Theor Ecol DOI 10.1007/s12080-015-0283-7 ORIGINAL PAPER The “edge effect” phenomenon: deriving population abundance patterns from individual animal movement decisions Jonathan R. Potts1 · Thomas Hillen2 · Mark A. Lewis2,3 Received: 16 June 2015 / Accepted: 22 October 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract Edge effects have been observed in a vast spec- Keywords Advection-diffusion · Animal movement · trum of animal populations. They occur where two conjoin- Edge effect · Landscape complexity · Mathematical ecology · ing habitats interact to create ecological phenomena that are Partial differential equations · Step selection · not present in either habitat separately. On the individual- Transport equations level, an edge effect is a change in behavioral tendency on or near the edge. On the population-level, it is a pattern of pop- ulation abundance near an edge that cannot be explained in Introduction terms of either habitat in isolation. That these two levels of description exist suggests there ought to be a mathematical link between them. Here, we make inroads into providing Edges separating habitats affect the behavior and demo- such a link, deriving analytic expressions describing oft- graphics of animal populations across a wide range of taxa, observed population abundance patterns from a model of from fish to birds, from insects to mammals [e.g., Lidicker movement decisions near edges. Depending on the model and William (1999), Laurance et al. (2004), Batary´ et al. parameters, we can see positive, negative, or transitional (2009), and Macreadie et al. (2010)]. These so-called “edge edge effects emerge. Importantly, the distance over which effects” are multi-faceted and depend on the nature of the animals make their decisions to move between habitats turns edge itself, the habitats on either side, and the behavioral out to be a key factor in quantifying the magnitude of certain tendencies of the animals (Ries et al. 2004). However, there observed edge effects. is a common feature: that the edge between two habitats can affect animal behavior in ways that cannot be explained in terms of either habitat in isolation. Jonathan R. Potts The implication of this phenomenon is that ecosystem [email protected] services provided by a landscape cannot be understood Thomas Hillen purely by assessing (i) how much of each type of habitat [email protected] is present in the landscape, and (ii) which ecosystem ser- Mark A. Lewis vices each habitat provides. Rather, the interaction between [email protected] habitats, as well as the geometric details of how they tes- sellate the landscape, can have a large effect. Moreover, the 1 School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK effects can be complex and difficult to unravel. For exam- ple, the question “in which situations it is better to build a 2 Centre for Mathematical Biology, Department Single Large conservation area Or Several Small ones?” is of Mathematical and Statistical Sciences, 632 CAB, the subject of ongoing debate (the SLOSS debate), despite University of Alberta, Edmonton T6G 2G1, Canada many decades of research. Much of the debate boils down 3 Department of Biological Sciences, University of Alberta, to understanding the myriad possible effects of edges of Edmonton, Canada the conservation areas on the animals living there (Burkey Theor Ecol 1989; McNeill and Fairweather 1993; Salomon et al. 2002; input parameters, we observe either positive, negative, or Tjrve 2010). transitional edge effects. Importantly, the distance an animal As well as these concerns, understanding how and why is likely to move between successive habitat-selection deci- edge effects emerge is vital for accurately estimating pop- sions is key to the scale of these patterns. We compare ulation demographics. Resource selection analysis (Manly our results to two other approaches to deriving PDEs from et al. 2002), though very widely used, traditionally assumes individual-level decisions. that population abundance is a function of the underlying We focus on edge effects caused by habitat selection habitat quality. On the other hand, an edge effect means that decisions made as the animal moves. The model uses a population abundance in a given habitat is different near simple landscape consisting of two distinct habitats, with the edge to within the middle of a large patch of the same the “edge” between them being implicitly defined (Fig. 1). habitat-type. Some recent studies (Barnett and Moorcroft Sometimes, edge effects occur because the edge provides 2008; Potts et al. 2014a) point to the idea that this contra- a unique habitat in and of itself. In such cases, the cor- diction might be resolved by incorporating movement into rect model would have at least three habitats, with a thin the resource selection procedure. After all, animals are not “edge habitat” squeezed between the other two. Indeed, able to assess the whole landscape before deciding where more complicated models could incorporate larger numbers to reside, as they are limited by their ability to move and to of habitats in a variety of geometric configurations. While perceive the surrounding landscape. Therefore, the distance our modeling framework can be readily extended to explore over which animals are making their decisions to move, as these scenarios numerically [see, e.g., Potts et al. (2014a)], well as the frequency of those decisions, will likely have a we focus here on a case simple enough to gain clear analytic big effect on the locations at which they tend to be observed. insight. Indeed, the phrase “edge effect” is used interchange- The paper is organized as follows. The “Modelling move- ably to refer to individual-level movement decisions and ment near edges” section sets up the movement model. “A population-level patterns. On the individual-level, an edge partial differential equation approximation” section derives effect means a change in an animal’s movement patterns the population abundance patterns. “Comparison with other near an edge (Laurance et al. 2004; Schultz et al. 2012). For example, this can be a propensity either to cross edges—a positive edge effect, or avoid them—a negative edge effect. On the population level, a negative (resp. positive) edge effect is reported if the population abundance is lower (resp. higher) than average near an edge (Laurance et al. 2004; Batary´ et al. 2009). There are also transitional edge effects, where a sharp edge between habitats causes a much more gradual transition in population abundance [e.g. Lidicker and William (1999)]. This conflation of language suggests that individual-level decisions about how to move near edges, taking place on a relatively small spatio-temporal scale, ought to be responsible for patterns of abundance observed on a much larger scale. If so, it should be pos- sible to derive mathematically the various population-level patterns typically observed at edges from rules describing the underlying movement decisions of individual animals. The aim of this paper is to make inroads into providing such mathematical analysis. We set up a stochastic model of individual movement decisions near a habitat edge that explicitly incorporates the frequency those decisions—i.e., how far into the future is an animal looking as it makes its decision to move in a particular direction. We use this model to derive a partial differential equation (PDE) describing the population abun- dance distribution that arises from many animals moving according to such movement rules. The steady-state of the Fig. 1 Model habitat. To model a habitat edge, we use a simple envi- ronment with two habitats, A and B. The weighting for moving from A PDE is solved exactly to give an analytic expression of the to B is given by β, whereas α denotes the weighting for moving from predicted population abundance patterns. Depending on the BtoA Theor Ecol approaches” section examines other approaches to deriv- which habitats to move from and to. Consequently, given ing abundance patterns, comparing them to ours. Discus- the speed at which the animal moves, the standard deviation sion and concluding remarks are given in “Discussion and of φτ reflects the spatial scale over which such a decision is conclusions” section. made. A model of movement near edges Modelling movement near edges To gain analytic insight into how the scale of behavioral The general movement kernel framework decisions affect the population-level patterns near edges, we examine a particular one dimensional (1D) version of Eq. 1. The model is based on a discrete-time movement kernel For simplicity, we assume that the resource independent kτ (z|x,θ0,E), which describes the probability of an animal movement kernel φτ (z|x,θ0) = ρτ (z−x) is only dependent moving through an environment E to position z at a time on the difference z−x.Furthermore,weassumethatρτ (r) is τ in the future, given that it is currently at a position x and normally distributed with mean 0 and standard deviation σ. has arrived there on a bearing θ0. These often appear in the The landscape is modeled as a unit interval consisting of ecological literature as step selection functions and can be two habitats, A and B. Habitat A consists of the left-hand readily fitted to movement data using techniques that are half of the interval, [0, 1/2], and habitat B consists of the now relatively standard (e.g., Fortin et al. 2005;Foresteret right-hand half, (1/2, 1],sothatH(x) = A for x ∈[0, 1/2] al. 2009;Merkleetal.2014; Thurfjell et al. 2014). They and H(x) = B for x ∈ (1/2, 1]. The weighting func- enable the use of every location in a data set, together tion W[H(z),H(x)] is given by W[A, A]=W[B,B]= with environmental information, to uncover the effects of 1 (the weighting given for staying in the same habitat), landscape covariates on animal movement.