Sufficient condition for dispersal-induced growth on dynamic networks

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.

💡 Research Summary

The paper studies a linear metapopulation model in which a finite set of habitats (sites) are linked by a time‑varying directed network of migrations. The population at site i, denoted x_i(t), obeys the non‑autonomous differential equation

 dx_i/dt = r_i(t/T) x_i + m ∑j ℓ{ij}(t/T) x_j, i = 1,…,n,

where r_i(·) and ℓ_{ij}(·) are 1‑periodic functions, T is the environmental period (e.g., a year), and m ≥ 0 is a scalar measuring overall migration intensity. The migration matrix L(t) = (ℓ_{ij}(t)) is a Metzler matrix (non‑negative off‑diagonal entries, zero column sums), guaranteeing that solutions with positive initial data remain positive.

The authors introduce the notion of a p‑dynamic network: the interval