Modeling Warfare in Social Animals: A "Chemical" Approach

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 (Lasius neglectus and Lasius paralienus), 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 interaction and mortality. The interaction experiments are designed to obtain information on the combat dynamics and lasted one hour. The mortality 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 (e.g., 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 (e.g., spatial ones), with the goal of distinguishing collective effects from the strategic ones.


Integration of differential equations
In order to integrate this system of ordinary differential equations we use the numerical method of Cash and Karp [1] checking that the fourth and fifth order solutions provide the same results. This method is a member of the Runge-Kutta family of ordinary differential equation solvers.
This method explicitly simulates each reaction, giving a stochastic formulation of chemical kinetics based on the theory of collisions. We have to compute, for each reaction, the number of ways in which it can be realized (multiplicity) and the probability density function of its occurrence during time (an exponential function), normalized over the sum of all reactions.
In our case, for each reaction i (out of the 15 of Eqs. 11) with reaction constant k i , the propensity function is defined as F i = k i g i , where g i (the multiplicity factor) is the number of distinct ways in which the individual reactants of each species can be sampled in the population. The multiplicity factors g i are reported in Table 1.
Considering a time interval ∆t, the reaction probability density function of reaction i is P (∆t, i) = P 1 (∆t)P 2 (j) (due to time homogeneity), with The two terms forming the probability density function are an exponential distribution of time reactions P 1 and the normalized propensity function P 2 . The Gillespie algorithm can be implemented by choosing the next event time interval ∆t as and the reaction i as the integer for which where r 1 and r 2 are uniform random numbers between 0 and 1.

Nonexistence of attractors
As expected, this system does not exhibit fixed points, i.e., the stationary solution (ẋ i = 0) always depends on the initial conditions as can be seen in Fig. 1. The stationary condition is either x = 0 and y = z = u = v = 0 or y = 0 and x = z = u = v = 0. We can denote these two conditions as absorbing states for the dynamics. This is consistent with the biological point of view: the winning species and the the number of survivors depend on the initial number of individuals involved in the battle. Another view to illustrate the absence of the fixed point is to calculate directly the variations of the total number of A and B, denoted a and b. From Eqs. (11), we have After deriving and substituting Eqs. (11), we obtaiṅ that simply show, as expected, the dependence of the total number of A and B from the mortality terms that depends only from the groups AB, ABB and ABBB. Since z, u and v are either positive or zero, the only stationary statesȧ =ḃ = 0 is given by the absence of any cluster (z = u = v = 0) and since with this conditionż = k 1 xy, u = k 10 xy 2 , v = 0, we have either x = 0 or y = 0 for the stationary state.