DOI: 10.5061/dryad.n8pk0p2xr
title: Code and data for: Emergence of spatially structured populations by area-concentrated search

description: The idea that populations are spatially structured has become a very
 powerful concept in ecology, raising interest in many research areas.
 However, despite dispersal being a core component of the concept, it
 typically does not consider the movement behavior underlying any
 dispersal. Using individual-based simulations in continuous space, we
 investigate the emergence of a spatially structured population in
 landscapes with spatially heterogeneous resource distribution and with
 organisms following simple area-concentrated search (ACS); individuals do
 not, however, perceive or respond to any habitat attributes per se but
 only to their foraging success. We investigated effects of different
 resource clustering patterns in landscapes (single large cluster vs. many
 small clusters) and different resource densities on spatial structure of
 populations and movement between resource clusters of individuals. As the
 results, we found that foraging success increased with increasing resource
 density and decreasing number of resource clusters. In a wide parameter
 space, the system exhibited attributes of a spatially structured
 population with individuals concentrated in areas of high resource
 density, searching within areas of resources, and 'dispersing'
 in a straight line between resource patches. 'Emigration' was
 more likely from patches that were small or of low quality (low resource
 density), but we observed an interaction effect between these two
 parameters. With the ACS implemented, individuals tended to move deeper
 into a resource cluster in scenarios with moderate resource density than
 in scenarios with high resource density. 'Looping' from patches
 was more likely if patches were large and of high quality. Our simulations
 demonstrate that spatial structure in populations may emerge if critical
 resources are heterogeneously distributed and if individuals follow simple
 movement rules (such as ACS). Neither the perception of habitat nor an
 explicit decision to emigrate from a patch on the side of acting
 individuals is necessary for the emergence of spatial structure.
description: We implement a simple model simulating the movement of
 resource-searching individuals (ACS) in a continuous landscape with
 heterogeneous resource distribution; both, the position of individuals and
 resources are thus continuous point coordinates. We investigate how
 resource distribution affects the spatial distribution (density) of
 individuals and the movement (dispersal) of individuals between resource
 clusters. Our simulation ignores birth and death events, but the model
 implicitly accounts for the diffuse effect of competition over resources
 on foraging behavior.  <strong>Spatial
 distribution of resources and scenarios</strong><br>We
 simulated foraging movement in a square landscape of area
 $4\times10\textsuperscript{6}$ ($2000 \times 2000$) squared spatial units
 with resources distributed within it. In the simulations we created $k$
 resource clusters within the landscape as continuous spatial point pattern
 with points generated by the Mat\'{e}rn Cluster Point Process, using
 R version 3.5.2, library spatstat version 1.58-2 (Baddeley and Turner,
 2005). Clusters were generated with daughter points (resources)
 distributed according to a random uniform distribution on a disk around
 parent points with $g$ as radius of the clusters and $u$ as resource
 density per area unit so that  $\bar R=g_i^2 \pi \times u$  was the
 expected number of resource items per cluster, and the expected number of
 resources items in the landscape was thus $k \times \bar R$. Parent points
 were distanced at least $3g$ units apart from each other to avoid cluster
 overlapping. The landscapes were wrapped into a torus in both dimensions
 to avoid edge effects and mimic a landscape of infinite dimension. Across
 scenarios, the number of resource clusters was increased from $k=1$ to
 $k=16$ clusters whereas the radius of clusters ($g$) was reduced from 320
 (at $k=1$) to 80 (at $k=16$) so that the total area covered by resource
 clusters was identical in all scenarios (\textit{c.} 8\% of total area).
 The average resource density in resource clusters was varied from $u=0.01$
 to $u=1.27$ resources per unit area. A summary of all model and simulation
 parameters and their values can be found in Table 1. <br>  
  <br><strong>Movement rule</strong><br>The
 movement of each individual was modeled as an ACS. Here we implemented the
 simplest of such possible rules, assuming that individual $i$ moved
 straighter, the longer the time interval in which it did not find a food
 item was, i.e. the longer the searching time $\Delta_{S,i}$ was (see
 Benhamou 1992, reviewed in Barto\'{n} and Hovestadt 2013); generally,
 such models have been shown to be efficient foraging strategies (e.g.
 Benhamou, 1992; Pyke 2015). Comparable movement was, for example, observed
 in starved amoeboid cells that move rather straight whereas well-fed cells
 moved changed direction much more frequently (van Haastert and Bosgraaf
 2009) but just as well in mammal species (Auger-Méthé et al. 2016). At any
 moment $t$, and for any moving individual $i$, the turning angle between
 two consecutive steps was determined by drawing a random value from a
 wrapped normal distribution (Jammalamadaka and SenGupta 2001) with mean 0
 and standard deviation $d_{i,t} (\Delta_{S,i})$ calculated
 as<br>\begin{equation}<br>d_{i,t}
 (\Delta_{S,i})=d_{min}+(d_{max}-d_{min})⋅(1-\frac{\Delta_{S,i}^\alpha}{(\Delta_{S,i}^\alpha+h^\alpha)})    <br>\end{equation}   <br><br>Consequently, $d_{i,t}$ ranges between $d_{min}=0.01$ (nearly straight-line movement) when $\Delta_{S,i} &gt;&gt; h$ and $d_{max}=1$ when $\Delta_{S,i}=0$, i.e. when the individual just found a food item. In the latter case, the movement became highly uncorrelated, and the individual performed area-concentrated search. The parameter $\alpha$ is a shape parameter (in our simulations always $\alpha = 3$), and $h$ is the half-saturation constant (always $h = 200$). The effects of parameter $\alpha$ and $h$ on $d_{i,t}$ and on foraging success were described in Barto\'{n} and Hovestadt (2013). We also tested different values of \(h\) and $\alpha$ in this study but found that these two parameters did not strongly affect results. We thus kept these two parameter values constant in all simulations. <br><br><strong>Foraging</strong><br> At each time step, each individual moved one step according to the movement rule described above. Individuals were moved in a random sequence to avoid priority benefits. The step length of movement ($p$) was constant and equal to 1 spatial unit. After movement, an individual immediately found all resource items within its perception radius ($c=1$ spatial unit, identical to the step length). All resource items within this radius were 'foraged' and removed (the individual maintained its position, however). Following a movement step, the value of $\Delta_{S,i}$ for each individual was increased to $\Delta_{S,i}+1$ in case an individual did not find a resource item, but was reset to $\Delta_{S,i}=0$ whenever the individual found a food item, thus initiating the ACS as described above.<br>    <br>After movement of all individuals, removed resource items were replaced by the same number of new items placed randomly as daughter points of randomly selected parent points according to the rules explained above (global replacement). With this global replacement, we implemented a global equilibrium assumption between resource production (regrowth) and consumption yet nonetheless allowing for the more short-term depletion (competition) effects due to intense local harvesting.<br><br><strong>Simulations and analysis</strong><br>For each parameter combination (resource density and cluster size, see above), we carried out ten replicates on ten independently created landscapes. In each simulation, eighty individuals were released at random coordinates within resource clusters. Their $\Delta_{S,i}$ value was set to $\Delta_{S,i}=500$ at the beginning of a simulation so that individuals started with nearly straight line movement. The initial direction of each individual was randomly selected from a uniform distribution between 0 and $2\pi$. At each time step, individuals moved and foraged resource items as described above. All individuals were allowed to move for 10,000 steps, but all analyses described below are based on data collected over the last 2000 movement steps only. <br>    <br>At the beginning of each simulation, the expected number of resource items per cluster was equal to $\bar R$ (see above). Due to the global replacement of foraged resource items, the total number of resource items in the landscape was kept constant and consequently, the average number of resource items per cluster remained at $\bar R$. However, the number of items in a single cluster could vary over time and degrade if the cluster was harvested intensively, i.e. by many individuals at the same time.