  The idea of a metapopulation has become canonical in ecology. Its original
mean field form provides the important intuition that migration and extinction
interact to determine the dynamics of a population composed of subpopulations.
From its conception, it has been evident that the very essence of the
metapopulation paradigm centers on the process of local extinction. We note
that there are two qualitatively distinct types of extinction, gradual and
catastrophic, and explore their impact on the dynamics of metapopulation
formation using discrete iterative maps. First, by modifying the classic
logistic map with the addition of the Allee effect, we show that catastrophic
local extinctions in subpopulations are a pre-requisite of metapopulation
formation. When subpopulations experience gradual extinction, increased
migration rates force synchrony and drive the metapopulation below the Allee
point resulting in migration induced destabilization of the system across
parameter space. Second, a sawtooth map (an extension of the Bernoulli bit
shift map) is employed to simultaneously explore the increasing and decreasing
modes of population behavior. We conclude with four generalizations. 1. At low
migration rates, a metapopulation may go extinct faster than completely
unconnected subpopulations. 2. There exists a gradient between stable
metapopulation formation and population synchrony, with critical transitions
from no metapopulation to metapopulation to synchronization, the latter
frequently inducing metapopulation extinction. 3. Synchronization patterns
emerge through time, resulting in synchrony groups and chimeric populations
existing simultaneously. 4. There are two distinct mechanisms of
synchronization: i. extinction and rescue and, ii.) stretch reversals in a
modification of the classic chaotic stretching and folding.
