How animal movement influences wildlife-vehicle collision risk: a mathematical framework for range-resident species

Abstract

Wildlife-vehicle collisions (WVC) threaten both biodiversity and human safety worldwide. Despite empirical efforts to characterize the major determinants of WVC risk and optimize mitigation strategies, we still lack a theoretical framework linking traffic, landscape, and individual movement features to collision risk. Here, we introduce such a framework by leveraging recent advances in movement ecology and reaction-diffusion stochastic processes with partially absorbing boundaries. Focusing on range-resident terrestrial mammals -- responsible for most fatal WVCs -- we model interactions with a single linear road and derive exact expressions for key survival statistics, including mean collision time and road-induced lifespan reduction. These quantities are expressed in terms of measurable parameters, such as traffic intensity or road width, and movement parameters that can be robustly estimated from relocation data, such as home-range crossing times, home-range sizes, or distance between home-range center and road. Therefore, our work provides an effective theoretical framework integrating movement and road ecology, laying the foundation for data-driven, evidence-based strategies to mitigate WVCs and promote safer, more sustainable transportation networks.

Figures (5)

• Figure 1: A) Schematic representation of the wildlife-vehicle interaction problem. The green shaded area represents the animal space utilization function, with darker green accounting for regions with a higher occupancy probability. A linear road of width $\Delta d$, where cars appear following a Poisson point process with rate $\nu_0$, crosses the animal home range at distance $d$ from its center. We are interested in computing the collision times between these cars and animal trajectories starting from a random initial condition, $\bm{r}_0$, sampled from the animal utilization function. (B) This encounter process can be mapped to a time process in which traffic intensity is defined by a local Poisson process with rate $\nu_0$ on the road, and the features of animal trajectories in space define the time statistics of its road crossing events. Within this framework, the animal may cross the road several times (blue lines in panels A and B) before colliding with a car (red line).
• Figure 2: Scheme of the decomposition of the interaction time. Depending on the ratio between the first-hitting time to the road, $T$ and the collision time conditioned on initial condition at the road $K$, we distinguish three regimes. For $T\ll K$ (left panel), the encounter reduces to a first-passage problem; for $T\gg K$ (right panel), the collision time is fully dominated by dynamics close to the road and thus proportional to the road occupation probability. Between these two limits (central panel),the encounter process can be studied using excursion theory
• Figure 3: (Log-log scale). Validation of theoretical results for the mean collision time, $\langle R \rangle$, as a function of the home-range area $\sigma^2$ and varying the distance between the road and the home-range center. Solid color lines show the exact result, while solid-black and dashed-black lines show, respectively, the large-$d/\sigma$ and small-$d/\sigma$ expansions. Gray dots are obtained from direct numerical simulations of the reaction-diffusion stochastic dynamics (see Appendix \ref{['sec:micro']} for details on the implementation). Parameters: $\eta = 10\,$km/day, $\tau$ varies such that $\sigma/\tau=5\,$km/day.
• Figure 4: A) Regime separation in the mean collision time $\langle R \rangle$ as a function of the effective traffic intensity $\eta$. For a parameter set $\sigma^2=100\,\mathrm{km}^2$ and $\tau=2\,\mathrm{day}$, animal-vehicle collisions can be studied using the animal range distribution when the average time between car appearances is lower than $15\,\mathrm{min}$, assuming a road width of $10\,\mathrm{m}$. B) $\langle K \rangle / \langle T \rangle$ as a function of the dimensionless parameters $\alpha\equiv d/\sigma$ and $\beta \equiv \sigma/(\eta \tau)$. When this quantity is large (red region), animal-vehicle collisions can be studied in terms of animal range distributions.
• Figure 5: A) (Log-linear scale) Probability of premature death, or equivalently, reduction in average lifespan, as a function of the intensity of the hazards that can cause animal death: effective traffic intensity $\eta$ and intrinsic death rate, $\delta$. Parameters like in Fig. \ref{['fig:lim']}A: $\sigma^2=100\,\mathrm{km}^2$, $\tau=2\,\mathrm{day}$, $\Delta d = 10\,\mathrm{m}$, $d=4\,\mathrm{km}$. B) (Log-log scale) Probability of premature death as a function of potential collision rate and home-range area. Parameters: $\delta = 0.2\, \mathrm{year}^{-1}$, $d=4\,\mathrm{km}$, $\Delta d = 10\,\mathrm{m}$, and $\tau$ varies with $\sigma$ to keep a characteristic animal velocity $\sigma/\tau= 5\,\mathrm{km/day}$ constant.