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Permanence of age-structured populations in a spatio-temporal variable environment
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

It is widely recognized that various biotic and abiotic factors cause changes in the size of a population and its age distribution. Population structure, intra-specific competition, temporal variability and spatial heterogeneity are identified as the most important factors that, alone or in combination, influence population dynamics. Despite being well-known, these factors are difficult to study, both theoretically and empirically. However, in an increasingly variable world, permanence of a growing number of species is threatened by climate changes, habitat fragmentation or reduced habitat quality. For purposes of conservation of species and land management, it is crucially important to have a good analysis of population dynamics, which will increase our theoretical knowledge and provide practical guidelines.

One way to address the problem of population dynamics is to use mathematical models. The choice of a model depends on what we want to study or what we aim to achieve. For an extensive theoretical study of population processes and for obtaining qualitative results about population growth or decline, analytical models with various level of complexity are used. The competing interests of realism and solvability of the model are always present. This means that, on one hand, we always aim to make a model that will truthfully reflect reality, while on the other hand, we need to keep the model mathematically solvable. This prompts us to carefully choose the most prominent ecological factors relevant to the problem at hand and to incorporate them into a model. Ideally, the results give new insights into population processes and complex interactions between the mentioned factors and population dynamics.

The objective of the thesis is to formulate, analyze, and apply various mathematical models of population dynamics. We begin with a classical linear age-structured model and gradually add temporal variability, intra-specific competition and spatial heterogeneity. In this way, every subsequent model is more realistic and complex than the previous one. We prove existence and uniqueness of a nonnegative solution to each boundary-initial problem, and continue with investigation of the large time behavior of the solution.

In the ecological terms, we are establishing conditions under which a population can persist in a certain environment. Since our aim is a qualitative analysis of a solution, we often examine upper and lower bounds of a solution. Their importance is in the fact that they are obtained analytically and parameters in their expression have biological meaning. Thus, instead of analyzing an exact solution (which often proves to be difficult), we analyze the corresponding upper and lower solutions.

We apply our models to demonstrate the influence of seasonal changes (or some other periodic temporal variation) and spatial structure of the habitat on population persistence. This is particularly important in explaining behavior of migratory birds or populations that inhabits several patches, some of which are of low quality. Our results extend the previously obtained results in some aspects and point out that all factors (age structure, density dependence, spatio-temporal variability) need to be considered when setting up a population model and predicting population growth.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. , 40 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1781
National Category
Mathematics Computational Mathematics Evolutionary Biology
URN: urn:nbn:se:liu:diva-130927DOI: 10.3384/diss.diva-130927ISBN: 9789176857069 (Print)OAI: diva2:956827
Public defence
2016-09-29, Planck, Fysik huset, Campus Valla, Linköping, 10:15 (English)
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2016-09-26Bibliographically approved
List of papers
1. Estimating effective boundaries of population growth in a variable environment
Open this publication in new window or tab >>Estimating effective boundaries of population growth in a variable environment
2016 (English)In: Boundary Value Problems, ISSN 1687-2762, E-ISSN 1687-2770, 1-28 p., 2016:172Article in journal (Refereed) Published
Abstract [en]

We study the impact of age-structure and temporal environmental variability on the persistence of populations. We use a linear age-structured model with time-dependent vital rates. It is the same as the one presented by Chipot in (Arch. Ration. Mech. Anal. 82(1):13-25, 1983), but the assumptions on the vital rates are slightly different. Our main interest is in describing the large-time behavior of a population provided that we know its initial distribution and transient vital rates. Using upper and lower solutions for the characteristic equation, we define time-dependent upper and lower boundaries for a solution in a constant environment. Moreover, we estimate solutions for the general time-dependent case and also for a special case when the environment is changing periodically.

Place, publisher, year, edition, pages
SpringerOpen, 2016
age-structure, time-dependency, environmental variability, population growth, upper and lower bounds, periodic oscillations
National Category
Probability Theory and Statistics Computational Mathematics
urn:nbn:se:liu:diva-131558 (URN)10.1186/s13661-016-0681-9 (DOI)
Available from: 2016-09-26 Created: 2016-09-26 Last updated: 2016-10-03Bibliographically approved
2. Persistence analysis of the age-structured population model on several patches
Open this publication in new window or tab >>Persistence analysis of the age-structured population model on several patches
2016 (English)In: Proceedings of the 16th International Conference on Mathematical Methods in Science and Engineering, July 4-8, Rota, Cadiz, Spain, Vol. III / [ed] J. Vigo-Aguiar, Universidad de Cádiz , 2016, Vol. 3, 717-727 p.Chapter in book (Refereed)
Abstract [en]

We consider a system of nonlinear partial differential equations that describes an age-structured population living in changing environment on $N$ patches. We prove existence and uniqueness of solution and analyze large time behavior of the system in time-independent case and for periodically changing environment. Under the assumption that every patch can be reached from every other patch, directly or through several intermediary patches, and that net reproductive operator has spectral radius larger than one, we prove that population is persistent on all patches. If the spectral radius is less or equal one, extinction on all patches is imminent.

Place, publisher, year, edition, pages
Universidad de Cádiz, 2016
age-structure, persistence, Kermack-McKendrick equation, Lotcka-Volterra equation
National Category
Biological Sciences Mathematics
urn:nbn:se:liu:diva-130231 (URN)9788460860822 (ISBN)
Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2016

Associate Editors

P. Schwerdtfeger (New Zealand), W. Sprößig (Germany), N. Stollenwerk (Portugal), Pino Caballero (Spain), J. Cioslowski (Poland), J. Medina (Spain), I. P. Hamilton (Canada), J. A. Alvarez-Bermejo (Spain)

Available from: 2016-07-21 Created: 2016-07-21 Last updated: 2016-08-31Bibliographically approved

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