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Finding analytical approximations for discrete, stochastic, individual-based models of ecology
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.ORCID iD: 0000-0002-8745-4480
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology.ORCID iD: 0000-0002-1436-9103
2023 (English)In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 365Article in journal (Refereed) Published
Abstract [en]

Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

Place, publisher, year, edition, pages
Elsevier, 2023. Vol. 365
National Category
Computational Mathematics Other Mathematics Probability Theory and Statistics
Research subject
Mathematics with specialization in Applied Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-455245DOI: 10.1016/j.mbs.2023.109084ISI: 001103942100001OAI: oai:DiVA.org:uu-455245DiVA, id: diva2:1600612
Available from: 2021-10-05 Created: 2021-10-05 Last updated: 2024-02-21Bibliographically approved
In thesis
1. Mathematical models of biological interactions
Open this publication in new window or tab >>Mathematical models of biological interactions
2021 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Mathematical models are used to describe and analyse different types of biological interactions.  From self-propelled particle models capturing the collective motion of fish schools to models in mathematical neuroscience describing the interactions between neurons to individual-based models of ecological interactions. A question that arises for all such models is how we scale from one level to another. How do we scale from fish interactions to the movement of the school of fish? How do we scale from neuronal interactions to the functioning of the brain?  How do we scale from animal competition to population dynamics? It is approaches to this question that we study in this thesis for two different systems. 

In paper I,  we study a class of spatially explicit individual-based models with contest competition. Based on measures of the spatial statistics, we develop two new approximate descriptions of the spatial population dynamics. The first is based on local interactions of the individuals and approximates the individual-based model well for small dispersal distances. The second approximates the long-range interactions of the individual-based model. Both approximations incorporate the demographic stochasticity from the individual-based model and show that dispersal stabilizes the population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the classical mean field approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Taken together, our results deepen the understanding of spatial population dynamics and introduces new approximate analytical descriptions.

In paper II,  we propose a model of social burst and glide motion in pairs of fish by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion.  Our model, in which visual stimuli of the position of the other fish affect the internal burst or glide state of the fish, captures a rich set of swimming dynamics found in many species of fish. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and pendulum-like tit-for-tat turn taking. Unlike self-propelled particle models, which assume that fish move at a constant speed, the model produces realistic motion of individual fish. Moreover, unlike previous studies where a random component is used for leadership switching to occur, we show that leadership switching, both periodic and chaotic, can be the result of from a deterministic interaction.  We give several empirically testable predictions on how fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity. 

Place, publisher, year, edition, pages
Uppsala: Uppsala University, 2021. p. 11
Series
U.U.D.M. report / Uppsala University, Department of Mathematics, ISSN 1101-3591 ; 2021:4
National Category
Other Mathematics Computational Mathematics Probability Theory and Statistics
Research subject
Mathematics with specialization in Applied Mathematics
Identifiers
urn:nbn:se:uu:diva-455554 (URN)
Presentation
2021-10-29, 4001, Ångströmlaboratoriet, 13:15 (English)
Opponent
Supervisors
Available from: 2021-10-12 Created: 2021-10-08 Last updated: 2021-10-12Bibliographically approved
2. The Art of Modelling Oscillations and Feedback across Biological Scales
Open this publication in new window or tab >>The Art of Modelling Oscillations and Feedback across Biological Scales
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers in the field of mathematical biology. All papers aim to advance our understanding of biological systems through the development and application of innovative mathematical models. These models cover a diverse range of biological scales, from the nuclei of unicellular organisms to the collective behaviours of animal populations, showcasing the broad applicability and potential of mathematical approaches in biology. While the first three papers study mathematical models of very different applications and at various scales, all models contribute to the understanding of how oscillations and/or feedback mechanisms on the individual level give rise to complex emergent patterns on the collective level. In Paper I, we propose a mathematical model of basal cognition, inspired by the true slime mould, Physarum polycephalum. The model demonstrates how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner. In Paper II, we propose a model of social burst-and-glide motion in pairs of swimming fish by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. Our model, in which visual stimuli of the position of the other fish affect the internal burst or glide state of the fish, captures a rich set of swimming dynamics found in many species of fish. In Paper III, we study a class of spatially explicit individual-based models with contest competition. Based on measures of the spatial statistics, we develop two new approximate descriptions of the spatial population dynamics. Paper IV takes a reflective turn, advocating from a philosophical perspective the importance of developing new mathematical models in the face of current scientific challenges.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2024. p. 48
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 135
Keywords
mathematical biology, mathematical modelling, oscillations, feedback mechanisms, dynamical systems, individual-based models, complex systems
National Category
Mathematics
Research subject
Applied Mathematics and Statistics
Identifiers
urn:nbn:se:uu:diva-523639 (URN)978-91-506-3039-8 (ISBN)
Public defence
2024-04-12, Sonja Lyttkens (101121), Ångströmlaboratoriet, Uppsala, 09:15 (English)
Opponent
Supervisors
Available from: 2024-03-19 Created: 2024-02-21 Last updated: 2024-03-19

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Gyllingberg, LinnéaSumpter, David J. T.

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