Agent-Based Models for Collective Animal Movement: Proximity-Induced State Switching

Abstract

Animal movement is a complex phenomenon where individual movement patterns can be influenced by a variety of factors including the animal’s current activity, available terrain and habitat, and locations of other animals. Motivated by modeling grizzly bear movement in the Greater Yellowstone Ecosystem, this article presents an agent-based model represented in a state-space framework for collective animal movement. The novel contribution of this work is a collective animal movement model that captures interactions between animals that can trigger changes in movement patterns, such as when a dominant grizzly bear may cause another subordinate bear to temporarily leave an area. The modeling framework enables learning different movement patterns through a state-space representation with particle-MCMC methods for fully Bayesian model fitting and the prediction of future animal movement behaviors.Supplementary materials accompanying this paper appear online.

Appendix

1.1 P-MCMC Algorithm

Using conjugate priors, the sampling largely consists of Gibbs steps along with a particle approach for the state parameters.

• 1. Propose new set of state parameters using marginal particle proposal. Specifically, a particle filter is used to propose ${\mathcal {S}}$, ${\mathcal {X}}$ ,$\mathcal {\delta }$, and state conditional on the remaining parameters in the model. The script notation refers to all of the variables of that type, across agents, years, and time points. Proposals are accepted with the typical Metropolis–Hastings ratio as detailed in Andrieu et al. (2010). This procedure updates one agent at a time, within a given year, but the procedure can be parallelized across years.

• 2. The probability parameters associated with the state transitions are fit using conjugate priors from a beta distribution.

• 3. The step size ($\mu _{u,j}$) and variance parameters ($\sigma ^2_{u,j}$) in the lognormal distribution for each state can be sampled using a Gibbs sampler.

• 4. The variance associated with the measurement error ($\sigma _\varepsilon$) can be sampled from an inverse gamma distribution.

• 5. The final piece is taking samples from the projected normal, and Nuñez-Antonio et al. (2015) outline a Griddy–Gibbs approach (Ritter and Tanner 1992) for this procedure. With this approach, the angular data ($\theta$) are converted to Cartesian coordinates, where $x = r \cos (\theta )$ and $y = r \sin (\theta )$. Integrating out r and using the x and y data enable Gibbs samples for the mean of the projected normal. Wang and Gelfand (2013) present a more general approach for sampling from a projected normal distribution.

1.2 6.1 Model Specification and Priors for Data Analysis

1.2.1 Observation Equation

$$\begin{aligned} \mathbf {{\underline{z}}_t} \sim N \left( \mathbf {H_t} \mathbf {{\underline{s}}_t} + \mathbf {{\underline{\epsilon }}_t}, \sigma ^2_\epsilon I \right) \end{aligned}$$

1.2.2 Evolution Equation

$$\begin{aligned} \mathbf {{\underline{s}}_{i,t}}&= \mathbf {{\underline{s}}_{i-1,t}} + u_{i,j,t}\mathbf {{\underline{\delta }}_{i,j,t}}\\ \mathbf {{\underline{\delta }}_{i,j,t}}&= (\cos (\theta _{i,j,t} + \nu _{i,j,t}), \sin (\theta _{i,j,t} + \nu _{i,j,t})) \\ \theta _{i,j,t}&\sim PN(\underline{\mu _{\theta ,j}}, I)\\ u_{i,j,t}&\sim LN(\mu _{u,j}, \sigma ^2_{u,j})\\ p_{12[c]}&= Pr[j_t =1 \rightarrow j_{t+1} = 2 | d_t< thr]\\ p_{12[f]}&= Pr[j_t =1 \rightarrow j_{t+1} = 2 | d_t> thr]\\ p_{22[c]}&= Pr[j_t =2 \rightarrow j_{t+1} = 2 | d_t < thr]\\ p_{22[f]}&= Pr[j_t =2 \rightarrow j_{t+1} = 2 | d_t > thr], \end{aligned}$$

where $j = 1$ is the model state with shorter steps that tend to stay in the same area, $j=2$ is the model state with larger steps that tend to maintain the existing heading, and $\theta _{i,j,t}$ is the stochastic component of the heading, where $\nu _{i,j,t}$ aligns the heading to a collection of previous locations for state i or the most recent heading for state j. The switching component of the model is determined by whether the distance to the nearest animal, $d_t$, is less than a threshold, thr.

1.2.3 Prior Distributions

In general, the prior distributions are weakly informative and aligned with biological understanding.

$$\begin{aligned} \sigma ^2_{\epsilon }&\sim IG(5, 50000) \end{aligned}$$

The prior for $\sigma ^2_{\epsilon }$ has a mean that roughly corresponds to a standard deviation of 100 meters.

$$\begin{aligned} \mu _\theta&\sim N(\underline{(1,0)}^T, .1 I) \end{aligned}$$

The prior for $\mu _\theta$ which is centered at $(\underline{(1,0)}^T)$, with relatively weak precision, has little impact on the posterior.

$$\begin{aligned} \mu _{u1}&\sim N(\log (500),.5 ) \\ \mu _{u2}&\sim N(\log (5000), .5) \\ \sigma ^2_{u1}&\sim IG(5, 1.8) \\ \sigma ^2_{u2}&\sim IG(5, 0.6) \end{aligned}$$

The mean step size parameters are centered at 500 meters and 5000 meters (after accounting for lognormal parameterization), but have small enough precision that they can be largely informed by the data. Similarly, the standard deviation values are not particularly influential.

$$\begin{aligned} p_{12[c]}&\sim Beta(1.9, .1) \\ p_{12[f]}&\sim Beta(.1 , 1.9) \\ p_{22[c]}&\sim Beta(1.9, .1) \\ p_{22[f]}&\sim Beta(1.9, .1) \\ \end{aligned}$$

The priors for the transition probabilities suggest inertia (95 percent probability of staying in same state); except for when another bear is in close proximity and a bear is in state 1, then the prior probability of switching is 95 percent. However, the prior probabilities correspond to 2 data points, so there is little impact on the posterior probability.