Homogenization of Large-Scale Movement Models in Ecology

Abstract

A difficulty in using diffusion models to predict large scale animal population dispersal is that individuals move differently based on local information (as opposed to gradients) in differing habitat types. This can be accommodated by using ecological diffusion. However, real environments are often spatially complex, limiting application of a direct approach. Homogenization for partial differential equations has long been applied to Fickian diffusion (in which average individual movement is organized along gradients of habitat and population density). We derive a homogenization procedure for ecological diffusion and apply it to a simple model for chronic wasting disease in mule deer. Homogenization allows us to determine the impact of small scale (10-100 m) habitat variability on large scale (10-100 km) movement. The procedure generates asymptotic equations for solutions on the large scale with parameters defined by small-scale variation. The simplicity of this homogenization procedure is striking when compared to the multi-dimensional homogenization procedure for Fickian diffusion,and the method will be equally straightforward for more complex models.

How Individual Movement Response to Habitat Edges Affects Population Persistence and Spatial Spread

“…The persistence conditions for our model differ from the ones found by Cantrell and Cosner (1999) only by a factor of D or D 1/2 , depending on k. Hence, the results with respect to a are robust across model implementations, and we conjecture that there is some deeper mathematical connection between the two approaches than first meets the eye. In particular, the derivation of skew Brownian motion might depend on using Fickian flux or biological diffusion in reaction-diffusion models (Turchin 1998;Garlick et al 2011).…”

Mathematical Model of Seed Dispersal by Frugivorous Birds and Migration Potential of Pinyon and Juniper in Utah

“…One would expect step sizes in the random walks that dispersers follow, and therefore their diffusions rates, to vary strongly with habitat type. Recent advances in the use of homogenization [33] make integration of reaction-diffusion models with highly variable constants surprisingly easy, so building SDKs with variable diffusion and applying asymptotic techniques like homogenization will be our future concentration of research. 1 1 , n n n n j j j j t t P P S S P S t t…”

“…To estimate R 0 we refer to [24], who determined that only 6 pinyon survived the nineteenth-century fire on their site. The number of pinyon pines increased to 1051 over the next 145 years giving R 0 = 2.04/generation for pinyon from equation (33). [7] observed that Clark's Nutcracker fly from 4000 to 5000 meters while caching seeds, taking 15 -30 minutes.…”

“…To estimate R 0 we follow [24], who observed that only 109 junipers survived the nineteenth-century fire on their site. The number increased to 172 over the next 145 years giving R 0 = 1.17/generation for juniper from Equation (33). [6] observed that American robins forage in the range of 10 -100 meters with mean 55 meters and average 4 minutes between trees.…”

Temporal variation and scale in movement-based resource selection functions

“…The formal mathematical description of animal movement is quite old, dating back hundreds of years, and even though the seminal text on the topic written by Turchin [47] is quite relevant, it is currently out of print and lacks a contemporary statistical perspective. Despite the existence of highly technical literature describing ways to model animal movement (e.g., [7,15]), most applied studies that have actual management or conservation objectives have sought to focus more on what is termed ''space use'', in which they seek to characterize the geographical and/or environmental space used by either individual animals or populations or both.…”