On the formulation and analysis of general deterministic structured population models: II. Nonlinear theory

O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A J Metz, Horst Thieme

Research output: Contribution to journalArticle

133 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)157-189
Number of pages33
JournalJournal of Mathematical Biology
Volume43
Issue number2
StatePublished - Aug 2001

Fingerprint

Structured Populations
Population Model
Formulation
Feedback Law
Population
Nonlinear Model
Nonlinear Dynamics
Linear Model
nonlinear models
Feedback
Contraction Mapping Principle
environmental factors
Semiflow
Linear Models
Fixed Point Problem
Population Size
Linear equations
Nonlinear Problem
Fixed point
Population Density

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

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.

In: Journal of Mathematical Biology, Vol. 43, No. 2, 08.2001, p. 157-189.

Research output: Contribution to journalArticle

Diekmann, O, Gyllenberg, M, Huang, H, Kirkilionis, M, Metz, JAJ & Thieme, H 2001, 'On the formulation and analysis of general deterministic structured population models: II. Nonlinear theory', Journal of Mathematical Biology, vol. 43, no. 2, pp. 157-189.
Diekmann, O. ; Gyllenberg, M. ; Huang, H. ; Kirkilionis, M. ; Metz, J. A J ; Thieme, Horst. / On the formulation and analysis of general deterministic structured population models : II. Nonlinear theory. In: Journal of Mathematical Biology. 2001 ; Vol. 43, No. 2. pp. 157-189.
@article{42276babeb09413d9b3242378ce69bd3,
title = "On the formulation and analysis of general deterministic structured population models: II. Nonlinear theory",
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.",
keywords = "Cannibalism, Deterministic at population level, Nonlinear feedback via the environment, Physiological structure, Population dynamics",
author = "O. Diekmann and M. Gyllenberg and H. Huang and M. Kirkilionis and Metz, {J. A J} and Horst Thieme",
year = "2001",
month = "8",
language = "English (US)",
volume = "43",
pages = "157--189",
journal = "Journal of Mathematical Biology",
issn = "0303-6812",
publisher = "Springer Verlag",
number = "2",

}

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 -