Model framework
The capture-recapture model applied here is the hierarchical model for stratified populations proposed by Royle et al.48. The model aims to estimate local population size or community structure49 using capture-recapture data from multiple independent locations. In the following, we briefly describe the model in our context, including addressing heterogeneity in detection probability.
Let us consider that we establish S independent camera stations in a survey area. Then, we install K camera traps at each station to monitor exactly the same focal area (totally S × K camera traps will be used). We assume that these camera traps detect animals within the focal areas NT times in total. For animal pass i (i = 1, 2, 3, …, NT), we will obtain (1) at the station where the animal is detected (hereafter station identity; gi), and (2) how many of the K cameras at the station were successful in detecting the animal pass (hereafter detection history; yi). The hierarchal capture-recapture model uses these two data, gi and yi.
Let the number of the animal passes at station s be Ns (s = 1, 2, 3, …, S). Then, we assume that Ns follows a Poisson distribution with a parameter λ. In this case, the probability of passage i occurring at station s is expected to be (frac{lambda }{lambda times S}). Thus, station identity, gi, can be modelled as follows:
$$g_{i} sim {text{ Categorical}}; left(frac{lambda }{lambda times S}right)$$
When the number of the animal passes at station s, Ns, may have larger variation than expected from the Poisson case, we may assume a negative binomial distribution model or may give a random effect to the parameter of the Poisson distribution at the camera station level.
The detection history Y with elements yi can be modelled using a data augmentation procedure47. Specifically, the original detection Y is artificially augmented by many M – n passes with all-zero histories (i.e. not detected by any camera). The augmented data W with elements wi (y1, y2…yNT, 0, 0, … 0) will consist of the passage that occurred but was not detected by any camera (false zero), which occurs with probability ψ, and the passage that did not occur (structural zeros) with the probability 1 − ψ. A set of latent augmentation binary variables, z1, z2, … zM, is introduced, which denotes the false zero (z = 1) and the structural zero (z = 0). That is
$$z_{i} sim {text{ Bernoulli }}left( psi right).$$
The elements of the augmented data, wi, can be modelled conditional on the latent variables zi. There would be two alternative approaches to modelling the wi.
The simplest one may regard wi as random binomial variables. That is
$$w_{i} |z_{i} = , 1sim {text{ Binomial }}left( {K,p} right)$$
When accounting for the heterogeneity of detection among animal passes, it can be accommodated using a beta distribution as follows;
$$w_{i} |z_{i} = , 1sim {text{ Binomial }}left( {K,p_{i} } right)$$
$$p_{i} sim {text{ Beta}}left( {alpha ,beta } right)$$
The expected detection probability can be derived from (widehat{alpha }/(widehat{alpha }+widehat{beta })) and the correlation coefficients can be calculated by (1/(widehat{alpha }+widehat{beta }+1)).
Alternatively, we can regard wi as a categorical variable that takes values from zero to K.
$$w_{i} sim {text{ Categorical }}left( pi right)$$
where π is a probability vector of length K + 1. For simplicity, let us consider two camera traps installed at each station, and those cameras have equal detection probability. Then, wi can take either 0 (i.e. zi = 0 or both camera traps missed animals with conditional on zi = 1), 1 (i.e. only one camera trap detected animals with conditional on zi = 1), or 2 (i.e. both camera traps detected animals with conditional on zi = 1). Thus, when we define the probability that wi takes 0, 1, 2 with conditional on zi = 1, as φm (m = 1, 2, 3), the elements of π is equal to {zi × φ0 + (1 − zi)}, {zi × φ1}, {zi × φ2}, respectively.
We then take different modelling approaches depending on whether detection probability among animal passes is heterogeneous or not. When two camera traps at a station detect animals independently with the same probability ρ, φ0, φ1, and φ2 can be expressed as a function of ρ, i.e. (1 − ρ)2, 2 × ρ × (1 − ρ)2, ρ2, respectively (Clare et al.47). On the other hand, when detections by the two camera traps are correlated, we need to estimate three real parameters φm that designate the probabilities of all outcomes wi|zi = 1. We assume that ρm follows the Dirichlet distribution with the parameter γm (m = 1, 2, 3). That is
$$varphi_{m} sim {text{ Dirichlet}}left( {gamma_{1} ,gamma_{2} , , gamma_{3} } right)$$
In this approach, the expected detection probability can be derived from ({widehat{varphi }}_{1}/2+{widehat{varphi }}_{2}) and the correlation coefficients can be calculated by ({widehat{varphi }}_{2}-{({widehat{varphi }}_{1}/2+{widehat{varphi }}_{2})}^{2}).
Compared to the beta-binomial distribution approach, the approach using categorical-Dirichlet distribution might be more flexible in accommodating detection heterogeneity while it might be more challenging to estimate the model parameters. In either approach, the expected total number of animal passes can be expressed as (lambda times S). Thus, ψ can be fixed as follows:
$$psi = frac{lambda times S}{M}$$
For more details of the models, see Royle et al.48 and Clare et al.44.
Testing the effectiveness of the hierarchical capture-recapture model
We performed Monte Carlo simulations to evaluate the effectiveness of the hierarchical capture-recapture model. Because the model reliability has been confirmed well48, we here focused on the effects of heterogeneity in detection probability on the accuracy and precision of the estimates.
We assumed that the number of detections by camera traps followed a negative binomial distribution with a mean of 5.0 and dispersion parameter 1.27, which derived the actual data on an ungulate in African rainforests34. We also assumed two camera traps each at 30 stations (i.e. 60 camera traps in total). We generated detection histories (i.e. the number of camera traps successfully detecting animals in each animal passage) using a beta-binomial distribution with the expected detection probability at 0.8 or 0.4. We varied the correlation coefficients (= 1/(α + β + 1)), from 0.1 to 0.5 in 0.1 increments. The scale parameters of the beta distributions for each scenario are shown in Table 1. Additionally, to determine the effects of sample sizes on the accuracy and precision of estimates, we increased the number of camera stations at 100. Since this setting requires much computation time, we only assumed a detection probability of 0.4 and a correlation coefficient of 0.3.
We estimated the parameters of the hierarchical capture-recapture models assuming a beta-binomial distribution and a categorical-Dirichlet distribution using the Markov chain Monte Carlo (MCMC) implemented in JAGS (version 3.4.0) in all the simulations. We assumed that the number of animal passes followed a negative binomial distribution. For the model assuming a beta-binomial distribution, we transformed the scale parameters, α and β as p*phi and p*(1 − phi), respectively (p is an expected detection probability). Then we used a weakly informative prior (gamma distribution with shape = 10 and rate = 2) for phi and a non-informative uniform distribution from 0 to 1 for the detection probability49. For the model assuming a categorical-Dirichlet distribution, the Dirichlet prior distribution was induced by treating each γm ~ Gamma(1, 1) and calculating each probability by ({varphi }_{m}={{gamma }_{m}}/{sum }_{m=1}^{M}{gamma }_{m}) followingv and Clare et al.44. We generated three chains of 3000 iterations after a burn-in of 1000 and thinned by 5. The convergence of models was determined using the Gelman–Rubin statistic, where values < 1.1 indicated convergence. These procedures were repeated 300 times. We report the mean of estimated median detection probability and the expected number of animal passes, and their 95% credible interval (CI) coverage of the densities. The R code to implement this simulation is available as supplementary material (Supplementary R1).
Determining a suitable camera installation
The above simulations suggested that a key to safely applying the hierarchical capture-recapture model may avoid correlated detections. To determine a preferred survey design to secure independent detections, we performed additional simulations. Specifically, we tested how the position and the trigger speed of camera traps may affect the independence of detections by considering the process by which cameras detect moving animals.
The simulation was performed using similar procedures taken by Rowcliffe et al.17. We assumed that the sensor of camera traps has a two-dimensional detection surface defining the instantaneous ‘risk’ (as analogous to the risk of mortality in survivorship analysis) of an animal being detected at any given location within the camera’s field of view (FOV). The instantaneous risk landscape was defined with respect to distance r and angle θ relative to the camera as follows:
$$h = aleft( {1 – exp left( { – left( {{r / s}} right)^{ – b} } right)} right)exp left( { – {theta / c}} right)$$
where a defines the maximum risk close to the sensor, s and b define the position and shape of decline in risk with distance, respectively, and c defines the rate of decline in risk with angle. We set a, s, b, and c at 0.2, 3.0, 5.0, and 0.5, respectively. Although there is no empirical evidence for this particular function form, the simulated results were comparable to those observed. Note that our interest is not in estimating the actual values of the correlation but instead in obtaining information that will help us decide what cameras to install and how to install them.
We made an animal pass through the risk landscape in a straight line in a random direction. The animal movement speed (ms-1) followed a log-normal distribution with a mean (± SD) of 0.18 ± 0.15 m, which roughly accord with the speed of red duikers in our study sites (Y Nakashima, unpublished data). We then generated a random animal’s position to be detected, considering the cumulative risk of detection and the camera trap’s trigger speed (for the details, see Rowcliffe et al.19).
We considered three designs of camera placements (Fig. 1). The first one is to install two camera traps at the same position (i.e. mounted on the same tree) and in the same direction to monitor the entire field of view within 10 m from the cameras (ins. 1). The second and third one is assumed to monitor a specific small area within the FOV. The focal area was a small equilateral triangle with a side length of 1.9 m and was centred within the FOV. The nearest vertex was set to be 1.9 m away from the cameras. This area corresponds to the highest detection probability in the camera model used in our field study. This focal area is monitored from the same direction (ins. 2) or different angles of 60 degrees (ins. 3). We then considered using camera models with a fast trigger speed (0.1 s) and a slow (1.5 s) for each installation. Finally, we generated the detection history of 500 times animal passes and calculated the correlation coefficients between the two camera traps for each scenario.
Field surveys
We conducted a field survey in and around the Boumba-Bek and Nki National Parks in southeast Cameroon (October 2018 to January 2019) and the Boso Peninsula in Japan (August-November 2018). The study area of Cameroon consists primarily of evergreen and semi-deciduous forests. The annual rainfall is approximately 1,500 mm, and the mean annual temperature is ca. 24 °C. Typically, the dry season occurs from December to February and the rainy season is from March to November. The Boso Peninsula was in the southern part of Chiba Prefecture in central Japan (35 N °N, 140E). The vegetation consists of either broad-leaved evergreen forests (Castanopsis sieboldii and Quercus spp.) or coniferous plantations (Cryptomeria japonica and Chamaecyparis obtusa). The monthly mean temperature was 20.6 ± 5.9 °C during the study period, with the highest in August (27.4 °C) and the lowest (13.6 °C) in November.
The simulations suggest that camera traps monitor the predefined focal area from different positions (see below). According to the results, we used the camera traps with a high trigger speed (0.15 s) (Browning Strike Force Pro, BTC-5HDP, Browning, Missouri, US) at both study sites. We regarded two camera traps to monitor the same equilateral triangle area as in the simulation from different directions by 60 degrees (the right panel of Fig. 1). We surrounded the focal area with a white rope and manually filmed it with camera traps as a reference. The rope was removed after filming to avoid disturbing the animal behaviour. We set camera traps at approximately 0.7 m above ground without baits or lures. We used the ‘video mode’ and designated the video length as 20 s and the delay period between videos at 1 s (minimum delay period in this product). In Japan, we established seven camera stations at least 2 km apart and installed two camera traps at each station. In Cameroon, we set 26 camera stations at least 2-km apart from each other. Since the number of camera stations is not enough to estimate the expected number of animal passes in Japan, we focused on the detection probability.
We determined whether animals passed within or outside the focal area by superimposing videos and the reference image. We used only images of animals crossing the focal area for subsequent analyses. We then matched each detection from the two cameras to determine whether two camera traps successfully recorded an animal pass. We applied the model to the detection probability of the species detected more than 10 times in both Japan and Cameroon. The analysis was limited to images in which the animal species were reliably identified. We plotted the estimated detection probability against the median value of body mass (kg), drawn from Ohdachi et al.50 for animals in Japan and Kingdon51 for those in Cameroon.
The survey in Cameroon was conducted with approval from the Ministry of Scientific Research and Innovation (MINRESI, N°0190/ MINRESI/Projet COMECA/PM/07/2018) and the Ministry of Forestry and Wildlife (MINFOF, N°1527/L/MINFOF/SETAT/SG/DFAP/SDCF/SEP/EP). The installation of the camera in the Boso Peninsula was done with the permission of the landowner.
Source: Ecology - nature.com
