2012, 9(1): 75-96. doi: 10.3934/mbe.2012.9.75

Nonlinear functional response parameter estimation in a stochastic predator-prey model

1. 

Dipartimento di Scienze Biomediche e Biotecnologie, Università di Brescia, Viale Europa 11, 25125 Brescia, Italy

2. 

CNR-IMATI, Via Bassini 15, 20133 Milano, Italy, Italy

Received  May 2010 Revised  March 2011 Published  December 2011

Parameter estimation for the functional response of predator-prey systems is a critical methodological problem in population ecology. In this paper we consider a stochastic predator-prey system with non-linear Ivlev functional response and propose a method for model parameter estimation based on time series of field data. We tackle the problem of parameter estimation using a Bayesian approach relying on a Markov Chain Monte Carlo algorithm. The efficiency of the method is tested on a set of simulated data. Then, the method is applied to a predator-prey system of importance for Integrated Pest Management and biological control, the pest mite Tetranychus urticae and the predatory mite Phytoseiulus persimilis. The model is estimated on a dataset obtained from a field survey. Finally, the estimated model is used to forecast predator-prey dynamics in similar fields, with slightly different initial conditions.
Citation: Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75
References:
[1]

H. J. Barclay, Models for pest control using predator release, habitat management and pesticide release in combination, J. Applied Ecology, 19 (1982), 337-348. doi: 10.2307/2403471.

[2]

A. A. Berryman, J. Michaelski, A. P. Gutierrez and R. Arditi, Logistic theory of food web dynamics, Ecology, 76 (1995), 336-343. doi: 10.2307/1941193.

[3]

G. Buffoni, M. P. Cassinari, M. Groppi and M. Serluca, Modelling of predator-prey trophic interactions. I. Two trophic levels, J. Math. Biol., 50 (2005), 713-732. doi: 10.1007/s00285-004-0312-4.

[4]

G. Buffoni and G. Gilioli, A lumped parameter model for acarine predator-prey population interactions, Ecological Modelling, 170 (2003), 155-171. doi: 10.1016/S0304-3800(03)00223-0.

[5]

M. K. Cowles and B. P. Carlin, Markov chain Monte Carlo convergence diagnostics: A comparative review, Journal of the American Statistical Association, 91 (1996), 883-904. doi: 10.2307/2291683.

[6]

G. B. Durham and A. R. Gallant, Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes, Journal of Business & Economic Statistics, 20 (2002), 297-338. doi: 10.1198/073500102288618397.

[7]

O. Elerian, S. Chib and N. Shephard, Likelihood inference for discretely observed nonlinear diffusions, Econometrica, 69 (2001), 959-993. doi: 10.1111/1468-0262.00226.

[8]

B. Eraker, MCMC analysis of diffusion models with application to finance, Journal of Business & Economic Statistics, 19 (2001), 177-191. doi: 10.1198/073500101316970403.

[9]

M. L. Flint and R. van den Bosch, "Introduction to Integrated Pest Management,'' Plenum Press, New York, 1981.

[10]

D. Gamerman and H. F. Lopes, "Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference," Second edition, Texts in Statistical Science Series, Chapman & Hall, Boca Raton, FL, 2006.

[11]

G. Gilioli and V. Vacante, Aspetti della dinamica di popolazione del sistema. Tetranychus urticae - Phytoseiulus persimilis, in "Pieno Campo: Implicazioni per le Strategie di Lotta Biologica," in Atti del Convegno La Difesa delle Colture in Agricoltura Biologica, Grugliasco - Torino, 5-6 Settembre 2001, Notiziario sulla Protezione delle Piante, 13 (nuova serie), (2001), 95-99.

[12]

G. Gilioli, S. Pasquali and F. Ruggeri, Bayesian inference for functional response in a stochastic predator-prey system, Bulletin of Mathematical Biology, 70 (2008), 358-381. doi: 10.1007/s11538-007-9256-3.

[13]

W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds., "Markov Chain Monte Carlo in practice,'' Interdisciplinary Statistics, Chapman & Hall, London, 1996.

[14]

A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximation, Biometrics, 61 (2005), 781-788. doi: 10.1111/j.1541-0420.2005.00345.x.

[15]

A. Golightly and D. J. Wilkinson, Bayesian inference for nonlinear multivariate diffusion models observed with error, Computational Statistics & Data Analysis, 52 (2008), 1674-1693. doi: 10.1016/j.csda.2007.05.019.

[16]

A. Golightly and D. J. Wilkinson, Markov chain Monte Carlo algorithms for SDE parameter estimation, in "Learning and Inference for Computational Systems Biology" (eds. N. D. Lawrence, M. Girolami, M. Rattray and G. Sanguinetti), MIT Press, (2010), 253-275.

[17]

A. P. Gutierrez, "Applied Population Ecology. A Supply-Demand Approach,'' John Wiley & Sons, New York, 1996.

[18]

V. S. Ivlev, "Experimental Ecology of the Feeding of Fishes,'' Yale University Press, New Haven, 1961.

[19]

C. Jost and R. Arditi, Identifying predator-prey process from time-series, Theoretical Population Biology, 57 (2000), 325-337. doi: 10.1006/tpbi.2000.1463.

[20]

C. Jost and R. Arditi, From pattern to process: Identifying predator-prey models from time-series data, Population Ecology, 43 (2001), 229-243. doi: 10.1007/s10144-001-8187-3.

[21]

P. Kareiva, Population dynamics in spatially complex environments: Theory and data, Phil. Trans. R. Soc. Lond., 330 (1990), 175-190. doi: 10.1098/rstb.1990.0191.

[22]

H. McCullum, "Population Parameters. Estimation for Ecological Models,'' Blackwell, Oxford, 2000.

[23]

B. Øksendal, "Stochastic Differential Equations: An Introduction with Applications,'' $5^{th}$ edition, Springer, Berlin, 1998.

[24]

F. D. Parker, Management of pest populations by manipulating densities of both host and parasites through periodic releases, in "Biological Control" (ed. C. B. Huffaker), Plenum Press, (1971), 365-376.

[25]

M. A. Pascual and K. Kareiva, Predicting the outcome of competition using experimental data: Maximum likelihood and bayesian approaches, Ecology, 77 (1996), 337-349. doi: 10.2307/2265613.

[26]

G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm, Biometrika, 88 (2001), 603-621. doi: 10.1093/biomet/88.3.603.

[27]

T. Royama, A comparative study of models for predation and parasitism, Research on Population Ecology, Supp. 1 (1971), 1-91. doi: 10.1007/BF02511547.

[28]

M. A. Tanner and W. H. Wong, The calculation of posterior distributions by data augmentation, Journal of the American Statistical Association, 82 (1987), 528-550. doi: 10.2307/2289457.

show all references

References:
[1]

H. J. Barclay, Models for pest control using predator release, habitat management and pesticide release in combination, J. Applied Ecology, 19 (1982), 337-348. doi: 10.2307/2403471.

[2]

A. A. Berryman, J. Michaelski, A. P. Gutierrez and R. Arditi, Logistic theory of food web dynamics, Ecology, 76 (1995), 336-343. doi: 10.2307/1941193.

[3]

G. Buffoni, M. P. Cassinari, M. Groppi and M. Serluca, Modelling of predator-prey trophic interactions. I. Two trophic levels, J. Math. Biol., 50 (2005), 713-732. doi: 10.1007/s00285-004-0312-4.

[4]

G. Buffoni and G. Gilioli, A lumped parameter model for acarine predator-prey population interactions, Ecological Modelling, 170 (2003), 155-171. doi: 10.1016/S0304-3800(03)00223-0.

[5]

M. K. Cowles and B. P. Carlin, Markov chain Monte Carlo convergence diagnostics: A comparative review, Journal of the American Statistical Association, 91 (1996), 883-904. doi: 10.2307/2291683.

[6]

G. B. Durham and A. R. Gallant, Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes, Journal of Business & Economic Statistics, 20 (2002), 297-338. doi: 10.1198/073500102288618397.

[7]

O. Elerian, S. Chib and N. Shephard, Likelihood inference for discretely observed nonlinear diffusions, Econometrica, 69 (2001), 959-993. doi: 10.1111/1468-0262.00226.

[8]

B. Eraker, MCMC analysis of diffusion models with application to finance, Journal of Business & Economic Statistics, 19 (2001), 177-191. doi: 10.1198/073500101316970403.

[9]

M. L. Flint and R. van den Bosch, "Introduction to Integrated Pest Management,'' Plenum Press, New York, 1981.

[10]

D. Gamerman and H. F. Lopes, "Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference," Second edition, Texts in Statistical Science Series, Chapman & Hall, Boca Raton, FL, 2006.

[11]

G. Gilioli and V. Vacante, Aspetti della dinamica di popolazione del sistema. Tetranychus urticae - Phytoseiulus persimilis, in "Pieno Campo: Implicazioni per le Strategie di Lotta Biologica," in Atti del Convegno La Difesa delle Colture in Agricoltura Biologica, Grugliasco - Torino, 5-6 Settembre 2001, Notiziario sulla Protezione delle Piante, 13 (nuova serie), (2001), 95-99.

[12]

G. Gilioli, S. Pasquali and F. Ruggeri, Bayesian inference for functional response in a stochastic predator-prey system, Bulletin of Mathematical Biology, 70 (2008), 358-381. doi: 10.1007/s11538-007-9256-3.

[13]

W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds., "Markov Chain Monte Carlo in practice,'' Interdisciplinary Statistics, Chapman & Hall, London, 1996.

[14]

A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximation, Biometrics, 61 (2005), 781-788. doi: 10.1111/j.1541-0420.2005.00345.x.

[15]

A. Golightly and D. J. Wilkinson, Bayesian inference for nonlinear multivariate diffusion models observed with error, Computational Statistics & Data Analysis, 52 (2008), 1674-1693. doi: 10.1016/j.csda.2007.05.019.

[16]

A. Golightly and D. J. Wilkinson, Markov chain Monte Carlo algorithms for SDE parameter estimation, in "Learning and Inference for Computational Systems Biology" (eds. N. D. Lawrence, M. Girolami, M. Rattray and G. Sanguinetti), MIT Press, (2010), 253-275.

[17]

A. P. Gutierrez, "Applied Population Ecology. A Supply-Demand Approach,'' John Wiley & Sons, New York, 1996.

[18]

V. S. Ivlev, "Experimental Ecology of the Feeding of Fishes,'' Yale University Press, New Haven, 1961.

[19]

C. Jost and R. Arditi, Identifying predator-prey process from time-series, Theoretical Population Biology, 57 (2000), 325-337. doi: 10.1006/tpbi.2000.1463.

[20]

C. Jost and R. Arditi, From pattern to process: Identifying predator-prey models from time-series data, Population Ecology, 43 (2001), 229-243. doi: 10.1007/s10144-001-8187-3.

[21]

P. Kareiva, Population dynamics in spatially complex environments: Theory and data, Phil. Trans. R. Soc. Lond., 330 (1990), 175-190. doi: 10.1098/rstb.1990.0191.

[22]

H. McCullum, "Population Parameters. Estimation for Ecological Models,'' Blackwell, Oxford, 2000.

[23]

B. Øksendal, "Stochastic Differential Equations: An Introduction with Applications,'' $5^{th}$ edition, Springer, Berlin, 1998.

[24]

F. D. Parker, Management of pest populations by manipulating densities of both host and parasites through periodic releases, in "Biological Control" (ed. C. B. Huffaker), Plenum Press, (1971), 365-376.

[25]

M. A. Pascual and K. Kareiva, Predicting the outcome of competition using experimental data: Maximum likelihood and bayesian approaches, Ecology, 77 (1996), 337-349. doi: 10.2307/2265613.

[26]

G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm, Biometrika, 88 (2001), 603-621. doi: 10.1093/biomet/88.3.603.

[27]

T. Royama, A comparative study of models for predation and parasitism, Research on Population Ecology, Supp. 1 (1971), 1-91. doi: 10.1007/BF02511547.

[28]

M. A. Tanner and W. H. Wong, The calculation of posterior distributions by data augmentation, Journal of the American Statistical Association, 82 (1987), 528-550. doi: 10.2307/2289457.

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