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An approximation of populations on a habitat with large carrying capacity

N. Bauman, P. Chigansky, F. Klebaner

Abstract

We consider stochastic dynamics of a population which starts from a small colony on a habitat with large but limited carrying capacity. A common heuristics suggests that such population grows initially as a Galton-Watson branching process and then its size follows an almost deterministic path until reaching its maximum, sustainable by the habitat. In this paper we put forward an alternative and, in fact, more accurate approximation which suggests that the population size behaves as a special nonlinear transformation of the Galton Watson process from the very beginning.

An approximation of populations on a habitat with large carrying capacity

Abstract

We consider stochastic dynamics of a population which starts from a small colony on a habitat with large but limited carrying capacity. A common heuristics suggests that such population grows initially as a Galton-Watson branching process and then its size follows an almost deterministic path until reaching its maximum, sustainable by the habitat. In this paper we put forward an alternative and, in fact, more accurate approximation which suggests that the population size behaves as a special nonlinear transformation of the Galton Watson process from the very beginning.
Paper Structure (15 sections, 8 theorems, 106 equations)

This paper contains 15 sections, 8 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Large initial colony
  4. Small initial colony
  5. This paper's contribution
  6. The main result
  7. Proof of Theorem \ref{['thm:main']}
  8. Properties of $\mathbf H$
  9. The auxiliary Galton-Watson process
  10. The approximation
  11. Contribution of the initial condition
  12. Contribution of $R_n(K)$
  13. Contribution of $\varepsilon^{(3)}$
  14. Contribution of $\varepsilon^{(2)}$
  15. Contribution of $\varepsilon^{(1)}$

Key Result

Theorem 2

Let $n_1:=n_1(K)=[\log_\rho K]$ then

Theorems & Definitions (19)

  • Remark 1
  • Theorem 2: ChJK19
  • Remark 3
  • Remark 4
  • Theorem 5
  • Example 6
  • Example 7
  • Lemma 8
  • Lemma 9
  • proof
  • ...and 9 more