Modeling Collective Animal Movement Through Interactions in Behavioral States

Abstract

Animal movement often exhibits changing behavior because animals often alternate between exploring, resting, feeding, or other potential states. Changes in these behavioral states are often driven by environmental conditions or the behavior of nearby individuals. We propose a model for dependence among individuals’ behavioral states. We couple this state switching with complex discrete-time animal movement models to analyze a large variety of animal movement types. To demonstrate this method of capturing dependence, we study the movements of ants in a nest. The behavioral interaction structure is combined with a spatially varying stochastic differential equation model to allow for spatially and temporally heterogeneous collective movement of all ants within the nest. Our results reveal behavioral tendencies that are related to nearby individuals, particularly the queen, and to different locations in the nest.

Appendices

Appendix A: Model Parameters

A summary of all the model parameters is presented in Table 3.

Table 3 Description of model parameters.

Prior distributions for all parameters are given in Table 4.

Table 4 Prior distributions.

Appendix B: Details on Markov Chain Monte Carlo Inference

As discussed in Sect. 4, estimating the discrete behavioral states improves the computational efficiency of Bayesian inference. If we assume that the latent states are known, inference can be separated into several separate parts (five in this particular case.) Each of the parts corresponds to modeling one particular aspect of ant movement. Details for updating all model parameters within separate Markov chains for the ant movement data are given below. Full-conditional distributions are given for parameters when Gibbs updates are used. Each separate part can be adjusted depending on the particular movement that is being modeled.

• 1. Parameters for moving ants

  • $\beta$—Gibbs update

    $$\begin{aligned} \beta&\sim \text {N}_{+}\left( \frac{a}{b} , \frac{c}{b} \right) [0, \infty ]\\ a&=10^4 \sum _{s_{i,j}=M} M^{-2}(x_{t-1,j}, y_{t-1,j}) * \\&\left( \left( x_{t-1,j}\left( 1 + \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} \right) - x_{t,j} - x_{t-2,j}\frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} \right) \right. \\&\left( x_{t-1,j} \Delta \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} - x_{t-2,j} \Delta \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} \right. \\&\quad \left. - \Delta ^2 M(x_{t-1,j},y_{t-1,j}) \frac{\mathrm{d}H(x_{t-2,j}, y_{t-2,j})}{\mathrm{d}x} \right) \\&+\left( y_{t-1,j}\left( 1 + \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} \right) - y_{t,j} - y_{t-2,j}\frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} \right) \\&\left. \left( y_{t-1,j} \Delta \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} - y_{t-2,j} \Delta \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} \right. \right. \\&\quad \left. \left. - \Delta ^2 M(x_{t-1,j},y_{t-1,j}) \frac{\mathrm{d}H(x_{t-2,j}, y_{t-2,j})}{\mathrm{d}y} \right) \right) + \sigma ^2 \Delta ^3\\&b=10^4 \sum _{s_{i,j}=M} M^{-2}(x_{t-1,j}, y_{t-1,j}) * \\&\left( \left( x_{t-1,j} \Delta \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} - x_{t-2,j} \Delta \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})}\right. \right. \\&\quad \left. \left. - \Delta ^2 M(x_{t-1,j},y_{t-1,j}) \frac{\mathrm{d}H(x_{t-2,j}, y_{t-2,j})}{\mathrm{d}x} \right) ^2 \right. \\&+\left( y_{t-1,j} \Delta \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} - y_{t-2,j} \Delta \frac{M(x_{t-1,j},y_{t-1,j})}{M(x_{t-2,j},y_{t-2,j})} \right. \\&\left. \left. - \Delta ^2 M(x_{t-1,j},y_{t-1,j}) \frac{\mathrm{d}H(x_{t-2,j}, y_{t-2,j})}{\mathrm{d}y} \right) ^2 \right) + \sigma ^2 \Delta ^3\\ c&=10^4 2\sigma ^2 \Delta ^3 \end{aligned}$$

  • $\varvec{\gamma }$—Gibbs update

    $\varvec{\gamma } \sim \text {N}(\cdot , \cdot )$ with mean $\left( \sum _{s_{t,j}=M} \varvec{a}_{i,j} + \tau _{\gamma } \left( \varvec{D} -\rho _{\gamma } \varvec{Q} \right) \right) ^{-1} \left( \sum _{s_{t,j}=M} \varvec{b}_{i,j} \right)$ and variance $\left( \sum _{s_{t,j}=M} \varvec{a}_{i,j} + \tau _{\gamma } \left( \varvec{D} -\rho _{\gamma } \varvec{Q} \right) \right) ^{-1}$

    $$\begin{aligned} \varvec{a}_{i,j}&=\frac{\beta ^2 \Delta }{\sigma ^2} \Phi '_x (x_{t-2,j}, y_{t-2,j})\Phi _x (x_{t-2,j}, y_{t-2,j}) \\&\quad + \frac{\beta ^2 \Delta }{\sigma ^2} \Phi '_y (x_{t-2,j}, y_{t-2,j})\Phi _y (x_{t-2,j}, y_{t-2,j})\\ \varvec{b}_{i,j}&=\frac{\beta }{\sigma ^2 \Delta M(x_{t-1,j}, y_{t-1, j})} * \\&\left( \Phi '_x (x_{t-2,j}, y_{t-2,j}) \left( x_{t,j} - x_{t-1,j}(2-\beta \Delta ) -x_{t-2,j} (\beta \Delta -1) \right. \right. \\&\quad \left. \left. - \beta \Delta ^2 M(x_{t-1,j}, y_{t-1, j}) \frac{d}{dx}R(x_{i,j}, y_{i,j}) \right) \right. \\&\quad \left. +\Phi '_y (x_{t-2,j}, y_{t-2,j})\left( y_{t,j} - y_{t-1,j}(2-\beta \Delta ) -y_{t-2,j} (\beta \Delta -1) \right. \right. \\&\quad \left. \left. - \beta \Delta ^2 M(x_{t-1,j}, y_{t-1, j}) \frac{d}{dy}R(x_{i,j}, y_{i,j}) \right) \right) \end{aligned}$$

    where $\Phi _x (x_{t,j}, y_{t,j})$ and $\Phi _y (x_{t,j}, y_{t,j})$ are the derivatives of the B-spline Basis functions $\Phi _ (x_{t,j}, y_{t,j})$ with respect to x and y.

  • $\tau _{\gamma }$—Gibbs update

    $\tau ^2_{\gamma } \sim \text {IG}( \frac{1}{2}N_{Basis} + 1, \frac{1}{2}\varvec{\gamma }' \left( \varvec{D} -\rho _{\gamma } \varvec{Q} \right) \varvec{\gamma } + 1 )$

    where $N_{Basis}$ is the number of basis functions used for estimating the potential surface

  • $\rho _{\gamma }$—Metropolis-Hastings Update

  • $r_1$ - Metropolis-Hastings Update

  • $\alpha _1$ - Metropolis-Hastings Update

  • $\alpha _2$ - Metropolis-Hastings Update

• 2. Parameter for non-moving ants

  • $\kappa ^2$—Gibbs update

    $\kappa ^2 \sim \text {IG}(N_{N-M}+ 10^{-6}, \sum _{s_{t,j} = N} \left( (x_{t,j} - x_{t-1,j} )^2 + (y_{t,j} - y_{t-1,j})^2 \right) + 10^{-6} )$

    where $N_{N-M}$ is the total number of observations of non-moving ants.

• 3. Parameters for ants entering the observation window.

  • $\mu _{x_O}$—Gibbs update

    $\mu _{x_O} \sim \text {N} \left( \frac{10^4\sum _{RE} x_{t,j}}{10^4 N_{RE} + 1}, \frac{10^4}{10^4 N_{RE} + 1} \right)$

    where the sum is taken over all re-entry observations and $N_{RE}$ represents the number of re-entry observations.

  • $\mu _{y_O}$—Gibbs update

    $\mu _{y_O} \sim \text {N} \left( \frac{10^4\sum _{RE} y_{t,j}}{10^4 N_{RE} + 1}, \frac{10^4}{10^4 N_{RE} + 1} \right)$

    where the sum is taken over all re-entry observations and $N_{RE}$ represents the number of re-entry observations.

  • $\kappa ^2_{x_O}$—Gibbs update

    $\kappa ^2_{x_O} \sim \text {IG}\left( \frac{1}{2} N_{RE}+ 10^{-2}, \frac{1}{2} \sum _{RE}( x_{t,j}- \mu _{x_O})^2 + 10^{-2} \right)$

    where the sum is taken over all re-entry observations and $N_{RE}$ represents the number of re-entry observations.

  • $\kappa ^2_{y_O}$—Gibbs update

    $\kappa ^2_{y_O} \sim \text {IG}\left( \frac{1}{2} N_{RE}+ 10^{-2}, \frac{1}{2} \sum _{RE}( y_{t,j}- \mu _{y_O})^2 + 10^{-2} \right)$

    where the sum is taken over all re-entry observations and $N_{RE}$ represents the number of re-entry observations.

  • $p_{OM}$—Gibbs update

    $\kappa ^2_{x_O} \sim \text {Beta}\left( N_{RE}+ 10^{-3}, N_{OO} + 10^{-3} \right)$

    where $N_{RE}$ represents the number of re-entry observations and $N_{OO}$ represents the number of individuals that stay outside for consecutive observations.

• 4. Parameters for behavioral state transitions of moving ants

  • $\varvec{\eta }_M$—Gibbs update

    See Albert and Chib (1993) for additional details on auxiliary variable method

• 5. Parameters for behavioral state transitions of non-moving ants.

  • $\varvec{\eta }_N$—Gibbs update

    See Albert and Chib (1993) for additional details on auxiliary variable method