### Abstract

Abstract: Structured compartmental models in mathematical biology track age classes, stage classes, or size classes of a population. Structured modeling becomes important when mechanistic formulations or intraspecific interactions are class-dependent. The classic derivation of such models from partial differential equations produces time delays in the transition rates between classes. In particular, the transition from juvenile to adult has a delay equal to the maturation period of the organism. In the literature, many structured compartmental models, posed as ordinary differential equations, omit this delay. We reviewed occurrences of continuous-time compartmental models for age- and stage-structured populations in the recent literature (2000–2016) to discover which papers did so. About half of the 249 papers we reviewed used a maturation delay. Papers with ecological models were more likely to have the delay than papers with disease models, and mathematically focused papers were more likely to have the delay than biologically focused papers. Recommendations for Resource Managers: Interacting populations often are modeled with systems of ordinary differential equations in which the state variables are numbers of individuals of each species and interaction terms depend only on the current state of the system. Single-population continuous-time models with age- or stage-structure, in which state variables represent numbers of individuals in classes such as juveniles and adults, often but not always contain maturation time delays in the transition rates between classes. The exclusion of the delay typically changes the model dynamics. Managers should be aware of the maturation delay issue when considering the results of continuous-time models of structured populations. Discrete-time models have an inherent time delay, set by the census time step chosen by the modeler, and for that reason are convenient for modeling maturation and other biological delays.

Original language | English (US) |
---|---|

Article number | e12160 |

Journal | Natural Resource Modeling |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2018 |

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### Keywords

- age structure
- compartmental model
- continuous-time population model
- delay differential equation
- maturation rate
- McKendrick–von Foerster partial differential equation
- ordinary differential equation
- stage structure

### ASJC Scopus subject areas

- Modeling and Simulation
- Environmental Science (miscellaneous)

### Cite this

*Natural Resource Modeling*,

*31*(1), [e12160]. https://doi.org/10.1111/nrm.12160

**A matter of maturity : To delay or not to delay? Continuous-time compartmental models of structured populations in the literature 2000–2016.** / Robertson, Suzanne L.; Henson, Shandelle M.; Robertson, Timothy; Cushing, Jim M.

Research output: Contribution to journal › Review article

*Natural Resource Modeling*, vol. 31, no. 1, e12160. https://doi.org/10.1111/nrm.12160

}

TY - JOUR

T1 - A matter of maturity

T2 - To delay or not to delay? Continuous-time compartmental models of structured populations in the literature 2000–2016

AU - Robertson, Suzanne L.

AU - Henson, Shandelle M.

AU - Robertson, Timothy

AU - Cushing, Jim M

PY - 2018/2/1

Y1 - 2018/2/1

N2 - Abstract: Structured compartmental models in mathematical biology track age classes, stage classes, or size classes of a population. Structured modeling becomes important when mechanistic formulations or intraspecific interactions are class-dependent. The classic derivation of such models from partial differential equations produces time delays in the transition rates between classes. In particular, the transition from juvenile to adult has a delay equal to the maturation period of the organism. In the literature, many structured compartmental models, posed as ordinary differential equations, omit this delay. We reviewed occurrences of continuous-time compartmental models for age- and stage-structured populations in the recent literature (2000–2016) to discover which papers did so. About half of the 249 papers we reviewed used a maturation delay. Papers with ecological models were more likely to have the delay than papers with disease models, and mathematically focused papers were more likely to have the delay than biologically focused papers. Recommendations for Resource Managers: Interacting populations often are modeled with systems of ordinary differential equations in which the state variables are numbers of individuals of each species and interaction terms depend only on the current state of the system. Single-population continuous-time models with age- or stage-structure, in which state variables represent numbers of individuals in classes such as juveniles and adults, often but not always contain maturation time delays in the transition rates between classes. The exclusion of the delay typically changes the model dynamics. Managers should be aware of the maturation delay issue when considering the results of continuous-time models of structured populations. Discrete-time models have an inherent time delay, set by the census time step chosen by the modeler, and for that reason are convenient for modeling maturation and other biological delays.

AB - Abstract: Structured compartmental models in mathematical biology track age classes, stage classes, or size classes of a population. Structured modeling becomes important when mechanistic formulations or intraspecific interactions are class-dependent. The classic derivation of such models from partial differential equations produces time delays in the transition rates between classes. In particular, the transition from juvenile to adult has a delay equal to the maturation period of the organism. In the literature, many structured compartmental models, posed as ordinary differential equations, omit this delay. We reviewed occurrences of continuous-time compartmental models for age- and stage-structured populations in the recent literature (2000–2016) to discover which papers did so. About half of the 249 papers we reviewed used a maturation delay. Papers with ecological models were more likely to have the delay than papers with disease models, and mathematically focused papers were more likely to have the delay than biologically focused papers. Recommendations for Resource Managers: Interacting populations often are modeled with systems of ordinary differential equations in which the state variables are numbers of individuals of each species and interaction terms depend only on the current state of the system. Single-population continuous-time models with age- or stage-structure, in which state variables represent numbers of individuals in classes such as juveniles and adults, often but not always contain maturation time delays in the transition rates between classes. The exclusion of the delay typically changes the model dynamics. Managers should be aware of the maturation delay issue when considering the results of continuous-time models of structured populations. Discrete-time models have an inherent time delay, set by the census time step chosen by the modeler, and for that reason are convenient for modeling maturation and other biological delays.

KW - age structure

KW - compartmental model

KW - continuous-time population model

KW - delay differential equation

KW - maturation rate

KW - McKendrick–von Foerster partial differential equation

KW - ordinary differential equation

KW - stage structure

UR - http://www.scopus.com/inward/record.url?scp=85040984362&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85040984362&partnerID=8YFLogxK

U2 - 10.1111/nrm.12160

DO - 10.1111/nrm.12160

M3 - Review article

AN - SCOPUS:85040984362

VL - 31

JO - Natural Resource Modelling

JF - Natural Resource Modelling

SN - 0890-8575

IS - 1

M1 - e12160

ER -