The EID-mediated hyperpredation hypothesis

In a broad sense, hyperpredation can be defined as an enhanced predation pressure on a secondary prey due to either: a) an increase in the abundance of the predator population caused by an abrupt increase in the abundance of its main prey; or b) a sudden drop in the abundance of the main prey [12], [13]. Theoretical evidence suggest that this enhanced predation, which does not necessarily mean very high, but rather ‘higher-than-normal’ levels of consumption, could be enough to result in demographic effects in prey populations [13], with the only condition of predation being additive to other mortality causes. To date the available empirical evidence on hyperpredation is restricted to cases in which the introduction of a feral prey caused overexploitation of native prey [e.g.], [ 12]–[17]. Based on the population dynamics theory of infectious diseases, and on the modelled effects of the RHD on the demography of the European rabbit [18], we can predict three dynamic phases in the interaction between the RHD and the rabbit: 1) A pre-outbreak phase, with a large population density on average; 2) A post-outbreak phase during which a large portion of the individuals become infected and die, hence causing a population drift towards a low density period. The intensity and duration of this phase depends, among other things, on the rate of spatial spread of the disease; and 3) A post-immunization phase, in which large-scale mortality rates decrease due to the immunization of rabbits and cause a return towards medium-high population densities. Due to the large numbers of predatory species shared by the European rabbit and the red-legged partridge in Spain, we can hypothesize that the RHD can potentially impact indirectly on the dynamics of the latter through hyperpredation. The RHD outbreak could thus potentially cause synchronized dynamics in the abundance of the infected host and the hyperpredated species. In this paper we test this prediction using both feeding data of shared predators and population dynamics data of the prey species.

Data set

To model the population dynamics of the European rabbit and the red-legged partridge, we gathered hunting bag data from the Spanish Ministry of Agriculture, Fisheries and Food. Available data span the 50 Spanish provinces from 1986 to 2003. However, the rabbit is native only to the 45 continental provinces, so we used data for those areas. Because hunting effort can influence the number of rabbits and partridges hunted [22], [23], we divided hunting bag data by the number of hunting licenses per year to obtain a reliable estimate of population size [7], [23].

To asses the numeric role of rabbits and partridges as prey for the Iberian predators (raptors and carnivorous mammals), we used our own field data and also conducted a bibliographic review of published data on feeding ecology of their predators (References S1). Egg predation was not considered in our review. Any predator including more than 5% of relative frequency of rabbit/partridge in its diet was considered as a rabbit/partridge consumer. We then examined whether rabbit consumers and partridge consumers were shared (Table S1). Overall, we gathered spatially segregated information on comparative diet of the main rabbit and partridge predators both before (in the “high prey density phase”) and after (in a “low prey density phase”) the RHD outbreak (Table S2). This allowed us to assess the impact of the viral outbreak in the relative diet composition of the main shared predators.

Population dynamics modeling and cross-species correlation

In order to test for cross-species correlation in the population dynamics, we fitted a log-linear (Gompertz) model of population growth to the abundance of each species [24]. Recent evidence suggests a pervasive effect of rainfall on the population dynamics of a bird species at a national scale in Spain during the same study period [25]. Thus, we gathered information on the monthly variations in precipitation throughout the study area [25] and extended the simple Gompertz model by including rainfall as an environmental covariate. We used a state-space approach to control for sampling variability during parameter estimation [24], [26]. Be X^T = [X₁, X₂, …, X_T] the vector of the time series for the hidden (unobserved or latent) log-transformed abundances, X_i. These are the states of the true population system. Be r^T = [r₁, r ₂, … , r _T] the vector of the time series for the amount of rainfall during each year (measured from December to December). The state equation for the temporal evolution of this sequence can be written as(1)were is the one-step ahead predicted population size, which is drawn from a normal distribution with the hidden population size at time t, X_t, as the mean, and process variance σ². Through a simple factorization, eqn. (1) can be simplified to(2)where β₁ = (1+b) is the coefficient for first-order density dependence [24], γ is a parameter measuring the effect of rainfall on population size, and β₀, also known as the drift parameter, is a constant measuring the density- and environmental-independent growth rate (that is, when β₁ = 1, γ = 0). Since the hidden abundances are, by definition, unobserved, we must link them to the observations through an observation model:(3)were Y_t is the log-transformed hunting bag record at time t and τ² is the sampling variance. Both eqns. (2) and (3) are called the Gompertz state-space population abundance model [24], extended through the inclusion of an environmental covariate. Since the numeric interaction between predator and prey populations can involve several time-lags [27], we tested for delayed density-dependence. Therefore, the final model becomes:(4)were β₂ is the coefficient for second-order density dependence. Solving linear state-space models through Maximum Likelihood techniques is feasible, but the estimation process can yield an extremely complex likelihood profile and the convergence of parameters and states cannot always be guaranteed [24]. Here we adopt a Bayesian approach and estimate parameters and states through Markov Chain Monte Carlo (MCMC, [28], [29]). This approach uses an iterative simulation scheme, so that the model need not be resolved analytically (see [30] for an introduction to MCMC techniques in ecology). A Bayesian model consists of a likelihood (or data) model, including the information contained in the data, and a prior (or parameter) model, containing previous information on parameter values. A posterior distribution for both parameters and states is obtained through the confrontation of both models, following the Bayes theorem [28]. We fitted the saturated model in eqn. 4 and a family of alternative models constructed by dropping some parameters from the saturated one: 1) a random walk with drift (no density-dependence, no climate effect); 2) a density-independent model with climatic effect; 3) a direct density-dependent model (AR1 model; [19]); 4) a direct density-dependence model with climatic effect; 5) a second-order density-dependence model; and 6) a second-order density-dependence model with climate effect. We compared the relative performance of each model using the Bayesian Information Criterion (BIC; [31]), with the model minimizing this quantity selected as the best descriptor of the dataset. A difference in BIC >1 was considered sufficient evidence in favour of the model with a smaller BIC. Since we had no previous information on parameter values, we constructed a prior model with vague distributions [28]. Values for the scale parameters were drawn from extremely flat (platikurtic) normal distributions: β₀, β₁, β₂, γ∼N(0, 10⁶), while the variance parameters were given inverse-gamma distributions: σ², τ²∼IG(10⁻³, 10³). Through this approach, we let the likelihood to dominate the prior in the posterior distribution. We initiated the MCMC simulation scheme for each model using three chains with different sets of initial values, and ran each chain for 300,000 iterations. We discarded the first 100,000 samples as a burn-in period, and used the remaining 200,000 to construct the posterior distributions of parameters and states. We used standard diagnostic tests for MCMC simulation [32] to check that the Markov chains were both stationary and uncorrelated, and to asses that the posterior density kernels conformed to the underlying statistical distributions.

We tested for cross-species correlation in the population dynamics of the partridge and rabbit populations using the point estimates for the hidden states in X^T. We used model averaging to obtain a final estimate of the state evolution, constructed by weighting each point estimate by the individual weight of each fitted model (see [33]); however, we emphasize that the correlation in the state estimates across models is very high (r>0.9), so the results are consistent among models. To test for time lags we performed the following analyses with the hidden time series of the red-legged partridge lagged up to two years with respect to the rabbit. Two different analyses were performed. First, we correlated directly the estimates in X^T for both species. This yielded estimates of cross-correlation while controlling for both sampling and process error, but not density-dependence and climate effects. Second, we used the residuals of the state equation in (4), . This provided estimates of cross-species correlation while controlling for sampling error, process error, density-dependence and climate. In either case, a positive cross-correlation coefficient will indicate that the population abundances of the prey species are fluctuating synchronously through time.