Modeling Warfare in Social Animals: A "Chemical" Approach

<div><p>We present here a general method for modelling the dynamics of battles among social animals. The proposed method exploits the procedures widely used to model chemical reactions, but still uncommon in behavioural studies. We applied this methodology to the interpretation of experimental observations of battles between two species of ants (<i>Lasius neglectus</i> and <i>Lasius paralienus</i>), but this scheme may have a wider applicability and can be extended to other species as well. We performed two types of experiment labelled as <i>interaction</i> and <i>mortality</i>. The <i>interaction</i> experiments are designed to obtain information on the combat dynamics and lasted one hour. The <i>mortality</i> ones provide information on the casualty rates of the two species and lasted five hours. We modelled the interactions among ants using a chemical model which considers the single ant individuals and fighting groups analogously to atoms and molecules. The mean-field behaviour of the model is described by a set of non-linear differential equations. We also performed stochastic simulations of the corresponding agent-based model by means of the Gillespie event-driven integration scheme. By fitting the stochastic trajectories with the deterministic model, we obtained the probability distribution of the reaction parameters. The main result that we obtained is a dominance phase diagram, that gives the average trajectory of a generic battle, for an arbitrary number of opponents. This phase diagram was validated with some extra experiments. With respect to other war models (<i>e.g.</i>, Lanchester's ones), our chemical model considers all phases of the battle and not only casualties. This allows a more detailed description of the battle (with a larger number of parameters), allowing the development of more sophisticated models (<i>e.g.</i>, spatial ones), with the goal of distinguishing collective effects from the strategic ones.</p></div

Fitting abundance of species B.

<p>Blue curve: average of 100 simulations of the stochastic model with Gillespie algorithm. Green curve: the average of the experimental data. Red dotted and dash-dotted curves indicated the variance. The black line is the solution of the deterministic chemical model for the species .</p

Distribution of the reaction coefficient.

<p>Coefficients (A), (B), (C), (D), (E) and (F), obtained with our stochastic procedure; the best fit is achieved by means of Log-normal distribution comparing the likelihoods.</p

Supremacy phase diagram.

<p>Evolution of the trajectories of the deterministic model starting from different initial conditions, projected on the plane determined by the total number of A individuals () and the total number of B individuals (). Initial conditions indicated by diamonds lead to the supremacy of 's (), while the x-marks lead to the supremacy of the species (). The red dashed line is the separatix between the two phases, where both species die; the green square marks the region where fitting has been performed and where non-linear effects are most effective. The magenta and the blue lines indicate experimental trajectories, each one given by an average over 5 experiments.</p

Means of 5 experiments for 7 different initial conditions for the duration of 5 hour (Set 1, Set 2, etc.); <i>L. paralienus</i> (), <i>L. neglectus</i> ().

<p>Means of 5 experiments for 7 different initial conditions for the duration of 5 hour (Set 1, Set 2, etc.); <i>L. paralienus</i> (), <i>L. neglectus</i> ().</p

Stochastic time series.

<p>Temporal variation of the (A) and (B) chemical species, obtained from three simulations of the stochastic model. Predictions of the deterministic model are also shown (black line).</p

An example of experimental data.

<p>The first column shows the occurrence time (in seconds) of a given reaction (event). The other columns (2–6) report the total number of each chemical species (, , , and .</p><p>An example of experimental data.</p

Comparison among estimated reaction constants (), from all sets of data, averages from Gillespie simulations and experimental averages over single experiment.

<p>In column <b><i>ExpT</i></b> the reaction constants extracted by overlapping all datasets (20 observations). In column <b><i>Gill</i></b> we report the estimated reaction constants from Gillespie simulations, fitting each single experiment and obtaining , and then computing , according to the lognormal distribution. In column <b><i>Exp</i></b> we report the same calculation for the experiments.</p><p>Comparison among estimated reaction constants (), from all sets of data, averages from Gillespie simulations and experimental averages over single experiment.</p

The likelihood obtained from the first set of 50 stochastic simulation; the likelihood of the second set of 50 stochastic simulation evaluated using the distribution fit parameters achieved with the first set ; the likelihood from the first set of 10 experiments; the likelihood of the second set of 10 experiments evaluated using the distribution fit parameters achieved with the first set ; the Wilcoxon test respectively, between the first stochastic set and the first experimental one, the second stochastic and the second experimental, the first stochastic and the second experimental, and finally the vice versa of the latter (0 accepted the zero hypothesis of the same distribution in which something belongs with significance level of 0.05, while if 1 the zero hypothesis is refused).

<p>The likelihood obtained from the first set of 50 stochastic simulation; the likelihood of the second set of 50 stochastic simulation evaluated using the distribution fit parameters achieved with the first set ; the likelihood from the first set of 10 experiments; the likelihood of the second set of 10 experiments evaluated using the distribution fit parameters achieved with the first set ; the Wilcoxon test respectively, between the first stochastic set and the first experimental one, the second stochastic and the second experimental, the first stochastic and the second experimental, and finally the vice versa of the latter (0 accepted the zero hypothesis of the same distribution in which something belongs with significance level of 0.05, while if 1 the zero hypothesis is refused).</p