Animal Population Dynamics, with Particular Reference to Univoltine Species. Ecological Monographs, 51, 473-493. http://dx.doi.org/10.2307/2937325

[13] McArdle, B.H. (1989) Bird Population Densities. Nature, 338, 628.

http://dx.doi.org/10.1038/338628a0

[14] Lundberg, P., Ranta, E., Ripa, J. and Kaitala, V. (2000) Population Variability in Space and Time. Trends in Ecology & Evolution, 15, 460-464.

http://dx.doi.org/10.1016/S0169-5347(00)01981-9

[15] Jonzén, N., Ripa, J. and Lundberg, P. (2002) A Theory of Stochastic Harvesting in Stochastic Environments. American Naturalist, 159, 427-437.

http://dx.doi.org/10.1086/339456

[16] Lande, R., Engen, S., Sather, B.-E., Filli, F., Matthysen, E. and Weimerskirch, H. (2002) Estimating Density Dependence from Population Time Series Using Demographic Theory and Life-History Data. American Naturalist, 159, 321337.

http://dx.doi.org/10.1086/338988

[17] Edelstein-Keshet, L. (2005) Mathematical Models in Biology. Society for Industrial and Applied Mathematics.

http://dx.doi.org/10.1137/1.9780898719147

[18] Shepherd, J.G. and Cushing, D.H. (1990) Regulation in Fish Populations: Myth or Mirage? Philosophical Transactions of the Royal Society of London B, 330, 151-164.

http://dx.doi.org/10.1098/rstb.1990.0189

[19] Walters, C. and Parma, A.M. (1996) Fixed Exploitation Rate Strategies for Coping with Effects of Climate Change. Canadian Journal of Fisheries and Aquatic Sciences, 53, 148-158.

http://dx.doi.org/10.1139/f95-151

[20] Dixon, P.A., Milicich, M.J. and Sugihara, G. (1999) Episodic Fluctuations in Larval Supply. Science, 283, 1528-1530.

http://dx.doi.org/10.1126/science.283.5407.1528

[21] Niwa, H.S. (2006) Recruitment Variability in Exploited Aquatic Populations. Aquatic Living Resources, 19, 195-206.

http://dx.doi.org/10.1051/alr:2006020

[22] von Neumann, J. (1941) Distribution of the Ratio of the Mean Square Successive Difference to the Variance. Annals of Mathematical Statistics, 12, 367-395.

http://dx.doi.org/10.1214/aoms/1177731677

[23] Bulmer, M.G. (1975) The Statistical Analysis of Density Dependence. Biometrics, 31, 901-911.

http://dx.doi.org/10.2307/2529815

[24] Kubo, R., Toda, M. and Hashitsume, N. (1991) Statistical Physics II. Nonequilibrium Statistical Mechanics. 2nd Edition, Springer-Verlag, Berlin.

http://dx.doi.org/10.1007/978-3-642-58244-8

[25] Niwa, H.S. (2007) Random-Walk Dynamics of Exploited Fish Populations. ICES Journal of Marine Science, 64, 496502.

http://dx.doi.org/10.1093/icesjms/fsm004

[26] Sparholt, H., Bertelsen, M. and Lassen, H. (2007) A Meta-Analysis of the Status of ICES Fish Stocks during the Past Half Century. ICES Journal of Marine Science, 64, 707-713.

http://dx.doi.org/10.1093/icesjms/fsm038

[27] ICES (2008) Report of the ICES Advisory Committee, ICES Advice 2008, Books 1-10.

http://www.ices.dk/publications/our-publications/Pages/ICES-Advice.aspx