# Phase-Type Distributions of Animal Trajectories with Random Walks
## 1 Introduction
Animals often need to be captured to attach uncomfortable monitoring devices before being reintroduced into their natural habitat, causing stress to the animals and potential risks to humans, as observed in the case of ocelots in Barro Colorado [14]. Commonly, WSNs are used for monitoring without the need for attached sensors, particularly in large areas. However, factors such as energy consumption of the nodes and other parameters must be taken into account. The use of large vehicles such as trucks or helicopters incurs high costs and requires highly specialized human intervention [15; 16; 17]. Moreover, the natural habitats of animals are likely to be disturbed. Methods such as remote-controlled aerial or terrestrial small vehicles offer reduced monetary costs for data collection, but the observation area may be limited or not accessible, and the lifetime of small unmanned vehicles is also restricted. Figure 1: WSN for animal monitoring from a base station using cluster heads and cluster members. Given these considerations, it is crucial to know the animal trajectories in advance before installing, building, or acquiring the necessary devices, tools, or programs to optimize the monitoring system's performance. For example, if a group of scientists is studying mangabey monkeys' migration in the jungle [18], which spans large areas, they can attach sensors to the animals that will be detected by a WSN consisting of static sensors randomly placed on the ground. This information allows them to predict the zone through which the animals will pass, even if it is an approximation. Consequently, they can position the sensors in a way that minimizes energy consumption [19; 20]. However, obtaining real animal trajectories is a challenging task since only a small number of animals are being tracked, and even fewer are reported in the literature. Therefore, it is often not feasible to consider thousands of different trajectories for Monte Carlo simulations or other realistic simulators such as the \(NS\)-\(3\) network simulator software [21; 22]. In this regard, the proposed methodology aims to generate as many virtual animal traces [23; 24; 25] as required by the research team to conduct statistically accurate simulations. The performance and accuracy of the monitoring methods described above highly depend on certain characteristics of the monitored animals, such as animal movement statistics [26]. Factors such as the duration of animal inactivity, walking patterns, and the chosen direction of movement directly influence the probability of animal detection, i.e., the likelihood of the animal being within or outside the coverage radius of a node in a WSN or camera. Therefore, it is essential to develop accurate random walks that closely resemble the movement of each animal. It is advisable for any animal tracking system to consider the mobility statistics of each specific animal [27; 28]. For illustrative purposes, four different animals are considered: African mangabey monkeys, ocelots of Barro Colorado, bats of Ghana, and Arctic long-tailed ducks. These animals are chosen for the following reasons:  
* The mangabey, a rare terrestrial monkey, is classified as endangered on the International Union for Conservation of Nature (IUCN) Red List of Threatened Species (RLTS). A recent assessment by the IUCN indicates that they are a step closer to extinction. The primate was known to inhabit specific sites in western Ghana, eastern Cote d'Ivoire, and southern Burkina Faso, but it was recently discovered in the Atewa Forest [29]. * The ocelot occupies a wide range of habitats, from mangroves to high-altitude cloud forests, but it is commonly associated with areas of thick vegetation. Classified as \"concern\" by the IUCN, the ocelot is protected in most countries within its distribution range. Hunting of this species is prohibited in many countries in North, Central, and South America. The ocelot has been exploited in the wild by the pet trade, often involving killing the mother to obtain the kittens.