# American Institute of Mathematical Sciences

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. doi: 10.2307/2403471. Google Scholar [2] A. A. Berryman, J. Michaelski, A. P. Gutierrez and R. Arditi, Logistic theory of food web dynamics,, Ecology, 76 (1995), 336. doi: 10.2307/1941193. Google Scholar [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. doi: 10.1007/s00285-004-0312-4. Google Scholar [4] G. Buffoni and G. Gilioli, A lumped parameter model for acarine predator-prey population interactions,, Ecological Modelling, 170 (2003), 155. doi: 10.1016/S0304-3800(03)00223-0. Google Scholar [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. doi: 10.2307/2291683. Google Scholar [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. doi: 10.1198/073500102288618397. Google Scholar [7] O. Elerian, S. Chib and N. Shephard, Likelihood inference for discretely observed nonlinear diffusions,, Econometrica, 69 (2001), 959. doi: 10.1111/1468-0262.00226. Google Scholar [8] B. Eraker, MCMC analysis of diffusion models with application to finance,, Journal of Business & Economic Statistics, 19 (2001), 177. doi: 10.1198/073500101316970403. Google Scholar [9] M. L. Flint and R. van den Bosch, "Introduction to Integrated Pest Management,'', Plenum Press, (1981). Google Scholar [10] D. Gamerman and H. F. Lopes, "Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference," Second edition,, Texts in Statistical Science Series, (2006). Google Scholar [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, (2001), 5. Google Scholar [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. doi: 10.1007/s11538-007-9256-3. Google Scholar [13] W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds., "Markov Chain Monte Carlo in practice,'', Interdisciplinary Statistics, (1996). Google Scholar [14] A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximation,, Biometrics, 61 (2005), 781. doi: 10.1111/j.1541-0420.2005.00345.x. Google Scholar [15] A. Golightly and D. J. Wilkinson, Bayesian inference for nonlinear multivariate diffusion models observed with error,, Computational Statistics & Data Analysis, 52 (2008), 1674. doi: 10.1016/j.csda.2007.05.019. Google Scholar [16] A. Golightly and D. J. Wilkinson, Markov chain Monte Carlo algorithms for SDE parameter estimation,, in, (2010), 253. Google Scholar [17] A. P. Gutierrez, "Applied Population Ecology. A Supply-Demand Approach,'', John Wiley & Sons, (1996). Google Scholar [18] V. S. Ivlev, "Experimental Ecology of the Feeding of Fishes,'', Yale University Press, (1961). Google Scholar [19] C. Jost and R. Arditi, Identifying predator-prey process from time-series,, Theoretical Population Biology, 57 (2000), 325. doi: 10.1006/tpbi.2000.1463. Google Scholar [20] C. Jost and R. Arditi, From pattern to process: Identifying predator-prey models from time-series data,, Population Ecology, 43 (2001), 229. doi: 10.1007/s10144-001-8187-3. Google Scholar [21] P. Kareiva, Population dynamics in spatially complex environments: Theory and data,, Phil. Trans. R. Soc. Lond., 330 (1990), 175. doi: 10.1098/rstb.1990.0191. Google Scholar [22] H. McCullum, "Population Parameters. Estimation for Ecological Models,'', Blackwell, (2000). Google Scholar [23] B. Øksendal, "Stochastic Differential Equations: An Introduction with Applications,'' $5^{th}$ edition,, Springer, (1998). Google Scholar [24] F. D. Parker, Management of pest populations by manipulating densities of both host and parasites through periodic releases,, in, (1971), 365. Google Scholar [25] M. A. Pascual and K. Kareiva, Predicting the outcome of competition using experimental data: Maximum likelihood and bayesian approaches,, Ecology, 77 (1996), 337. doi: 10.2307/2265613. Google Scholar [26] G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm,, Biometrika, 88 (2001), 603. doi: 10.1093/biomet/88.3.603. Google Scholar [27] T. Royama, A comparative study of models for predation and parasitism,, Research on Population Ecology, Supp. 1 (1971), 1. doi: 10.1007/BF02511547. Google Scholar [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. doi: 10.2307/2289457. Google Scholar

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##### 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. doi: 10.2307/2403471. Google Scholar [2] A. A. Berryman, J. Michaelski, A. P. Gutierrez and R. Arditi, Logistic theory of food web dynamics,, Ecology, 76 (1995), 336. doi: 10.2307/1941193. Google Scholar [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. doi: 10.1007/s00285-004-0312-4. Google Scholar [4] G. Buffoni and G. Gilioli, A lumped parameter model for acarine predator-prey population interactions,, Ecological Modelling, 170 (2003), 155. doi: 10.1016/S0304-3800(03)00223-0. Google Scholar [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. doi: 10.2307/2291683. Google Scholar [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. doi: 10.1198/073500102288618397. Google Scholar [7] O. Elerian, S. Chib and N. Shephard, Likelihood inference for discretely observed nonlinear diffusions,, Econometrica, 69 (2001), 959. doi: 10.1111/1468-0262.00226. Google Scholar [8] B. Eraker, MCMC analysis of diffusion models with application to finance,, Journal of Business & Economic Statistics, 19 (2001), 177. doi: 10.1198/073500101316970403. Google Scholar [9] M. L. Flint and R. van den Bosch, "Introduction to Integrated Pest Management,'', Plenum Press, (1981). Google Scholar [10] D. Gamerman and H. F. Lopes, "Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference," Second edition,, Texts in Statistical Science Series, (2006). Google Scholar [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, (2001), 5. Google Scholar [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. doi: 10.1007/s11538-007-9256-3. Google Scholar [13] W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds., "Markov Chain Monte Carlo in practice,'', Interdisciplinary Statistics, (1996). Google Scholar [14] A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximation,, Biometrics, 61 (2005), 781. doi: 10.1111/j.1541-0420.2005.00345.x. Google Scholar [15] A. Golightly and D. J. Wilkinson, Bayesian inference for nonlinear multivariate diffusion models observed with error,, Computational Statistics & Data Analysis, 52 (2008), 1674. doi: 10.1016/j.csda.2007.05.019. Google Scholar [16] A. Golightly and D. J. Wilkinson, Markov chain Monte Carlo algorithms for SDE parameter estimation,, in, (2010), 253. Google Scholar [17] A. P. Gutierrez, "Applied Population Ecology. A Supply-Demand Approach,'', John Wiley & Sons, (1996). Google Scholar [18] V. S. Ivlev, "Experimental Ecology of the Feeding of Fishes,'', Yale University Press, (1961). Google Scholar [19] C. Jost and R. Arditi, Identifying predator-prey process from time-series,, Theoretical Population Biology, 57 (2000), 325. doi: 10.1006/tpbi.2000.1463. Google Scholar [20] C. Jost and R. Arditi, From pattern to process: Identifying predator-prey models from time-series data,, Population Ecology, 43 (2001), 229. doi: 10.1007/s10144-001-8187-3. Google Scholar [21] P. Kareiva, Population dynamics in spatially complex environments: Theory and data,, Phil. Trans. R. Soc. Lond., 330 (1990), 175. doi: 10.1098/rstb.1990.0191. Google Scholar [22] H. McCullum, "Population Parameters. Estimation for Ecological Models,'', Blackwell, (2000). Google Scholar [23] B. Øksendal, "Stochastic Differential Equations: An Introduction with Applications,'' $5^{th}$ edition,, Springer, (1998). Google Scholar [24] F. D. Parker, Management of pest populations by manipulating densities of both host and parasites through periodic releases,, in, (1971), 365. Google Scholar [25] M. A. Pascual and K. Kareiva, Predicting the outcome of competition using experimental data: Maximum likelihood and bayesian approaches,, Ecology, 77 (1996), 337. doi: 10.2307/2265613. Google Scholar [26] G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm,, Biometrika, 88 (2001), 603. doi: 10.1093/biomet/88.3.603. Google Scholar [27] T. Royama, A comparative study of models for predation and parasitism,, Research on Population Ecology, Supp. 1 (1971), 1. doi: 10.1007/BF02511547. Google Scholar [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. doi: 10.2307/2289457. Google Scholar
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