Anomalously Slow Attrition Times for Asymmetric Populations with Internal Group Dynamics

The many-body dynamics exhibited by living objects include group formation within a population and the nonequilibrium process of attrition between two opposing populations due to competition or conflict. We show analytically and numerically that the combination of these two dynamical processes generates an attrition duration $T$ whose nonlinear dependence on population asymmetry $x$ is in stark contrast to standard mass-action theories. A minority population experiences a longer survival time than two equally balanced populations, irrespective of whether or not the majority population adopts such an internal grouping. Adding a third population with predefined group sizes allows $T(x)$ to be tailored. Our findings compare favorably to real-world observations.

Figure 1

(a) Duration $T$ of attrition (or equivalently, the extinction time or survival time of the smaller population) as a function of initial $A$ population $N_A(0)$ and fragmentation probability ${\nu }_A$ (${\nu }_B=0.3$). Solid lines are analytic [Eq. (2)] while the surface is a numerical simulation. $N_A(0)+N_B(0)=1000$. Qualitative features are unchanged by varying ${\nu }_B$. (b) Black curve: same as (a) with ${\nu }_A={\nu }_B=0.3$. Red dashed curve: $A$ contains rigid units (size 10) while $B$ features internal dynamical grouping (i.e., clustering) as explained in the text. Green curve: both $A$ and $B$ comprise rigid units (size 10). (c) Events in our model. Two groups of the same type can coalesce, e.g., $6+4=10$. Individual groups can fragment, e.g., $6\to 6\times 1$. Two groups of opposite type interact, e.g., a group of size $6-4=2$ remains.

Figure 2

Duration $T$ of human conflicts as a function of asymmetry $x$ between the two opposing military populations. $x=|N_A(0)-N_B(0)|/[N_A(0)+N_B(0)]$. Data are up to the end of 2008; hence, final data points for the three ongoing wars will lie above the positions shown, as indicated by arrows. The lower two blue lines are the mass-action results. The upper red curve [i.e., Eq. (2)] is generated using ${\nu }_A={\nu }_B=0.7$ and $[N_A(0)+N_B(0)]$ fixed [as in Fig. 1a]. Changing ${\nu }_A$ and ${\nu }_B$ changes the height of the theoretical peak but leaves qualitative features unchanged.

Figure 3

(a) Black curve as in Figs. 1a, 1b with $A$ and $B$ undergoing internal dynamical clustering. ${\nu }_A={\nu }_B=0.3$. Red dashed curve: $n_C=100$ third-party groups, each of size $s_C=1$. Green dotted curve: $n_C=1$ third-party group of size $s_C=100$. (b) Third-party blocking event. If neither $A$ nor $B$ clusters are bigger than the $C$ cluster, then the $C$ cluster blocks the interaction and permanently neutralizes both clusters.