A Next-Generation Approach to Calculate Source–Sink Dynamics in Marine Metapopulations

Next-generation matrices for marine metapopulations: the case of sea lice on salmon farms*

“…To calculate the next-generation matrix it is necessary to linearize a potentially density dependent model around the zero equilibrium so that the effect of density dependence at higher population sizes is ignored when calculating patch contribution. This approach of ignoring density dependence is common when determining persistence or patch contribution of marine metapopulations, as the focus is either on determining if a metapopulation can persist at all, or determining which habitat patches are acting as population sinks and which are acting as population sources [15, 10, 24, 23]. Alternatively it is also useful for determining patch contribution in metapopulations of species that are being actively controlled to remain at low densities, such as sea lice on salmon farms, which is our focus in section 2.4.…”

“…under the same framework of the next-generation matrix. The column sums can be used to measure the contribution of each habitat patch over a generation and the spectral radius can be used to evaluate metapopulation persistence [23]. Next-generation operators have previously been used in ecology to calculate source and sink regions in heterogeneous environments [24, 25, 26, 23], and to evaluate the level of control required to suppress invasive species [27].…”

A framework for studying transients in marine metapopulations

“…For the transient measure of the patch specific contribution to the initial growth of the total population we use λ j , previously defined in section 2. To measure the source-sink distribution we use the the classification derived from the next-generation matrix, K. This measure has previously been used to calculate the source-sink distribution in heterogenous environments [25,31,21,13]. In order to calculate the next-generation matrix for system (1) we decompose A = F − V , where F is a non-negative matrix with positive entries that describe the birth of new individuals in the metapopulation, and V is a non-singular M matrix with entries that describe the transfer of individuals between compartments or in this case habitat patches, and also includes the death of individuals.…”

“…For system (13), g-reactivity is equivalent to reactivity in our weighted norm, max j λp j , if C is a 2n × 2n identity matrix, but with the first n diagonal entries replaced with s 1 , . .…”

“…One of the new and easily tractable metrics that embodies this relationship comes from the theory of next-generation matrices and the basic reproduction number, R 0 . This framework, originally developed in epidemiology, has been used to characterize sources and sinks in populations of mussels, salmon, and sea lice on salmon farms [25,22,13].…”