Simulations

This protocol is extracted from research article:

Adaptive switch to sexually dimorphic movements by partner-seeking termites

**
Sci Adv**,
Jun 19, 2019;
DOI:
10.1126/sciadv.aau6108

Adaptive switch to sexually dimorphic movements by partner-seeking termites

Procedure

We developed an individual-based model to examine the efficiency of movement patterns observed in termites. We prepared two searching situations, a periodic boundary condition of size = *L* × *L* and a borderless continuous space where a female and a male initially separated by the distance *d*. The former was to simulate uninformed search before pair formation, and the latter was to simulate reunion search after separation. In each condition, we considered a female and a male walking until encountering another individual of the other sex. When the distance between the centers of the two individuals became smaller than φ, they were regarded to encounter. This φ value is based on the definition of tandem running, with 7 mm for *R. speratus* and 10 mm for *C. formosanus* (see Supplementary Text).

Individuals perform CRW with the parameters of speed and sinuosity, denoted by *v* and ρ, respectively. The speed parameter *v* was obtained as the mean value of the observation data for each sex and search scheme (Table 1), while the sinuosity parameter ρ was obtained as the estimated scale parameter from the data of turning angles during moving (table S1). On the basis of our behavioral analysis, each time step was adjusted to 0.2 s. Thus, each individual moved 0.2*v* mm in each time step. Turning angles followed wrapped Cauchy distribution with scale parameter ρ. After generating a uniform random number *u* (0 < *u* ≤ 1), the turning angles θ were derived from the following equation by applying the inversion method (*10*)$$\mathrm{\theta}=2\text{arctan}\left[\right(\frac{1-{\mathrm{\rho}}^{*}}{1+{\mathrm{\rho}}^{*}})\text{tan}[\mathrm{\pi}(u-0.5)]]$$

We initiated the simulation with a random bearing angle that fluctuated according to θ. At each step, the bearing angle was equal to the previous bearing angle plus the deviation θ such that the moving object always kept the previous direction, forming a CRW.

We added move-pause intermittency to the above CRWs to account for pausing behaviors. Observations showed that their distributions of the duration of moves and pauses followed either the truncated power-law or stretched exponential distributions. Thus, in the simulation, the duration of moves *t*_{m} and pauses *t*_{p} can be derived from the following equations (*47*). For truncated power-law distributions$${t}_{\mathrm{m}}\text{or}{t}_{\mathrm{p}}={({x}_{\text{max}}^{1-\mathrm{\mu}}-u({x}_{\text{max}}^{1-\mathrm{\mu}}-{x}_{\text{min}}^{1-\mathrm{\mu}}\left)\right)}^{\frac{1}{1-\mathrm{\mu}}}$$and for stretched exponential distributions$${t}_{\mathrm{m}}\text{or}{t}_{\mathrm{p}}={({x}_{\text{min}}^{\mathrm{\beta}}-(1/\mathrm{\lambda})\text{log}(u))}^{\frac{1}{\mathrm{\beta}}}$$where *x*_{min} is the minimum value of the data (0.2) and *x*_{max} is the maximum value of the data (table S2). Values of parameters are shown in table S2. In addition, we also performed simulations by randomly resampling the duration of moves and pauses from observed datasets directly to confirm that our model fitting worked well to describe the stochastic search rules in termites. These results were qualitatively the same with those using the fitted models (comparing Fig. 3 and fig. S8), indicating that our results are reliable even though goodness-of-fits of some datasets were low (table S2). For the initial condition, individuals were assumed to perform moving or pausing, depending on the proportion of pausing times during observations (fig. S4). When individuals finished pausing, they got a new angle from wrapped Cauchy distribution with a scale parameter for reorientation behavior (table S1), like the above method.

We compared the searching efficiency among four possible combinations of movement patterns—the observed sexually dimorphic movement after separation, the observed sexually monomorphic movement before pair formation, and two virtual monomorphic movements with both a female and a male moving like females or males after separation—for each searching condition in each species. Simulations were performed for 180 s (= 900 time steps). We ran 1,000,000 simulations and measured the efficiency as the probability to encounter a mating partner. The simulation program was implemented in Microsoft Visual Studio C++ 2017. The source codes for all simulations are available in the Supplementary Materials.

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