# Phase-Type Distributions of Animal Trajectories with Random Walks
## 1 Introduction  
In recent years, animal movement monitoring has been of great importance in various scientific areas, such as population control, endangered species monitoring, pest control, and biological diversity. Additionally, studies on the ecological impact of animals on specific regions, humans' effects on animal behavior, and virus dissemination among different species, among others [1, 2, 3, 4, 5], require the use of animal movement observations. Without reliable animal movement patterns, many of the aforementioned applications cannot be effectively implemented. For instance, sensor nodes may be placed in areas where animals do not cross, or cameras may be positioned in areas with low animal traffic, limiting the surveillance capabilities [6, 7]. Animals can be monitored by collecting specific information, such as vital signs (temperature, blood pressure), DNA, body mass, or GPS coordinates, depending on the requirements of the application [8, 9]. The most common methods for collecting this information include direct observation of species, the use of radar, GPS systems, thermal cameras, capture-mark-recapture techniques, and attaching or implanting monitoring devices (e.g., collars, leg bands, backpacks) to animals [10]. The key advantage of these methods lies in their high accuracy, primarily stemming from their reliance on human observations. However, a significant drawback lies in their invasiveness, potentially leadingto challenges due to the inherent difficulty and risks associated with accessing the natural habitats of the species. Remote methods can also be employed, such as utilizing wireless sensor networks (WSNs) to detect animal positions and extract biological parameters using biological sensors. The collected data from these devices can later be retrieved through ground and air vehicles such as trucks, remote-controlled cars, helicopters, satellites, or unmanned aerial vehicles (UAVs), as described in [11; 12], or a combination of these techniques. These methods offer the advantage of being non-invasive to animals, eliminating the need for a human presence on site or, at the very least, minimizing their proximity. However, a significant disadvantage lies in their susceptibility to reduced accuracy in animal monitoring, encompassing aspects such as animal counts and positions. This reduced accuracy stems from their reliance on sensors, such as camera traps, sound microphones, radio transceivers, and even artificial intelligence software designed for autonomous species detection. For example, consider a system where WSNs make use of drones to efficiently collect animal information. Figure 1 illustrates a clustered WSN formed by cluster heads (\(CHs\)) that collect data from their cluster members (\(CMs\)). The \(CMs\) store information from sensors attached to moving animals whenever they pass by the static nodes. Then, the UAV flies to the main cluster head (\(CH_{\textit{sink}}\)), which has previously gathered all the data from the other \(CHs\), collects the information, and finally, returns to its base station (BS). This approach reduces long-range transmissions from energy-constrained nodes, effectively increasing the system's lifetime. These methods have varying impacts on the animal environment and present different challenges in collecting information from the animals. For instance, the Arctic long-tailed duck, a newly considered endangered species [13], needs to be remotely monitored to obtain its geolocation data. This information is crucial for understanding the regions it visits and ensuring its safety from predators, hunters, or polluted environments. Many of these methods face difficulties, including dangerous environments for both humans and animals. Local cameras or radars need to be strategically installed, and data collection must take place on site, with the possibility that the animal may not pass within the transceiver's coverage.