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The multi-patch logistic equation with asymmetric migration

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari

Abstract

This paper considers a multi-patch model, where each patch follows a logistic law, and patches are coupled by asymmetrical migration terms. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic equation with a carrying capacity which in general is different from the sum of the n carrying capacities, and depends on the migration terms. Second, we determine, in some particular cases, the conditions under which fragmentation and asymmetrical migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities. Finally, for the three-patch model, we show numerically the existence of at least three critical values of the migration rate for which the total equilibrium population equals the sum of the carrying capacities.

The multi-patch logistic equation with asymmetric migration

Abstract

This paper considers a multi-patch model, where each patch follows a logistic law, and patches are coupled by asymmetrical migration terms. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic equation with a carrying capacity which in general is different from the sum of the n carrying capacities, and depends on the migration terms. Second, we determine, in some particular cases, the conditions under which fragmentation and asymmetrical migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities. Finally, for the three-patch model, we show numerically the existence of at least three critical values of the migration rate for which the total equilibrium population equals the sum of the carrying capacities.

Paper Structure

This paper contains 18 sections, 22 theorems, 102 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The mathematical model and preliminaries results
  3. Perfect mixing
  4. The fast dispersal limit
  5. Two time scale dynamics
  6. Comparison of $X_T^*(+\infty)$ with $\sum_iK_i$.
  7. Influence of asymmetric dispersal on total population size
  8. Asymmetric dispersal may be unfavorable to the total equilibrium population
  9. Asymmetric dispersal may be favorable to the total equilibrium population
  10. Independence of the total equilibrium population with respect to asymmetric dispersal
  11. Two blocks of identical patches
  12. Links between SIS and logistic patch models
  13. The SIS patch model
  14. Comparisons between the results on \ref{['m6']} and the results on \ref{['gao1']}
  15. Three-patch model
  16. ...and 3 more sections

Key Result

Proposition 2.1

The domain $\mathbb{R}_{+}^{n}=\left\lbrace (x_{1},\ldots, x_{n})\in \mathbb{R}^{n}/ x_{i}\geq 0, i=1,\ldots, n\right\rbrace$ is positively invariant for the system m6.

Figures (7)

  • Figure 1: Qualitative properties of Model (\ref{['m6']}) under the conditions (\ref{['samepatches']}) and (\ref{['samemigrations']}). In $\mathcal{J}_0$, patchiness has a beneficial effect on the total equilibrium population. This effect is detrimental in $\mathcal{J}_2$. In $\mathcal{J}_1$, the effect is beneficial for $\beta<\beta_0$ and detrimental for $\beta>\beta_0$.
  • Figure 2: The two graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ for which the migration matrix may be symmetric, if $\gamma_{ij}=\gamma_{ji}$.
  • Figure 3: The three graphs $\mathcal{G}_3, \mathcal{G}_4$ and $\mathcal{G}_5$ for which the migration matrix cannot be symmetric.
  • Figure 4: Total equilibrium population $X_{T}^{\ast}$ of the system \ref{['m6']}$(n=3)$ as a function of the migration rate $\beta$. The figure on the right is a zoom, near the origin, of the figure on the left. The parameter values are given in Table \ref{['t1']}.
  • Figure 5: Total equilibrium population $X_{T}^{\ast}$ of the system \ref{['m6']}$(n=3)$ as a function of the migration rate $\beta$. The figure on the right is a zoom, near the origin, of the figure on the left. The parameter values are given in Table \ref{['t1']}.
  • ...and 2 more figures

Theorems & Definitions (48)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 38 more