Abstract
Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; this last result is also proved theoretically under some regularity conditions on the model ingredients.
Original language | English |
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Pages (from-to) | 37-67 |
Number of pages | 31 |
Journal | Vietnam Journal of Mathematics |
Volume | 49 |
Issue number | 1 |
Early online date | 1 Jan 2020 |
DOIs | |
Publication status | Published - Mar 2021 |
Bibliographical note
Funding Information:The research of FS is funded by NSERC-Sanofi Industrial Research Chair. DB, FS and RV are members of the INdAM Research group GNCS. MG was supported by the Centre of Excellence in Analysis and Dynamics Research, Academy of Finland.
Funding Information:
The research of FS is funded by NSERC-Sanofi Industrial Research Chair. DB, FS and RV are members of the INdAM Research group GNCS. MG was supported by the Centre of Excellence in Analysis and Dynamics Research, Academy of Finland.
Publisher Copyright:
© 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
Funding
The research of FS is funded by NSERC-Sanofi Industrial Research Chair. DB, FS and RV are members of the INdAM Research group GNCS. MG was supported by the Centre of Excellence in Analysis and Dynamics Research, Academy of Finland.
Keywords
- Daphnia
- Equilibria
- First order partial differential equation
- Hopf bifurcation
- Numerical bifurcation analysis
- Periodic solutions
- Pseudospectral discretization
- Size-structured model
- Stability boundary
- Stem cells
- Transport equation