Modelling road mortality risks to persistence to a Western Toad ({\it Anaxyrus boreas}) population in British Columbia

TL;DR

The paper tackles the threat of road mortality to amphibian persistence by integrating a six-year field study of western toads with a tractable impulsive, stage-structured ODE model. The authors derive explicit periodic steady-state solutions and an extirpation threshold $m_z^c$ for highway-crossing mortality, showing that gravid female mortality markedly amplifies endangerment risk. Field data reveal a strong link between traffic and mortality (approximately $3.1\%$ more deaths per additional vehicle) and a high share of deaths among gravid females, underscoring vulnerability during peak crossing times. The work demonstrates a rapid, non-linear transition from healthy to endangered populations with increasing traffic and advocates targeted mitigation (e.g., toad tunnels and fencing) to maintain persistence, with broader applicability to similarly structured amphibian populations.

Abstract

Road mortality may be a significant factor in the global decline of amphibian populations, yet rigorous assessments of its effect on long-term population persistence are lacking. Here, we investigate population persistence through a field study and mathematical model of a western toad ({\textit{Anaxyrus Boreas}} {\RR(Baird and Girard, 1852)}) population within a highway corridor in the Selkirk Mountains of British Columbia. The analysis shows traffic levels strongly correlate with toad mortality, with each additional vehicle causing a 3.1\% $\pm$ 1.3\% ($p=0.020$) increase in toad deaths. Although the current risk of the population becoming threatened or endangered is low, it rises to 50\% if baseline road mortality increases from 10\% to 30\%. Gravid female mortality is higher than the baseline mortality and can increase the probability of endangerment by nearly two-fold at higher baseline mortality levels. We make the case that a small increase in vehicle traffic resulting from future development and recreational pressures could destabilize this apparently healthy toad population. The high sensitivity to traffic levels and rapid transition from healthy to endangered raises concerns for similar populations worldwide. Compensatory structures such as amphibian underpasses (toad tunnels) should be given high priority.

Figures (6)

• Figure 1: Fish and Bear Lakes study area is located in a mountain pass within the BC Highway 31A corridor of the Central Selkirk Mountains. Fish Lake provides the primary breeding and rearing habitat and consequently the highest toad mortality occurs on the highway segment directly adjacent to this lake. Adult toads migrating down from the mountains on the north side of the highway are most at risk. Very few toads approach the lake from the south. The area of interest (inset map: red circle) is approximately 7 km in diameter, and illustrates the potential maximum distance toads could travel between terrestrial foraging and hibernation areas and aquatic breeding sites cosewic:2012.
• Figure 2: Scatter plot of adult toad mortality rate per minute (all toads, i.e., male and female (gravid and non-gravid)) as a function of the number of vehicles on the road per minute. The fitted line shows a linear regression between these two variables, as well as the Pearson correlation ($R$) and the significance level ($p<0.05$). This plot shows the simplest model possible illustrating the relationship between the number of vehicles per minute and the number of dead toads per minute. The full analysis (Supplement \ref{['sec:regression']}) gives estimates for all of the model coefficients.
• Figure 3: Plot of $z_{min}$ as a function of highway-crossing mortality (per crossing, ranging from 0 to 0.8 (80%)) for breeding adults, and for different values of the carrying capacity $K$ and increased mortality factor for gravid females $\zeta_g$ (gravid vulnerability). The values of $K$ are indicated by line type (dashed for $K=3,000,000$, solid for $K=2,000,000$, and dotted for $K=1,000,000$), and the values of $\zeta_g$ are indicated by colour (red for $\zeta_g=1$, blue for $\zeta_g=2$, colour online). The upper and lower dotted black horizontal lines indicate, respectively, the "threatened" and "endangered" thresholds (see Section \ref{['sec:endangerment']}). The extirpation threshold $m_z^c\approx0.7$ for $\zeta_g=1$, and $\approx0.5$ for $\zeta_g=2$. Parameter values are at the default values listed in Table \ref{['tab:SymbolsTable']}.
• Figure 4: Relative risk of the toad population reaching a status of endangered (red), threatened (yellow), or healthy (purple) (colour online) as a function of baseline adult highway-crossing mortality ($m_z$), ranging from 0 to 1 (100%). The height of the bar at each level of mortality corresponds to the proportion of parameter sets, at that level of mortality, resulting in a minimum steady state population falling within the appropriate interval ($z_{min}\leq125$ (red), $125<z_{min}\leq500$ (yellow), $z_{min}>500$ (purple), colour online). Results are shown for (left) $K=1,000,000$ and (right) $K=3,000,000$. For the remaining parameter values, the parameter space was sampled 100,000 times, using a Monte Carlo sampling of parameter values (Section \ref{['sec:sampling']}) distributed uniformly over the ranges defined in Table \ref{['tab:SymbolsTable']}.
• Figure 5: Plots showing the effect of the gravid female mortality factor $\zeta_g$ on population persistence. (\ref{['fig:sustSn-zeta-zmin']}) Plot of $z_{min}$ versus $\zeta_g$, for different values of the carrying capacity $K$ (dot: $K=4,000,000$; solid: $K=2,000,000$; dashed: $K=1,000,000$) and non-gravid adult highway-crossing mortality $m_z$ (red: $m_z=10$%; blue: $m_z=30$%, colour online). The dotted black horizontal lines indicate the "threatened" and "endangered" thresholds (see Section \ref{['sec:endangerment']}). (\ref{['fig:sustSn-zeta-threat1']})--(\ref{['fig:sustSn-zeta-threat2']}) Risk of the toad population reaching a status of endangered (red), threatened (yellow), or neither (purple) (colour online) as a function of $\zeta_g$. The height of each coloured bar gives the proportion of parameter sets, at that value of $\zeta_g$, for which the population has the corresponding endangerment status. Results are shown for $K=1,000,000$, and adult highway-crossing mortality $m_z$ of 10%-30% (left) and 10%-50% (right). For the remaining parameter values, the parameter space was sampled $100,000$ times, using a Monte Carlo sampling of parameter values (Section \ref{['sec:endangerment']}) distributed uniformly over the ranges defined in Table 1.