Are there mathematical models that describe warfare tactics between competing clans of monkeys, perhaps similar to the predator-prey dynamics modeled by the Lotka–Volterra equations?
I am particularly interested in the formation of opposing lines and the subsequent dynamics observed in this video clip.
Has this type of behavior been studied mathematically before?
Yes, mathematical models have been developed to study animal conflict and territorial behaviors.
Smith and Price did an interesting work in 1973 where they applied game theory to study animal conflicts, particularly through the development of the Hawk-Dove Game. This model explains the evolution of aggressive and non-aggressive behaviors among animals of the same species competing for resources. This framework considers two strategies: the "hawk" strategy, where an individual always fights for a resource, and the "dove" strategy, where an individual avoids fighting and retreats if confronted.
Let the value of the resource be denoted by ( V > 0 ), and the cost of injury in a fight by ( C > 0 ). When two hawks meet, they fight, and each has a 50% probability of winning, resulting in an expected payoff of $ \frac{1}{2}(V - C) $. If a hawk meets a dove, the hawk wins the resource outright, gaining a payoff of ( V ), while the dove retreats without incurring any cost, receiving a payoff of ( 0 ). Finally, when two doves meet, they share the resource without fighting, resulting in an expected payoff of $\frac{V}{2}$ for each.
\begin{array}{c|c|c} & \text{Hawk} & \text{Dove} \\ \hline \text{Hawk} & \frac{1}{2}(V - C) & V \\ \text{Dove} & 0 & \frac{V}{2} \end{array}
The dynamics of the population can be modeled by considering the proportion of hawks, $x_n $ in generation n , and the proportion of doves $ 1 - x_n $. The average reproductive success of hawks and doves, $R_1(n)$ and $ R_2(n) $, respectively, is determined by the expected payoffs from encounters. The proportion of hawks in the next generation is given by:
$ x_{n+1} = \frac{x_n R_1(n)}{R(n)}$
where $R(n)$ is the average reproductive success of the entire population. The model predicts three possible steady states: a population entirely composed of hawks ( x = 1 ), a population entirely composed of doves ( x = 0 ), or a mixed population with a stable proportion of hawks and doves, $x^* = \frac{V}{C} $, provided ( V < C ).
So, in game-theoretical Models, groups choose tactics (e.g., attack, defend, retreat) based on expected payoffs.
As seen in your video, monkeys align into opposing formations, mathematical models for swarm/crowd dynamics can be adapted to include aggression (member of opposing groups) and cohesion forces (to be member of the same group).
References
"mathematical models" "primate" "conflict") just now and found some things that are at least conceivably related to what you want, such as https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3160838/ . You might do that also and then take a look at the bibliographies of Google's suggestions. – MJD Aug 08 '24 at 22:32