### Abstract

This paper is as much about a certain modelling methodology, as it is about the constructive definition of future population states from a description of individual behaviour and an initial population state. The key idea is to build a nonlinear model in two steps, by explicitly introducing the environmental condition via the requirement that individuals are independent from one another (and hence equations are linear) when this condition is prescribed as a function of time. A linear physiologically structured population model is defined by two rules, one for reproduction and one for development and survival, both depending on the initial individual state and the prevailing environmental condition. In Part I we showed how one can constructively define future population state operators from these two ingredients. A nonlinear model is a linear model together with a feedback law that describes how the environmental condition at any particular time depends on the population size and composition at that time. When applied to the solution of the linear problem, the feedback law yields a fixed point problem. This we solve constructively by means of the contraction mapping principle, for any given initial population state. Using subsequently this fixed point as input in the linear population model, we obtain a population semiflow. We then say that we solved the nonlinear problem.

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

Pages (from-to) | 157-189 |

Number of pages | 33 |

Journal | Journal of Mathematical Biology |

Volume | 43 |

Issue number | 2 |

State | Published - Aug 2001 |

### Fingerprint

### Keywords

- Cannibalism
- Deterministic at population level
- Nonlinear feedback via the environment
- Physiological structure
- Population dynamics

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

*Journal of Mathematical Biology*,

*43*(2), 157-189.

**On the formulation and analysis of general deterministic structured population models : II. Nonlinear theory.** / Diekmann, O.; Gyllenberg, M.; Huang, H.; Kirkilionis, M.; Metz, J. A J; Thieme, Horst.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 43, no. 2, pp. 157-189.

}

TY - JOUR

T1 - On the formulation and analysis of general deterministic structured population models

T2 - II. Nonlinear theory

AU - Diekmann, O.

AU - Gyllenberg, M.

AU - Huang, H.

AU - Kirkilionis, M.

AU - Metz, J. A J

AU - Thieme, Horst

PY - 2001/8

Y1 - 2001/8

N2 - This paper is as much about a certain modelling methodology, as it is about the constructive definition of future population states from a description of individual behaviour and an initial population state. The key idea is to build a nonlinear model in two steps, by explicitly introducing the environmental condition via the requirement that individuals are independent from one another (and hence equations are linear) when this condition is prescribed as a function of time. A linear physiologically structured population model is defined by two rules, one for reproduction and one for development and survival, both depending on the initial individual state and the prevailing environmental condition. In Part I we showed how one can constructively define future population state operators from these two ingredients. A nonlinear model is a linear model together with a feedback law that describes how the environmental condition at any particular time depends on the population size and composition at that time. When applied to the solution of the linear problem, the feedback law yields a fixed point problem. This we solve constructively by means of the contraction mapping principle, for any given initial population state. Using subsequently this fixed point as input in the linear population model, we obtain a population semiflow. We then say that we solved the nonlinear problem.

AB - This paper is as much about a certain modelling methodology, as it is about the constructive definition of future population states from a description of individual behaviour and an initial population state. The key idea is to build a nonlinear model in two steps, by explicitly introducing the environmental condition via the requirement that individuals are independent from one another (and hence equations are linear) when this condition is prescribed as a function of time. A linear physiologically structured population model is defined by two rules, one for reproduction and one for development and survival, both depending on the initial individual state and the prevailing environmental condition. In Part I we showed how one can constructively define future population state operators from these two ingredients. A nonlinear model is a linear model together with a feedback law that describes how the environmental condition at any particular time depends on the population size and composition at that time. When applied to the solution of the linear problem, the feedback law yields a fixed point problem. This we solve constructively by means of the contraction mapping principle, for any given initial population state. Using subsequently this fixed point as input in the linear population model, we obtain a population semiflow. We then say that we solved the nonlinear problem.

KW - Cannibalism

KW - Deterministic at population level

KW - Nonlinear feedback via the environment

KW - Physiological structure

KW - Population dynamics

UR - http://www.scopus.com/inward/record.url?scp=0035433541&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035433541&partnerID=8YFLogxK

M3 - Article

C2 - 11570590

AN - SCOPUS:0035433541

VL - 43

SP - 157

EP - 189

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 2

ER -