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Population Dynamics and Tree Growth Structure in Mathematical Ecology 2021 Isbn 978-91-7485-498-5 Isbn Issn 1651-4238 P.O

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Population Dynamics and Tree Growth Structure in Mathematical Ecology 2021 Isbn 978-91-7485-498-5 Isbn Issn 1651-4238 P.O Mälardalen University Doctoral Dissertation 331 Tin Nwe Aye Population Dynamics and Tree POPULATION DYNAMICS AND TREE GROWTH STRUCTURE IN MATHEMATICAL ECOLOGY Growth Structure in Mathematical Ecology Tin Nwe Aye Address: P.O. Box 883, SE-721 23 Västerås. Sweden ISBN 978-91-7485-498-5 2021 Address: P.O. Box 325, SE-631 05 Eskilstuna. Sweden E-mail: [email protected] Web: www.mdh.se ISSN 1651-4238 1 Mälardalen University Press Dissertations No. 331 POPULATION DYNAMICS AND TREE GROWTH STRUCTURE IN MATHEMATICAL ECOLOGY Tin Nwe Aye 2021 School of Education, Culture and Communication 2 Copyright © Tin Nwe Aye, 2021 ISBN 978-91-7485-498-5 ISSN 1651-4238 Printed by E-Print AB, Stockholm, Sweden 3 Mälardalen University Press Dissertations No. 331 POPULATION DYNAMICS AND TREE GROWTH STRUCTURE IN MATHEMATICAL ECOLOGY Tin Nwe Aye Akademisk avhandling som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras fredagen den 26 mars 2021, 10.00 i rum Zeta, Hus T och via Zoom, Mälardalens högskola, Västerås. Fakultetsopponent: Professor Christian Engström, Linnéuniversitetet Akademin för utbildning, kultur och kommunikation 4 Abstract This thesis is based on four papers related to mathematical biology, where three papers focus on population dynamics and one paper concerns tree growth and stem structure. The first two papers are mainly devoted to studying the dynamics of physiologically structured population models by using Escalator Boxcar Train (EBT) method. The third paper concerns a class of stage-structured population systems, in both deterministic and stochastic settings. The fourth paper explores how a branch thinning model can be utilized to describe the cross-sectional area of the stem of a tree, thus generalizing the classical pipe model. In Paper I, we present a merging procedure to reduce the increasing system of ordinary differential equations generated by the EBT method. In particular, we modify the EBT method to include merging of cohorts. The accuracy of this model is explored on a colony of Daphnia Pulex. In Paper II, we study the convergence rate of the modified EBT model, allowing a general class of non- linear merging procedures. We show that this modified EBT method induces a bounded number of cohorts, independent of the number of time steps. This in turn, improves the speed of the numerical algorithm for solving the population dynamics from polynomial time to linear time, that is, the time consumption to find the solution is proportional to the number of time steps. In Paper III, a class of non-linear two-stage structured population models is studied with different growth rates for the unstructured food resource under different harvesting rates in both deterministic and stochastic settings. In the stochastic setting, we develop methods to evaluate emergent properties equivalent to the properties investigated in the deterministic case. In addition, new emergent properties, e.g. probability of extinction, are also investigated. In Paper IV, we explore the stem model which is developed by combining the pipe model and the branch thinning model. The stem model provides estimates of the heartwood, sapwood and stem cross-sectional area at any height. We corroborate the accuracy of our model with empirical data and the cross validation of our results shows a very high goodness of fit for the stem model. ISBN 978-91-7485-498-5 ISSN 1651-4238 5 I dedicate this work to my aunt who has been supporting me throughout my life. 6 7 Acknowledgements This thesis becomes a reality with a kind support and help from many individuals. First of all, I would like to express my sincere gratitude to my supervisor Associate Professor (Docent) Linus Carlsson for the continuous support during my Ph.D study and research, for his patience, friendship, immense knowledge and care. He has taught me the methodology to carry out the research and the scientific tools for my research works. It was a great privilege and honor to work and study under his guidance. I am extending my thanks to his wife for her acceptance and patience during the discussion I had with him during research work. My sincere thanks also go to co-supervisor Professor Sergei Silvestrov for his encouragement, support, kind words and suggestions to improve thesis manuscript. I am grateful to co-supervisor Masood Aryapoor for providing suggestions on the research works and thesis manuscript improvement. I am extremely grateful to my parents for their love, caring and sacrifices for me. I am very much thankful to my aunt who strongly supports me throughout my life. I would like also to give much thanks to Professor Ohn Mar from Myanmar who suggested and encouraged me to start this wonderful job. My appreciation also goes to Professor Khin Myo Aye for her kind coordination of the SEAMaN project in my home country, the project which has supported me during my studies. My special thanks go to the International Science Programme (ISP) for financial support. I wish to show my gratitude to Pravina, Chris and Leif at ISP for being around me and not to let me to have any inconvenience for the whole time of my studies. I would like to extend my sincere obligation towards all the staffs and PhD students in Mathematics and Applied Mathematics, Mälardalen University. I am extremely thankful to all my friends, fellow PhD students under ISP and Sida, who encouraged and cared for me throughout my studies. I am very lucky to have you all around and appreciate for having such a wonderful family. 7 8 Population Dynamics and Tree Growth Structure in Mathematical Ecology At last, my thanks and appreciation go to all my colleague and people who have willingly helped me out with their abilities in so many ways. Västerås, March, 2021 Tin Nwe Aye 8 9 Populärvetenskaplig sammanfattning Matematisk biologi/ekologi är en snabbt växande, accepterad, intressant och mod- ern tillämpning av matematik. Ekologi är studien av interaktioner mellan organ- ismer, populationer och deras omgivning. En population är grupp av individer av en art som finns inom ett visst område vid en viss tid. Fysiologiskt strukturerade populationsmodeller (PSPMs) används för att un- dersöka populationsdynamiken, dvs. hur populationer och miljö förändras över tid. PSPMs är en klass av modeller som explicit sammankopplar populationsdy- namiken med individernas livshistoria, speciellt födointag, tillväxt, reproduktion och dödlighet. Dessa processer beror på individens tillstånd och den omgivande miljön. Modeller av ekologiska system med PSPMs har signifikant bidragit till vår förståelse av hur individers storlek påverkar populationsdynamiken. Individuell tillväxt, reproduktion och dödlighet är födoberoende och varierar med popula- tionstäthet och förändringar i den omgivande miljön. The Escalator Boxcar Train (EBT) är en numerisk metod för att lösa storleksberoende PSPMs. Den klassiska algoritmen för EBT genererar ett system av differentialekvationer vars antal ökar över tid och blir ohanterliga att lösa, även i en modern dator. Vi har modifierat EBT metoden så att systemet inte växer men bibehåller noggrannheten i lösningarna, och därför kan mer avancerade PSPMs lösas. En alternativ metod för att lösa PSPMs är att dela in populationsstorlekarna i unga och vuxna individer, så kallade storleksstrukturerade populationsmodeller. Vi studerar storleksstrukturerade, biomassabaserade konsument-resurs-modeller med olika födotillväxtsdynamik, både deterministisk och stokastisk, för att undersöka framträdande egenskaper i populationsdynamiken. Inom skogsindustri och skogsekologi har den klassiska rör-modellen använts för att uppskatta tvärsnittsarean i trädstammar. Men i originalrapporten, skriven för mer än ett halvt sekel sedan, säger författarna explicit att deras enkla rör- 9 10 Population Dynamics and Tree Growth Structure in Mathematical Ecology modell inte kan användas i detta syfte under trädkronan. Stammens tvärsnittsarea ökar markant med trädets storlek och ålder och är sammansatt huvudsakligen av kärnved och splintved. Sammansättningen av kärnved/splintved är viktig, både för industriella träprodukter, men även inom ekologisk skogsteori. Vi utvecklar en stammodell som förutsäger tvärsnittsarean av kärnved och splintved i stammen på träd. Vår modell överträffar den enkla rörmodellen. 10 11 Popular Science Summary Mathematical biology/ecology is a fast-growing, well-recognized and useful ex- citing modern application of mathematics. Ecology is the study of interactions between organisms, populations and their environment. A population is a group of interbreeding organisms of a particular species in the same geographic area over a period of time. Physiologically structured population models (PSPMs) investigate the popula- tion dynamics, that is, the study of change in populations and environment over time. PSPMs are a class of models which explicitly link population dynamics and individual life history, in particular feeding, development, reproduction, and mortality. These processes are dependent on the state of the individual organism itself and the environment in which it lives. Modelling ecological systems with PSPMs has contributed significantly to our understanding of how size-dependent individual life history processes affect the population dynamics. Individual growth reproduction is food-dependent and varies with population density and environ- mental changes. The Escalator Boxcar Train (EBT) is a numerical method, used to find solutions to size-dependent
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