Modeling Diffusive Spread in a Heterogeneous Population: A Movement Study with Stream Fish

Abstract

Using a mark-recapture technique in a small temperate stream, we described the movement of four fish species over a five-month period and developed a mathematical model that described the observed movement patterns. The movement distributions were generally leptokurtic, and two of the four species demonstrated some degree of upstream bias. There was little difference in movement among species or through time. There were no temporal correlations in probability of movement, movement direction, or distance moved. The spatial spread of the most abundant species, bluehead chubs, over a four-month period was characterized by upstream bias, diffusion-like spread, and persistent leptokurtosis. Bluehead chubs demonstrated complex relationships between probability of movement and size and growth, while creek chubs showed only an effect of size on probability of movement. Further, growth of individual bluehead chubs was correlated through time. These empirical results suggest the hypothesis that heterogeneity in phenotypic attributes, such as size and growth, is related to heterogeneity in movement behavior.A diffusion-advection model of bluehead chub movement, structured with two subgroups that dispersed at different rates (''fast fish'' and ''slow fish''), was parameterized and validated with the field data. This model with heterogeneity in movement rates generated the leptokurtic pattern observed in the field data, in contrast to the classic diffusion model without population heterogeneity, which produces a normal distribution.The results from this work suggest that heterogeneity in fitness-influencing attributes such as size and growth could explain heterogeneity in individual-level movement behavior and might underlie the leptokurtic patterns that have been observed at the population level in numerous field studies.

Movement in Corridors: Enhancement by Predation Threat, Disturbance, and Habitat Structure

“…Because our sampling interval was about two months, we present data as net movement per 60 days. We then analyzed such movement distributions by using the advection-diffusion framework for population movement in one dimension (Zabel and Anderson 1997, Turchin 1998, Skalski and Gilliam 2000, in which directional bias (the ''advection'' component), if any, is separated from population spread (the ''diffusion'' component), represented by the var-iance in distance moved. However, we never detected a directional bias (mean signed movement was never different from zero), and hereafter we concentrate on differences in variances of the movement distributions.…”

“…Among those sources are (a) temporal heterogeneity (water level or season), (b) spatial heterogeneity (isolated side pools and the river proper), and (c) morphological phenotype (body length). In principle, a simple diffusion process might describe the data well within any single source of heterogeneity, with possibly normally distributed movement distributions within each combination, and a leptokurtic distribution resulting when such normal distributions with unequal variances are summed (Skalski and Gilliam 2000). Further, Sokolowski et al (1997) have related movement to a genetic polymorphism, and we have discovered a source of phenotypic variation in Rivulus in which a laboratory assay for ''bold'' vs. ''shy'' behavior contributes to the heterogeneity of movement in field releases (D. F. Fraser and J. F. Gilliam, unpublished data).…”

Fish dispersal in fragmented landscapes: a modeling framework for quantifying the permeability of structural barriers

“…Population heterogeneity in dispersal behavior is common in nature (Skalski and Gilliam 2000, Rodrı´guez 2002, Gurarie et al 2009). The Laplace mixture accounts for population heterogeneity by combining two Laplace kernels, one for sedentary individuals and the other for mobile ones (Rodrı´guez 2002, Coombs andRodrı´guez 2007).…”

“…The models presented here assume that movement parameters are identical for upstream and downstream movements, implying symmetry and nondirectionality of displacements. When these assumptions do not hold, asymmetric Laplace distributions (Kotz et al 2001) or advectiondiffusion models (Skalski and Gilliam 2000) may provide viable alternatives. Similarly, separate permeability values can be used for upstream and downstream movements whenever a barrier is not expected to be equally permeable in either direction.…”

Disentangling the effects of habitat suitability, dispersal, and fragmentation on the distribution of river fishes

“…Recently developed methods and tools provide parameters to quantitatively describe the heterogeneous patterns of fish dispersal (e.g., Radinger and Wolter 2014) and quantitative approaches to account for barriers (e.g., permeability of barriers) influencing the shape of dispersal kernels (Pe´pino et al 2012). In addition, Radinger and Wolter (2014), as well as Skalski and Gilliam (2000), demonstrated that diffusion-like spread of species develops with time, resulting in increasingly flat dispersal kernels. Recent analytical advancements now allow for process-based modeling of species-specific fish dispersal in river networks by explicitly considering stationary and mobile components of a fish population, the location and passability of barriers, as well as the temporal component of dispersal .…”