Explaining empirical dynamic modelling using verbal, graphical and mathematical approaches.

Empirical dynamic modelling (EDM) is becoming an increasingly popular method for understanding the dynamics of ecosystems. It has been applied to laboratory, terrestrial, freshwater and marine systems, used to forecast natural populations and has addressed fundamental ecological questions. Despite its increasing use, we have not found full explanations of EDM in the ecological literature, limiting understanding and reproducibility. Here we expand upon existing work by providing a detailed introduction to EDM. We use three progressively more complex approaches. A short verbal explanation of EDM is then explicitly demonstrated by graphically working through a simple example. We then introduce a full mathematical description of the steps involved. Conceptually, EDM translates a time series of data into a path through a multi-dimensional space, whose axes are lagged values of the time series. A time step is chosen from which to make a prediction. The state of the system at that time step corresponds to a 'focal point' in the multi-dimensional space. The set (called the library) of candidate nearest neighbours to the focal point is constructed, to determine the nearest neighbours that are then used to make the prediction. Our mathematical explanation explicitly documents which points in the multi-dimensional space should not be considered as focal points. We suggest a new option for excluding points from the library that may be useful for short-term time series that are often found in ecology. We focus on the core simplex and S-map algorithms of EDM. Our new R package, pbsEDM, enhances understanding (by outputting intermediate calculations), reproduces our results and can be applied to new data. Our work improves the clarity of the inner workings of EDM, a prerequisite for EDM to reach its full potential in ecology and have wide uptake in the provision of advice to managers of natural resources.

Experimental evidence of size-selective harvest and environmental stochasticity effects on population demography, fluctuations and non-linearity.

Theory and analyses of fisheries data sets indicate that harvesting can alter population structure and destabilise non-linear processes, which increases population fluctuations. We conducted a factorial experiment on the population dynamics of Daphnia magna in relation to size-selective harvesting and stochasticity of food supply. Harvesting and stochasticity treatments both increased population fluctuations. Timeseries analysis indicated that fluctuations in control populations were non-linear, and non-linearity increased substantially in response to harvesting. Both harvesting and stochasticity induced population juvenescence, but harvesting did so via the depletion of adults, whereas stochasticity increased the abundance of juveniles. A fitted fisheries model indicated that harvesting shifted populations towards higher reproductive rates and larger-magnitude damped oscillations that amplify demographic noise. These findings provide experimental evidence that harvesting increases the non-linearity of population fluctuations and that both harvesting and stochasticity increase population variability and juvenescence.

Functional genetic diversity in an exploited marine species and its relevance to fisheries management.

The timing of reproduction influences key evolutionary and ecological processes in wild populations. Variation in reproductive timing may be an especially important evolutionary driver in the marine environment, where the high mobility of many species and few physical barriers to migration provide limited opportunities for spatial divergence to arise. Using genomic data collected from spawning aggregations of Pacific herring (Clupea pallasii) across 1600 km of coastline, we show that reproductive timing drives population structure in these pelagic fish. Within a specific spawning season, we observed isolation by distance, indicating that gene flow is also geographically limited over our study area. These results emphasize the importance of considering both seasonal and spatial variation in spawning when delineating management units for herring. On several chromosomes, we detected linkage disequilibrium extending over multiple Mb, suggesting the presence of chromosomal rearrangements. Spawning phenology was highly correlated with polymorphisms in several genes, in particular SYNE2, which influences the development of retinal photoreceptors in vertebrates. SYNE2 is probably within a chromosomal rearrangement in Pacific herring and is also associated with spawn timing in Atlantic herring (Clupea harengus). The observed genetic diversity probably underlies resource waves provided by spawning herring. Given the ecological, economic and cultural significance of herring, our results support that conserving intraspecific genetic diversity is important for maintaining current and future ecosystem processes.

Collapse, Tipping Points, and Spatial Demographic Structure Arising from the Adopted Migrant Life History.

The roles of dispersal and recruitment have long been a focal point in ecology and conservation. The adopted migrant hypothesis proposes a life history in which social learning transmits migratory knowledge between generations of iteroparous fish. Specifically, juveniles disperse from the parental spawning site, encounter and recruit to a local adult population, and learn migration routes between spawning and foraging habitats by following older, experienced fish. Although the adopted migrant life history may apply to many species of pelagic marine fishes, there is scant theoretical or empirical work on the consequent population dynamics. We developed and analyzed a mathematical model of this life history in which the recruitment of juveniles depends on the relative abundance of the local populations and recruitment overlap, which measures the ease with which juveniles are recruited by a nonparental population. We demonstrate that the adopted migrant life history can maintain spatial demographic structure among local populations, that it can also predispose local populations to collapse when a tipping point is crossed, and that recovery after collapse is impaired by reduced recruitment at small local population sizes.

Lessons from sea louse and salmon epidemiology.

Effective disease management can benefit from mathematical models that identify drivers of epidemiological change and guide decision-making. This is well illustrated in the host-parasite system of sea lice and salmon, which has been modelled extensively due to the economic costs associated with sea louse infections on salmon farms and the conservation concerns associated with sea louse infections on wild salmon. Consequently, a rich modelling literature devoted to sea louse and salmon epidemiology has been developed. We provide a synthesis of the mathematical and statistical models that have been used to study the epidemiology of sea lice and salmon. These studies span both conceptual and tactical models to quantify the effects of infections on host populations and communities, describe and predict patterns of transmission and dispersal, and guide evidence-based management of wild and farmed salmon. As aquaculture production continues to increase, advances made in modelling sea louse and salmon epidemiology should inform the sustainable management of marine resources.

Modeling parasite dynamics on farmed salmon for precautionary conservation management of wild salmon.

Conservation management of wild fish may include fish health management in sympatric populations of domesticated fish in aquaculture. We developed a mathematical model for the population dynamics of parasitic sea lice (Lepeophtheirus salmonis) on domesticated populations of Atlantic salmon (Salmo salar) in the Broughton Archipelago region of British Columbia. The model was fit to a seven-year dataset of monthly sea louse counts on farms in the area to estimate population growth rates in relation to abiotic factors (temperature and salinity), local host density (measured as cohort surface area), and the use of a parasiticide, emamectin benzoate, on farms. We then used the model to evaluate management scenarios in relation to policy guidelines that seek to keep motile louse abundance below an average three per farmed salmon during the March-June juvenile wild Pacific salmon (Oncorhynchus spp.) migration. Abiotic factors mediated the duration of effectiveness of parasiticide treatments, and results suggest treatment of farmed salmon conducted in January or early February minimized average louse abundance per farmed salmon during the juvenile wild salmon migration. Adapting the management of parasites on farmed salmon according to migrations of wild salmon may therefore provide a precautionary approach to conserving wild salmon populations in salmon farming regions.