Size-Structured Population Dynamics
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908601151389696 |
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| author | Diekmann, Odo Scarabel, Francesca |
| author_facet | Diekmann, Odo Scarabel, Francesca |
| contents | This chapter focuses on variable maturation delay or, more precisely, on the mathematical description of a size-structured population consuming an unstructured resource. When the resource concentration is a known function of time, we can describe the growth and survival of individuals quasi-explicitly, i.e., in terms of solutions of ordinary differential equations (ODE). Reproduction is captured by a (non-autonomous) renewal equation, which can be solved by generation expansion. After these preparatory steps, a contraction mapping argument is needed to construct the solution of the coupled consumer-resource system with prescribed initial conditions. As we shall show, this interpretation-guided constructive approach does in fact yield weak solutions of a familiar partial differential equation (PDE). A striking difficulty with the PDE approach is that the solution operators are, in general, not differentiable, precluding a linearized stability analysis of steady states. This is a manifestation of the state-dependent delay difficulty. As a (not entirely satisfactory and rather technical) way out, we present a delay equation description in terms of the history of both the $p$-level birth rate of the consumer population and the resource concentration. We end by using pseudospectral approximation to derive a system of ODE and demonstrating its use in a numerical bifurcation analysis. Importantly, the state-dependent delay difficulty dissolves in this approximation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03413 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Size-Structured Population Dynamics Diekmann, Odo Scarabel, Francesca Populations and Evolution This chapter focuses on variable maturation delay or, more precisely, on the mathematical description of a size-structured population consuming an unstructured resource. When the resource concentration is a known function of time, we can describe the growth and survival of individuals quasi-explicitly, i.e., in terms of solutions of ordinary differential equations (ODE). Reproduction is captured by a (non-autonomous) renewal equation, which can be solved by generation expansion. After these preparatory steps, a contraction mapping argument is needed to construct the solution of the coupled consumer-resource system with prescribed initial conditions. As we shall show, this interpretation-guided constructive approach does in fact yield weak solutions of a familiar partial differential equation (PDE). A striking difficulty with the PDE approach is that the solution operators are, in general, not differentiable, precluding a linearized stability analysis of steady states. This is a manifestation of the state-dependent delay difficulty. As a (not entirely satisfactory and rather technical) way out, we present a delay equation description in terms of the history of both the $p$-level birth rate of the consumer population and the resource concentration. We end by using pseudospectral approximation to derive a system of ODE and demonstrating its use in a numerical bifurcation analysis. Importantly, the state-dependent delay difficulty dissolves in this approximation. |
| title | Size-Structured Population Dynamics |
| topic | Populations and Evolution |
| url | https://arxiv.org/abs/2506.03413 |