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Stochastic analogues of deterministic single-species population models
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
2006 (English)In: Theoretical Population Biology, ISSN 0040-5809, Vol. 69, no 4, 442-451 p.Article in journal (Refereed) Published
Abstract [en]

Although single-species deterministic difference equations have long been used in modeling the dynamics of animal populations, little attention has been paid to how stochasticity should be incorporated into these models. By deriving stochastic analogues to difference equations from first principles, we show that the form of these models depends on whether noise in the population process is demographic or environmental. When noise is demographic, we argue that variance around the expectation is proportional to the expectation. When noise is environmental the variance depends in a non-trivial way on how variation enters into model parameters, but we argue that if the environment affects the population multiplicatively then variance is proportional to the square of the expectation. We compare various stochastic analogues of the Ricker map model by fitting them, using maximum likelihood estimation, to data generated from an individual-based model and the weevil data of Utida. Our demographic models are significantly better than our environmental models at fitting noise generated by population processes where noise is mainly demographic. However, the traditionally chosen stochastic analogues to deterministic models—additive normally distributed noise and multiplicative lognormally distributed noise—generally fit all data sets well. Thus, the form of the variance does play a role in the fitting of models to ecological time series, but may not be important in practice as first supposed.

Place, publisher, year, edition, pages
2006. Vol. 69, no 4, 442-451 p.
Keyword [en]
Population models, Stochastic population models, Ricker model, First principles
URN: urn:nbn:se:umu:diva-8157DOI: 10.1016/j.tpb.2006.01.006OAI: diva2:147828
Available from: 2008-01-15 Created: 2008-01-15 Last updated: 2010-01-22Bibliographically approved
In thesis
1. Modelling animal populations
Open this publication in new window or tab >>Modelling animal populations
2004 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [sv]

This thesis consists of four papers, three papers about modelling animal populations and one paper about an area integral estimate for solutions of partial differential equations on non-smooth domains. The papers are:

I. Å. Brännström, Single species population models from first principles.

II. Å. Brännström and D. J. T. Sumpter, Stochastic analogues of deterministic single species population models.

III. Å. Brännström and D. J. T. Sumpter, Coupled map lattice approximations for spatially explicit individual-based models of ecology.

IV. Å. Brännström, An area integral estimate for higher order parabolic equations.

In the first paper we derive deterministic discrete single species population models with first order feedback, such as the Hassell and Beverton-Holt model, from first principles. The derivations build on the site based method of Sumpter & Broomhead (2001) and Johansson & Sumpter (2003). A three parameter generalisation of the Beverton-Holtmodel is also derived, and one of the parameters is shown to correspond directly to the underlying distribution of individuals.

The second paper is about constructing stochastic population models that incorporate a given deterministic skeleton. Using the Ricker model as an example, we construct several stochastic analogues and fit them to data using the method of maximum likelihood. The results show that an accurate stochastic population model is most important when the dynamics are periodic or chaotic, and that the two most common ways of constructing stochastic analogues, using additive normally distributed noise or multiplicative lognormally distributed noise, give models that fit the data well. The latter is also motivated on theoretical grounds.

In the third paper we approximate a spatially explicit individual-based model with a stochastic coupledmap lattice. The approximation effectively disentangles the deterministic and stochastic components of the model. Based on this approximation we argue that the stable population dynamics seen for short dispersal ranges is a consequence of increased stochasticity from local interactions and dispersal.

Finally, the fourth paper contains a proof that for solutions of higher order real homogeneous constant coefficient parabolic operators on Lipschitz cylinders, the area integral dominates the maximal function in the L2-norm.

18 p.
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 29
population model, stochastic population model, population dynamics, discrete time model, Beverton-Holt model, Skellam model, Hassell model, Ricker model, first principles, coupled map lattice, CML, area integral, square function
Research subject
urn:nbn:se:umu:diva-205 (URN)91-7305-615-4 (ISBN)
Public defence
2004-03-29, 10:00 (English)
Available from: 2004-03-08 Created: 2004-03-08 Last updated: 2010-01-22Bibliographically approved

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