Comparison between the predicted frequency distribution of the least common behaviour (<i>F</i>), center of the thick gray line, and distributions obtained from Monte Carlo simulations (thin black lines) where the ratio, <i>c</i>, between the invaders' and the original behaviour' carrying capacities varied from A) <i>c</i> = 1.005 solid line, B) <i>c</i> = 1.5 thin dashed line and C) <i>c</i> = 5.0 dotted line.

<p>Increasing values of <i>c</i> implies increasing asymmetry in the logistic growth curve. The large width of the predicted gray line is more for help with the visual interpretation.</p

The expected transient time in years for a superior invader to increase in the original population from the fraction <i>F</i>_min to the fraction 1−<i>F</i>_min, where <i>s</i> is the rate of spread (fitness) of the superior invader and <i>F</i>_min is the smallest frequency at which one can observe the behaviour (user defined).

<p>The expected transient time in years for a superior invader to increase in the original population from the fraction <i>F</i>_min to the fraction 1−<i>F</i>_min, where <i>s</i> is the rate of spread (fitness) of the superior invader and <i>F</i>_min is the smallest frequency at which one can observe the behaviour (user defined).</p

Expected cumulative distribution function (CDF) (A) and the probability density function (PDF) (B) of the less common behaviour among populations with partial migration (frequencies between <i>F</i>_min and 1−<i>F</i>_min of the less common behaviour) as a function of the observed frequency of the less common behaviour (F).

<p>The thick line denotes <i>F</i>_min = 0.0001, thin solid line denotes <i>F</i>_min = 0.001, and the dashed line denotes <i>F</i>_min = 0.01.</p

The transient distribution describes the data better than the uniform distribution when the log-likelihood of the data based on the transient distribution (LLT) exceeds the log-likelihood based on the uniform distribution LLU and α<0.05.

<p>The transient distribution describes the data better than the uniform distribution when the log-likelihood of the data based on the transient distribution (LLT) exceeds the log-likelihood based on the uniform distribution LLU and α<0.05.</p

Cumulative distribution functions for the transient hypothesis for A) White perch (<i>Morone americana</i>) data from Kerr and Secor (2012) using F_min = 0.0151, B) Moose (<i>Alces alces</i>) data from Singh et al. (2012) using F_min = 0.026, and C) Red deer (<i>Cervus elaphus</i>) data from Mysterud et al. (2011) using F_min = 0.060.

<p>The dashed diagonal line denotes the cdf for the uniform distribution between F_min and 0.5.</p

Quantifying Migration Behaviour Using Net Squared Displacement Approach: Clarifications and Caveats

<div><p>Estimating migration parameters of individuals and populations is vital for their conservation and management. Studies on animal movements and migration often depend upon location data from tracked animals and it is important that such data are appropriately analyzed for reliable estimates of migration and effective management of moving animals. The Net Squared Displacement (NSD) approach for modelling animal movement is being increasingly used as it can objectively quantify migration characteristics and separate different types of movements from migration. However, the ability of NSD to properly classify the movement patterns of individuals has been criticized and issues related to study design arise with respect to starting locations of the data/animals, data sampling regime and extent of movement of species. We address the issues raised over NSD using tracking data from 319 moose (<i>Alces alces</i>) in Sweden. Moose is an ideal species to test this approach, as it can be sedentary, nomadic, dispersing or migratory and individuals vary in their extent, timing and duration of migration. We propose a two-step process of using the NSD approach by first classifying movement modes using mean squared displacement (MSD) instead of NSD and then estimating the extent, duration and timing of migration using NSD. We show that the NSD approach is robust to the choice of starting dates except when the start date occurs during the migratory phase. We also show that the starting location of the animal has a marginal influence on the correct quantification of migration characteristics. The number of locations per day (1–48) did not significantly affect the performance of non-linear mixed effects models, which correctly distinguished migration from other movement types, however, high-resolution data had a significant negative influence on estimates for the timing of migrations. The extent of movement, however, had an effect on the classification of movements, and individuals undertaking short- distance migrations can be misclassified as other movements such as sedentary or nomadic. Our study raises important considerations for designing, analysing and interpreting movement ecology studies, and how these should be determined by the biology of the species and the ecological and conservation questions in focus.</p></div

Comparison of migration model outputs at differing temporal resolutions of movement data and spatial resolution of starting locations using first recorded location and single location per day (SL), single location per day but a starting location that is the mean location during the first week (SLW), single location per day but a starting location that is the mean location during the first month (SLM) mean resolution data (MR) or high resolution data (HR).

<p>Comparison of migration model outputs at differing temporal resolutions of movement data and spatial resolution of starting locations using first recorded location and single location per day (SL), single location per day but a starting location that is the mean location during the first week (SLW), single location per day but a starting location that is the mean location during the first month (SLM) mean resolution data (MR) or high resolution data (HR).</p

The impact of “Starting Date” on the model outputs of migration parameters.

<p>Results are the averaged output for the 41 moose used in this study. The model outputs show estimates of distance (km²), timing (“From” is the start date of migration and “To” is the end date of migration) and duration (number of days) of the migrations. As per <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0149594#pone.0149594.g001" target="_blank">Fig 1</a>, Migration 1 or 2 can be either a spring or autumn migration depending on the starting date of the data. For moose in northern Sweden, spring migrations occur in May and June and autumn migrations occur between November and January. For the quantification of start and end of migration, we used the formulas mentioned in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0149594#pone.0149594.t001" target="_blank">Table 1</a>.</p

Effect of selection of “Starting Date” for different months and the predictability of migration.

<p>The table shows the percentage of moose individuals (<i>n</i> = 26) that are in their winter or summer range.</p

Mathematical equations and the description of parameters used in the Net Squared Displacement movement models in the study.

<p>Concordance criterion is used to estimate the model fits.</p