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Markovian Building Blocks for Individual-Based Modelling Downloaded from orbit.dtu.dk on: Sep 27, 2021 Markovian Building Blocks for Individual-Based Modelling Nilsson, Lars Anders Fredrik Publication date: 2007 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Nilsson, L. A. F. (2007). Markovian Building Blocks for Individual-Based Modelling. IMM-PHD-2008-171 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Markovian Building Blocks for Individual-Based Modelling L. A. Fredrik Nilsson Kongens Lyngby 2006 IMM-PHD-2006-XX Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800 Kongens Lyngby, Denmark Phone +45 45253351, Fax +45 45882673 [email protected] www.imm.dtu.dk IMM-PHD: ISSN 0909-3192 Summary The present thesis consists of a summary report, four research articles, one technical report and one manuscript. The subject of the thesis is individual- based stochastic models. The summary report is composed of three parts and a brief history of some basic models in population biology. This history is included in order to pro- vide a reader that has no previous exposure to models in population biology with a sufficient background to understand some of the biological models that are mentioned in the thesis. The first part of the rest of the summary is a description of the dramatic changes in the degree of aggregation of sprat or herring in the Baltic during the day, with special focus on the dispersion of the fish from schools at dusk. The next part is a brief introduction to Markovian arrival processes, a type of stochastic processes with potential applications as sub-models in population dynamical models. The last part introduces Markov additive processes as a means of simplifying some individual-based models. In the first part I present the background to article A and some extra material that were not included in the final article. The basic observation is that fish in schools migrate up toward the surface and disperse at dusk and aggregate in schools close to the bottom at dawn. This creates a periodically varying prey field to cod. Apart from humans, cod is the main predator of herring and sprat in the Baltic. In order to evaluate the consequences to cod of this variability it was necessary to describe this prey field. It was shown that the schools follow lines of constant light intensity and that they disperse below a critical light threshold. We propose that the dispersion is due to a random walk when light levels become sub-critical and provide time-scales for the dispersion of this type for different school geometries (random or on a regular square grid)—the time-scales are of the same order as those observed on the echosounder. ii Summary The second part is an introduction to Markovian arrival processes (MAPs), this is the background needed to understand papers C, B, E, and F, given some previous exposure to Markov chains in continuous time (see e.g. Grimmett and Stirzaker, 2001)). Markovian arrival processes are very general point processes that are relatively easy to analyse. They have, so far, been largely unknown to the ecological modelling community. The article C deals with a functional response in a heterogeneous environment. The functional response is a model of the mean ingestion rate of prey per predator as a function of prey and possibly predator density that appears in most models for populations. A previously pro- posed model for prey encounter in heterogeneous environments is reanalyzed, it is a stochastic process that easily can be implemented as a MAP. In article C we show that transferring a standard functional response to a heterogeneous envi- ronment does not preserve the functional form, contrary to previous assertions. In this simple case we provide a time-scale for when the heterogeneous environ- ment can be assumed to be well-mixed, or close to a Poisson process, for the predator. It is also shown that in some cases the variability may be more impor- tant than the mean, thus the mean rate does not necessarily provide sufficient information for the population dynamics. Article B provides the mathematical apparatus for evaluating any moment of a MAP, and also the means for eval- uating the conditional moments of a transient or terminating MAP. Transient MAPs are suitable as modelling tools when an important property of the sys- tem is that it can stop. This is the case for the young of many animals, where most of a large clutch die rather quickly, and yet it is the survivors that are interesting. The conditional moments can for instance be constructed such that one can evaluate the mean or the variance of the ingestion rate given that the animal did not die. Several different methods are used to obtain the formulas, which is an interesting aspect since some of these methods may be more suitable in situations where it is problematic to proceed using the standard formalism. I provide material on how to model periodic MAPs in paper E. These are, or could be, important since most animals live in a periodic environment and a pe- riodic system generally have dynamics that are different from the correspond- ing system with mean rates. The technical report F concerns how to model Markovian stomachs. Both aspects can be used in more advanced functional or numerical responses. The third part concerns a larger class of Markov processes, to which the above mentioned MAPs belong. These are the Markov additive processes, which are bivariate Markov processes (Xt,Nt) where the transition probabilities depend on the Xt process only. The Xt process is marginally a Markov process, and the Nt process is a process with conditionally independent increments given the state of the Xt process. This class is rich enough to provide substantial realism into individual-based models yet it is so simple that it is not a great extra burden to solve the partial differential equations (PDEs) that arise for the evaluation of the moments. They are particularly useful in oceanographic contexts since iii here the apparatus for solving the PDEs is usually present due to the need of solving fluid flow equations. The greatest benefit of the method is due to that it circumvents the need for statistical evaluation of the individual-based models. In all three parts further work has been proposed. iv Preface This thesis was prepared at the department of Informatics and Mathematical Modelling at the Technical University of Denmark in partial fulfillment of the requirements for acquiring the Ph.D. degree in engineering. The project has been carried out in collaboration with the Danish Institute for Fisheries Research. The research was supported by the SLIP research school un- der the Danish Network for Fisheries and Aquaculture Research (www.fishnet.dk) financed by the Danish Ministry of Food, Agriculture and Fisheries and the Danish Agricultural and Veterinary Research Council, and by the DTU. The subject of the thesis is the application of Markovian stochastic models to the dynamics of biological systems. The thesis consists of a summary report, four research papers written during the Ph.D. study (one published elsewhere, two accepted, one submitted), and two technical reports. Lyngby, August 2006 Fredrik Nilsson vi Papers included in the thesis [A] L. A. Fredrik Nilsson, Uffe Høgsbro Thygesen, Bo Lundgren, Bo Friis Nielsen, J. Rasmus Nielsen, Jan E. Beyer. Vertical migration and disper- sion of sprat (Sprattus sprattus) and herring (Clupea harengus) schools at dusk in the Baltic Sea. Aquatic Living Resources 2003. 16: 317–324. [B] Bo Friis Nielsen, Uffe Høgsbro Thygesen, L. A. Fredrik Nilsson, Jan E. Beyer. Higher order moments and conditional asymptotics of the Batch Markovian Arrival Process. Manuscript accepted for publication in Sto- chastic Models [C] L. A. Fredrik Nilsson, Bo Friis Nielsen, Jan E. Beyer, Uffe Høgsbro Thyge- sen. Heterogeneity may change the functional response—lessons from a simple stochastic encounter model. Manuscript to be submitted to Oikos. [D] Uffe Høgsbro Thygesen, L. A. Fredrik Nilsson, Ken H. Andersen. Eu- lerian techniques for individual-based models with additive components. Manuscript submitted to Journal of Marine Systems. [E] L. A. Fredrik Nilsson, Jan E. Beyer, Uffe Høgsbro Thygesen, Bo Friis Nielsen. A framework for functional and numerical responses in periodic environments. Manuscript. [F] L. A. Fredrik Nilsson. Stomachs as stochastic dams. Technical report. viii Acknowledgements I thank my supervisors for the large amount of support given despite being busy with their own research projects. Dr. Bo Friis Nielsen for remaining calm at times of stress and for sharing both his vast insight in queues and his good humour. For always finding time to discuss things, always being constructive, and postponing the summer vacation to the autumn! Dr. Jan Beyer for long talks about the project and for all interesting ideas. I learned a lot in these discussions. Dr. Uffe Høgsbro Thygesen for the always funny discussions and for his voodoo-like mathematical feeling, enthusiasm and talent.
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