Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ergodic and chaotic properties of some biological models

Ryszard Rudnicki

TL;DR

From properties of invariant measures, some chaotic properties of semiflows such as the existence of dense trajectories and strong instability of all trajectories are deduced.

Abstract

In this note we present two types of biological models which have interesting ergodic and chaotic properties. The first type are one-dimensional transformations, like a logistic map, which are used to describe the change in population size in successive generations. We study ergodic properties of such transformations using Frobenius--Perron operators. The second type are some structured populations models, for example a space-structured model, or a model of maturity-distribution of precursors of blood cells. These models are described by partial differential equations, which generate semiflows on the space of functions. We construct strong mixing invariant measures for these semiflows using stochastic precesses. From properties of invariant measures we deduce some chaotic properties of semiflows such as the existence of dense trajectories and strong instability of all trajectories.

Ergodic and chaotic properties of some biological models

TL;DR

From properties of invariant measures, some chaotic properties of semiflows such as the existence of dense trajectories and strong instability of all trajectories are deduced.

Abstract

In this note we present two types of biological models which have interesting ergodic and chaotic properties. The first type are one-dimensional transformations, like a logistic map, which are used to describe the change in population size in successive generations. We study ergodic properties of such transformations using Frobenius--Perron operators. The second type are some structured populations models, for example a space-structured model, or a model of maturity-distribution of precursors of blood cells. These models are described by partial differential equations, which generate semiflows on the space of functions. We construct strong mixing invariant measures for these semiflows using stochastic precesses. From properties of invariant measures we deduce some chaotic properties of semiflows such as the existence of dense trajectories and strong instability of all trajectories.
Paper Structure (4 sections, 2 theorems, 46 equations, 1 table)

This paper contains 4 sections, 2 theorems, 46 equations, 1 table.

Table of Contents

  1. Introduction
  2. Measure-preserving dynamical systems
  3. Population generation models
  4. Invariant measures and chaos of structured populations

Key Result

Theorem 3

Let $S\colon [0,L]\to [0,L]$ be a map satisfying the following conditions: Then the Frobenius--Perron operator $P_S$ is asymptotically stable, i.e., there exists $f^*\in D$ such that $\lim_{n\to\infty} P_S^nf=f^*$ for every $f\in D$.

Theorems & Definitions (3)

  • Theorem 3
  • Theorem 5
  • proof