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Sufficient condition for dispersal-induced growth on dynamic networks

Michel Benaïm, Claude Lobry, Tewfik Sari, Edouard Strickler

TL;DR

The paper addresses when a population distributed across a time-varying network can exhibit dispersal-induced growth (DIG) despite local sites being sinks in isolation. It introduces a framework based on dynamic networks and a growth index χ^C for p-circuits, proving a sufficient condition for DIG by minorizing growth along circuits using elementary linear-system comparisons. The results show that if a circuit has positive χ^C, DIG occurs for large environmental periods T and for migration strength m within an interval that remains nontrivial, with the threshold m^*(T) shrinking exponentially with T. Extensions to random season durations demonstrate that the DIG threshold remains exponentially small on average, and the approach yields practical insights across ecological scenarios, including migratory birds, and supports broader applications in epidemiology and complex dynamic networks.

Abstract

We consider a population spreading across a finite number of sites. Individuals can move from one site to the other according to a network (oriented links between the sites) that vary periodically over time. On each site, the population experiences a growth rate which is also periodically time varying. Recently, this kind of models have been extensively studied, using various technical tools to derive precise necessary and sufficient conditions on the parameters of the system (ie the local growth rate on each site, the time period and the strength of migration between the sites) for the population to grow. In the present paper, we take a completely different approach: using elementary comparison results between linear systems, we give sufficient condition for the growth of the population This condition is easy to check and can be applied in a broad class of examples. In particular, in the case when all sites are sinks (ie, in the absence of migration, the population become extinct in each site), we prove that when our condition of growth if satisfied, the population grows when the time period is large and for values of the migration strength that are exponentially small with respect to the time period, which answers positively to a conjecture stated by Katriel.

Sufficient condition for dispersal-induced growth on dynamic networks

TL;DR

The paper addresses when a population distributed across a time-varying network can exhibit dispersal-induced growth (DIG) despite local sites being sinks in isolation. It introduces a framework based on dynamic networks and a growth index χ^C for p-circuits, proving a sufficient condition for DIG by minorizing growth along circuits using elementary linear-system comparisons. The results show that if a circuit has positive χ^C, DIG occurs for large environmental periods T and for migration strength m within an interval that remains nontrivial, with the threshold m^*(T) shrinking exponentially with T. Extensions to random season durations demonstrate that the DIG threshold remains exponentially small on average, and the approach yields practical insights across ecological scenarios, including migratory birds, and supports broader applications in epidemiology and complex dynamic networks.

Abstract

We consider a population spreading across a finite number of sites. Individuals can move from one site to the other according to a network (oriented links between the sites) that vary periodically over time. On each site, the population experiences a growth rate which is also periodically time varying. Recently, this kind of models have been extensively studied, using various technical tools to derive precise necessary and sufficient conditions on the parameters of the system (ie the local growth rate on each site, the time period and the strength of migration between the sites) for the population to grow. In the present paper, we take a completely different approach: using elementary comparison results between linear systems, we give sufficient condition for the growth of the population This condition is easy to check and can be applied in a broad class of examples. In particular, in the case when all sites are sinks (ie, in the absence of migration, the population become extinct in each site), we prove that when our condition of growth if satisfied, the population grows when the time period is large and for values of the migration strength that are exponentially small with respect to the time period, which answers positively to a conjecture stated by Katriel.

Paper Structure

This paper contains 35 sections, 169 equations, 7 figures.

Table of Contents

  1. Model and notations
  2. The model
  3. Sites.
  4. Equations of the dynamic.
  5. Link, network.
  6. Seasonality.
  7. Migrations on dynamic networks.
  8. Exemple.
  9. Terminology: paths, circuits.
  10. Notations
  11. Path
  12. Dynamic path.
  13. Dynamic circuit.
  14. Main results
  15. Sources and sinks in a periodic system.
  16. ...and 20 more sections

Figures (7)

  • Figure 1: A migration network from KOL18 .
  • Figure 2: In red a 3-circuit in a dynamic network (black arrows).
  • Figure 3: Simulations of $\Sigma$ in the case $T =24$. One sees the logarithm of the population on site 1 (in red), on site 2 (in green) and site 3 (in blue)
  • Figure 4: Two possibilities for migration.
  • Figure 5: On the left: growth ‘‘migration to the source" \ref{['vlS']}. On the right growth of system ‘‘migration to the sink". Parameters:$r_1^1=r_2^2 = -2;\,r_2^1= r_1^2 = 1$
  • ...and 2 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Definition 6
  • Remark 2
  • Remark 3
  • Definition 7
  • ...and 7 more