Classifying area-restricted search (ARS) using a partial sum approach

Abstract

Many animals perform two distinct alternating movement strategies when foraging: intensive searches with low speed and high turning to cover a small area in high detail and extensive searches with high speed and low turning to cover a large area in low detail. Observed movement paths will tend to exhibit differences in speed and correlation between these different search strategies. Identifying transitions between strategies can enable one to acquire information regarding both the distribution of resources and the underlying behavioural mechanisms performed by a foraging animal. Methods such as the moving average, first-passage time, residence time and fractal landscape methods have been used to identify behavioural states of various real and simulated foragers. We provide a review of these current methods and identify a set of common limitations associated with each procedure. We develop a new mathematical approach: the partial sum method, which is designed to avoid these limitations. A comprehensive test is undertaken to evaluate and compare the performance of the partial sum and the existing methods using a carefully constructed set of computer-generated movement paths. Each simulated track was designed to replicate the possible paths performed by an animal under different foraging conditions. Our results provide strong evidence that the partial sum method is better than existing analytical methods for identifying transitions between two different search strategies.

Appendices

Appendix 1: The partial sum method

The PS method is an analytical approach designed specifically to identify transitions between two different behavioural states within an observed movement path of an animal. A continuous animal movement path in a two-dimensional space can be represented as a discrete ordered pair (x _t ,y _t ), for 1 ≤ t ≤ T, with x _t , y _t , representing the Cartesian coordinates of the location at time step t, where T denotes the total number of time steps in the observed movement path. Variables θ _t , l _t , d _t and v _t  = (l _t /τ _t ) denote the change in direction, distance, time and speed between successive observations (x _t − 1,y _t − 1) and (x _t ,y _t ).

• 1. Cumulative sum. Determine the cumulative sum

  $$ \begin{array}{rll} C_{1}&=&\bar{S}=\displaystyle\frac{\sum^{T}_{t=2}S_{t}}{(T-1)}, \\ C_{\tau}&=&\sum^{\tau}_{t=2}(S_{t}-\bar{S}), \end{array} $$

  for τ = 2,...,T, where C _τ denotes the cumulative sum of information at time step τ and S _t denotes one of the following statistical properties at time step t:

  • a. Speed

    $$ S_{t}=v_{t}=\frac{\sqrt{(x_{t}-x_{t-1})^{2}+(y_{t}-y_{t-1})^{2}}}{d_{t}}. $$

  • b. Absolute turning angle (i.e. the sign of the turning angle is not important)

    $$ S_{t}=\vert\theta_{t}\vert=\left\vert\arctan\left(\frac{x_{t}-x_{t-1}}{y_{t}-y_{t-1}}\right)\right\vert. $$

  • c. Sum of the normalised absolute turning angle and inverse of the normalised speed (i.e. these quantities are inversely proportional and are normalised as they use different scales)

    $$ S_{t}=\left(\frac{\vert\theta_{t}\vert-\mu_{\theta}}{\sigma_{\theta}}\right)+\left(\frac{\sigma_{v}}{v_{t}-\mu_{v}}\right), $$

    where σ _θ , σ _v and μ _θ , μ _v denote the sample standard deviation and sample mean (i.e. for time steps 2,...,T) for both θ and v, respectively.

• 2. Time series. Construct the time series C _τ vs. τ.

• 3. Termination criterion. Does a turning point exist within the generated time series?

  • Yes: proceed to 4.

  • No: one cannot effectively analyse this movement path; terminate procedure.

• 4. Max–min algorithm. Determine turning points of the time series using the max–min algorithm (see “Appendix 2” for full algorithm).

• 5. Conclusion. Classify turning points as either transitions from an extensive to an intensive search strategy or an intensive to an extensive search strategy.

Appendix 2: The max–min algorithm

The max–min (MM) algorithm is designed specifically to identify turning points within a time series generated from the PS method (see “ Appendix 1” for full method). Suppose a time series consisting of T steps is increasing. One aims to find the first turning point in the time series (i.e. a local maximum). One sets the current maximum value $C_{\tau_{\rm max}}$ equal to the cumulative sum at time τ (i.e C _τ ). Then for step τ + 1 we determine whether $C_{\tau+1}>C_{\tau_{\rm max}}$. If $C_{\tau+1}>C_{\tau_{\rm max}}$ then we set $C_{\tau_{\rm max}}=C_{\tau+1}$; however, if $C_{\tau+1}>C_{\tau_{\rm max}}$, then $C_{\tau_{\rm max}}$ is unchanged. If $C_{\tau_{\rm max}}$ remains unchanged up to a period of ϵ, time then the location of $C_{\tau_{\rm max}}$ (i.e. τ _max), is classified as a turning point within the time series. A similar procedure is also used to locate local minimum values.

• 1. Determine initial parameters.

  Set τ = 1 and choose a suitable threshold ϵ.

• 2. Increasing–decreasing criterion.

  Let C _τ denote the cumulative sum at time step τ, defined by Eq. 5 in the main text;

  • If C _τ  < C _τ + ϵ : increasing at τ, proceed to 3.

  • If C _τ  > C _τ + ε : decreasing at τ , proceed to 7.

• 3. Maximisation table.

  Construct maximisation table with headings: I, τ, C _τ , τ _max, $C_{\tau_{\rm max}}$ and ϵ. Enter initial values I = 1, τ = τ, C _τ  = C _τ , τ _max = τ, $C_{\tau_{\rm max}}=C_{\tau}$ and ν = 0, where I: iteration number, $C_{\tau_{\rm max}}$: current maximum value for C _τ , τ _max: time step of current $C_{\tau_{\rm max}}$ and ν: consecutive occurrences of current $C_{\tau_{\rm max}}$.

• 4. Termination criterion.

  Set I = I + 1;

  • If τ < T: set τ = τ + 1 and proceed to 5.

  • If τ ≥ T: terminate procedure.

• 5. Determine τ _max , $C_{\tau_{max}}$ and ϵ.

  • If $C_{\tau}>C_{\tau_{\rm max}}$: set τ _max = τ, $C_{\tau_{\rm max}}=C_{\tau}$, ν = 0 and proceed to 6.

  • If $C_{\tau}<C_{\tau_{\rm max}}$: leave $C_{\tau_{\rm max}}$ and τ _max unchanged, ν = ν + 1 and proceed to 6.

• 6. Maximum criterion.

  • If ν = ϵ: define maximum at τ _max with corresponding value $C_{\tau_{\rm max}}$, set τ = τ _max and proceed to 2.

  • If ν ≠ ϵ—proceed to 4.

• 7. Minimisation table.

  Construct minimisation table with headings: I, τ, C _τ , τ _min, $C_{\tau_{\rm min}}$ and ε. Enter initial values I = 1, τ = τ, C _τ  = C _τ , τ _min = τ, $C_{\tau_{\rm min}}=C_{\tau}$ and ε = 0, where I: iteration number, $C_{\tau_{\rm min}}$: current minimum value for C _τ , τ _min: time step of current $C_{\tau_{\rm min}}$ and ν: consecutive occurrences of current $C_{\tau_{\rm min}}$.

• 8. Termination criterion.

  Set I = I + 1;

  • If τ < T: set τ = τ + 1 and proceed to 9.

  • If τ ≥ T: terminate procedure.

• 9. Determine τ _min , $C_{\tau_{\min}}$ and τ.

  • If $C_{\tau}<C_{\tau_{\rm min}}$: set τ _min = τ, $C_{\tau_{\rm min}}=C_{\tau}$, ν = 0 and proceed to 10.

  • If $C_{\tau}>C_{\tau_{\rm min}}$: leave $C_{\tau_{\rm min}}$ and τ _min unchanged, ν = ν + 1 and proceed to 10.

• 10. Minimum criterion.

  • If ν = ϵ: define minimum at τ _min with corresponding value $C_{\tau_{\rm min}}$, set τ = τ _min and proceed to 2.

  • If ν ≠ ϵ: proceed to 8.