A model for a spatially structured metapopulation accounting for within patch dynamics

Synchronized Tick Population Oscillations Driven by Host Mobility and Spatially Heterogeneous Developmental Delays Combined

We consider a coupled system of delay differential equations for a single-species tick population dynamics, assuming feeding adult ticks are distributed by their hosts in a spatially heterogeneous environment consisting of two patches where egg ticks produced will complete their life cycles with different, normal and diapause, developmental delays. We show that the mobility of adult tick host and the diapause developmental delay combined drive a synchronized oscillation in the total tick populations around a uniquely defined positive equilibrium, and this synchronization makes the oscillatory patterns much simpler in comparison with multi-peak oscillations exhibited in the absence of host mobility.

A spatially structured metapopulation model within a stochastic environment

Populations often exist, either by choice or by external pressure, in a fragmented way, referred to as a metapopulation. Typically, the dynamics accounted for within metapopulation models are assumed to be static. For example, patch occupancy models often assume that the colonisation and extinction rates do not change, while spatially structured models often assume that the rates of births, deaths and migrations do not depend on time. While some progress has been made when these dynamics are changing deterministically, less is known when the changes are stochastic. It can be quite common that the environment a population inhabits determines how these dynamics change over time. Changes to this environment can have a large impact on the survival probability of a population and such changes will often be stochastic. The typical metapopulation model allows for catastrophes that could eradicate most, if not all, individuals on an entire patch. It is this type of phenomenon that this article addresses. A Markov process is developed that models the number of individuals on each patch within a metapopulation. An approximation for the original model is presented in the form of a piecewise-deterministic Markov process and the approximation is analysed to present conditions for extinction.

Species-level persistence probabilities for recovery and conservation status assessment

Recovery planning for species listed under the U.S. Endangered Species Act has been hampered by a lack of consistency and transparency, which can be improved by implementing a standardized approach for evaluating species status and developing measurable recovery criteria. However, managers lack an assessment method that integrates threat abatement and can be used when demographic data are limited. To help meet these needs, we demonstrated an approach for evaluating species status based on habitat configuration data. We applied 3 established persistence measures (patch occupancy, metapopulation capacity, and proportion of population lost) to compare 2 conservation strategies (critical habitat designated by the U.S. Fish and Wildlife Service and the Forest Service's Carbonate Habitat Management Strategy) and 2 threat scenarios (maximum limestone mining, removal of all habitat in areas with mining claims; minimum mining, removal of habitat only in areas with existing operations and high-quality ore) against a baseline of existing habitat for 3 federally listed plant species. Protecting all area within the designated critical habitat maintained a similar level (83.9-99.9%) of species persistence as the baseline, whereas maximum mining greatly reduced persistence (0.51-38.4% maintained). The 3 persistence measures provided complementary insights reflecting different aspects of habitat availability (total area, number of patches, patch size, and connectivity). These measures can be used to link recovery criteria developed following the 3 R principles (representation, redundancy, and resilience) to the resulting improvements in species viability. By focusing on amount and distribution of habitat, our method provides a means of assessing the status of data-poor species to inform decision making under the Endangered Species Act.

Limits for density dependent time inhomogeneous Markov processes

A new functional law of large numbers to approximate a time inhomogeneous Markov process that is only density dependent in the limit as an index parameter goes to infinity is developed. This extends previous results by other authors to a broader class of Markov processes while relaxing some of the conditions required for those results to hold. This result is applied to a stochastic metapopulation model that accounts for spatial structure as well as within patch dynamics with the novel addition of time dependent dynamics. The resulting nonautonomous differential equation is analysed to provide conditions for extinction and persistence for a number of examples. This condition shows that the migration of a species will positively impact the reproduction in less populated areas while negatively impacting densely populated areas.