## 5 Real World Application
### Non-Smooth and Non-Periodic Environments and Location Data  
In practical applications, environmental data often exhibit irregularities and a lack of periodicity. In this section, we address these challenges and demonstrate our approach using relevant environments for meerkats from the study conducted by [PERSON] et al. (2015) [17]. We refer the interested reader to that study for detailed information about the data acquisition process for the meerkat habitat [17]. Notably, the type of sand and the interfaces between different sand types are believed to influence meerkat movement. The environmental data collected from meerkat habitats in South Africa, representing sand types and sand-type interfaces, are depicted in Figure 14a,b, respectively. For ease of analysis, the data has been normalized to a range between zero and one. In the case of sand-type data (Figure 14a), meerkats seem to prefer ferrous sand over clay sand, thus zero  
Figure 13: Iterations of SGD minimizing the negative log-likelihood function to estimate \(\eta\) and \(b\) using data generated with \(M=300\), \(N=2\), \(b=0.25\), \(a=1\), and \(\eta=5\) and the equilibrium solutions to System (2) using the parameters used to generate the data and the estimated parameters. (**a**) Iterations of SGD and corresponding estimated \(b\) and \(\eta\) (**top, middle**) and negative log-likelihood value (**bottom**). (**b**) PDF used to generate data. (**c**) PDF from estimated parameters.  
corresponds to clay sand, and one corresponds to ferrous sand. In the case of edge data (Figure 13(b)), zero represents the interface between different sand types, while one represents a distinct sand type. To address the irregularities in the environmental data, we apply a smoothing technique by convolving it with an approximation to the identity, resulting in a regularized representation denoted as \(U\). This regularization process ensures uniformity throughout the data while preserving its influence on animal movement. The regularized environment functions are visualized in Figure 13(c,d). Additionally, it is unsurprising that meerkat habitats lack periodicity, which poses a challenge for our numerical scheme that assumes periodic boundary conditions. To mitigate boundary effects, we extend our domain and set \(U(x)=0\) for locations \(x\) in the extended part of the domain. This extension is depicted by the blue border in Figure 13(c,d).  
We incorporate meerkat location data from the Kalahari Meerkat Project for illustrative purposes. The data, as presented in [17], cover three distinct time periods: a steady period characterized by relatively stable territories, a period during which a social group divided into two new groups, and a period when territory locations underwent shifts. The dataset encompasses observations from thirteen meerkat groups, where each data point represents the time and location of a sighting of a group member. Figure 14(a) visualizes the location data from the steady period. In order to utilize initial conditions based on this dataset, we conduct kernel density estimation for each group. To achieve convergence in numerical simulations with these initial conditions, we mollified the kernel density estimation for each group and added a small constant of 0.001 to avoid densities close to zero. Note that this shifts all densities and has no bearing on the results. The initial conditions with the corresponding data are shown in Figure 14(b). Please note that Figure 14(b) is presented without the extended domain. However, when utilizing the extended domain, the initial  
Figure 14: The meerkat environmental data and their regularized counterpart on an extended domain.  
values for each group are further adjusted away from the boundaries, thereby reducing the impact of periodic boundary conditions.