### Dynamics of a single gang

Consider a simple model for a single gang in a bounded environment with *s* different discrete sites in which it might be active. These settings might be street corners (Taniguchi et al. 2011), street segments (Weisburd et al. 2012), police beats (Papachristos 2009), neighborhoods (Grannis 2009), block groups (Tita and Radil 2011) or even an arbitrary grid laid out over an urban landscape (Block 2000). Let \(p\) be the proportion of sites registering activity by the single gang at any one point in time (see Papachristos 2007). Let \(c\) be the rate at which activity spreads to sites in the environment. Let \(m\) be the rate at which activity ceases at occupied sites. Given these components we can construct a simple dynamical model describing the rate of change in the number of sites recording gang activity (Nee and May 1992; Tilman 1994):

$$\frac{dp}{dt} = cp\left( {1 - p} \right) - mp.$$

(1)

The interpretation of Eq. (1) is straightforward. The first term on the right-hand side states that the rate of spread of gang activities to different sites is dependent upon the current density of gang activity, captured by the product \(cp\). For a fixed rate \(c > 0\), the proportion of sites occupied increases exponentially in the existing density of activity. However, gang activity can only spread to sites that are currently unoccupied, captured by the term \(\left( {1 - p} \right)\). If the availability of sites were the only constraint, then gang activity would grow logistically to fill the entire environment. That is, the growth in \(p\) slows as the environment fills up, but eventually \(p = 1\).

Equation (1) goes one step further to assume that gang activity also ceases in locations currently occupied. This is captured by the second term on the right-hand side, \(mp\). The rate of activity cessation is also dependent upon the current proportion of sites presenting gang activity. If more sites show gang activity, then more sites will see gang activity cease. Gang activity also grows over time under these conditions, but towards an equilibrium below full saturation of the environment, reflecting a balance between activity spread and cessation. Setting Eq. (1) to zero and solving gives the equilibrium proportion of sites with gang activity (Tilman 1994).

$$\widehat{p} = 1 - \frac{m}{c}.$$

(2)

Equation (2) merits brief discussion. Note that if \(m \ge c\), then the rate at which gang activity ceases matches or exceeds the rate at which it spreads. This implies \(\widehat{p} < 0\) and ensures that gang activity will never take hold. Gang activity can only persist if \(m < c\), meaning that once gang activity has spread to a site it at least has some limited tenure there. The theoretical case of \(m = 0\) is intriguing (see also Tilman 1994). This implies gang activity *never* ceases once it is established at a site. This is the only circumstance under which an environment will be completely saturated with gang activity. Infinite persistence of gang activity at a site is theoretically possible if we allow the individuals to seamlessly replace one another over time. However, whether gang activity is considered persistence at a site depends substantially upon the scale of observation (see Mohler et al. 2019; Mohler et al. 2017). We assume that both \(c\) and \(m\) are intrinsic traits that do not vary through time, but may vary from one gang to another.

### Competition among two symmetric gangs

The single gang model may be extended to consider the dynamics of two gangs with competitive interactions. The first case to consider is competitive interaction between two gangs that are equal, or symmetric in their ability to hold any setting that they currently control. That is, a gang currently active at site *s* cannot be displaced by a rival that attempts to engage in activity at the site. No competitive hierarchy exists among the gangs.

We now index model parameters for each gang so that \(p_{1}\) and \(p_{2}\) are the proportion of sites *s* with activity attributed to gangs 1 and 2, respectively. Similarly, \(c_{1}\) and \(c_{2}\) reflect the rate of activity spread and \(m_{1}\) and \(m_{2}\) the rate of activity cessation for gangs 1 and 2, respectively. Putting these together we arrive at:

$$\frac{{dp_{1} }}{dt} = c_{1} p_{1} \left( {1 - p_{1} - p_{2} } \right) - mp_{1} ,$$

(3)

$$\frac{{dp_{2} }}{dt} = c_{2} p_{2} \left( {1 - p_{1} - p_{2} } \right) - mp_{2} .$$

(4)

Equations (3) and (4) are coupled ordinary differential equations describing the rate of change in the proportion of sites occupied by gang 1 and gang 2, respectively. They are coupled because the proportion of sites with activity attributed to gang 1 constrains the ability of gang 2 to occupy new sites and vice versa. Specifically, the proportion of open sites for new gang activity at any one time is \(\left( {1 - p_{1} - p_{2} } \right)\). At equilibrium, the proportions of space taken up by each gang are:

$$\widehat{p}_{1} = 1 - \frac{{m_{1} }}{{c_{1} }} - \widehat{p}_{2} ,$$

(5)

$$\widehat{p}_{2} = 1 - \frac{{m_{2} }}{{c_{2} }} - \widehat{p}_{1} .$$

(6)

Equations (3) and (4) leads to outcomes similar to those presented in Brantingham et al. (2012). Depending upon the activity spread and cessation rate of each gang, at equilibrium gangs can occupy exactly equal proportions of the environment (Fig. 1a), unequal but stable proportions (Fig. 1b), or one gang can eventually drive the other out of the environment (Fig. 1c). These outcomes are not about a gang’s ability to hold space in the face of direct challenges, since both gangs are equal in this regard. Rather, the outcomes depend on whether a gang is able to capitalize on vacant space in the environment. Using ecological terminology this would be a case of interference competition. Note then that a non-zero equilibrium density of gang activity \(\widehat{p}_{i} > 0\) for gang *i* requires \(m_{j} /c_{j} < 0.5\) for the other gang *j*. Each competitor must leave some space open for its rival if that rival is to persist (Tilman 1994).

### Competition among two asymmetric gangs

The second case to examine is two gangs with asymmetric competitive abilities. Consider a strict competitive hierarchy where gang 1 is *always* able to displace gang 2 at any site \(s\), but gang 2 is *never* able to displace gang 1. In the dyadic contest, we call the gang that is always able to displace its rival the superior competitor, while the gang that is never able to displace its rival is the inferior competitor. Referring to them as superior or inferior only denotes their competitive abilities with respect to spatial displacement, not any other attribute of the gangs we might wish to study.

We can write equations governing the dynamics of these two gangs as (Tilman 1994):

$$\frac{{dp_{1} }}{dt} = c_{1} p_{1} \left( {1 - p_{1} } \right) - m_{1} p_{1} ,$$

(7)

$$\frac{{dp_{2} }}{dt} = c_{2} p_{2} \left( {1 - p_{1} - p_{2} } \right) - m_{2} p_{2} - c_{1} p_{1} p_{2} .$$

(8)

Notice several key differences between Eqs. (7) and (8) and their symmetrical counterparts. Equation (7) describes the rate of change in the proportion of sites with activity attributed to gang 1. It is *not* coupled to the dynamics of gang 2, meaning that whatever the dynamics of gang 2 might be, it does not influence the dynamics of gang 1. This is a direct consequence of the strict competitive hierarchy. The equilibrium proportion of sites with gang 1 is actually no different than what would be the case if it were alone in the environment.

Equation (8) reflects quite different dynamics. The dynamics of gang 2 *are* coupled to the dynamics of gang 1. The term \(\left( {1 - p_{1} - p_{2} } \right)\) suggests that the rate of spread of gang 2 activities is limited to those sites currently left open by gang 2 *and* gang 1. This is analogous to the symmetrical case given in Eq. (4). The strict competitive hierarchy introduces another point of coupling, however. The term \(c_{1} p_{1} p_{2}\) captures the impact of competitive displacement events, where gang 1 encounters and competitively excludes gang 2. In probabilistic terms, \(c_{1} p_{1}\) is the probability that gang 1 spreads to a site and \(p_{2}\) is the probability that the site already hosts gang 2.

It is reasonable to suppose that the ecological conditions modeled by Eqs. (7) and (8) do not favor the persistence of gang 2. However, Tilman (1994) demonstrated that the inferior competitor can persist if it is able to take advantage of the sites left unoccupied by the superior competitor (see also Nee and May 1992). At equilibrium, there will be \(\left( {1 - \widehat{p}_{1} } \right)\) sites left open by the superior competitor at any one time. The inferior competitor must be able to find and exploit those sites before they are displaced completely.

There are two distinct ways in which open sites can be exploited by inferior competitors. The inferior competitor can persist if its rate of activity spread exceeds that of the superior competitor. Specifically, if

$$c_{2} > c_{1} \left( {\frac{{\widehat{p}_{1} }}{{1 - \widehat{p}_{1} }} + \frac{{m_{2} }}{{m_{1} }}} \right),$$

(9)

then the inferior gang will be able to invade the environment and maintain activity over some equilibrium proportion of sites. If we assume that the superior and inferior gangs have the same activity cessation rate \(m_{1} = m_{2} = m\), then Eq. (9) simplifies to (Tilman 1994):

$$c_{2} > c_{1} \left( {\frac{1}{{1 - \widehat{p}_{1} }}} \right).$$

(10)

Equation (10) is revealing. As the equilibrium proportion of sites occupied by the superior gang declines towards zero (i.e., \(\widehat{p}_{1} \to 0\)), the minimum spread rate needed to sustain the inferior gang approaches that of the superior one. Conversely, as the environment fills up with activity by the superior gang, the spread rate for the inferior gang must increase nonlinearly to ensure persistence. For example, when the superior gang is present in a proportion \(\widehat{p}_{1} = 0.25\) of sites given an activity spread rate of \(c_{1} = 0.2\), the inferior gang must have an activity spread rate of at least \(c_{2} > 0.2667\) to be able to persist, assuming equal activity cessation rates \(m\). The inferior gang must maintain at least a 33% faster activity spread rate. When the superior gang is present in a proportion \(\widehat{p}_{1} = 0.75\) of sites, given an activity spread rate of \(c_{1} = 0.2\), the inferior gang must have an activity spread rate of at least \(c_{2} > 0.8\). In this case, the inferior gang must spread at least 300% faster than the superior gang to capitalize on open space. In general, the greater the proportion of space occupied by a superior gang at equilibrium, the faster the inferior gang needs to spread to ensure survival.

The consequences of a faster rate of activity spread are shown in (Fig. 2a). Starting at low initial abundances, the inferior gang rapidly increases its presence, peaking at \(p_{2} = 0.675\) after about 57 time-steps. The higher activity spread rate allows it to capitalize on all of the empty space initially present. The inferior gang then starts to lose ground as the superior gang occupies more and more space. This reflects both the superior gang displacing the inferior gang and the superior gang preempting the inferior gang at some sites. Eventually, the superior gang surpasses the inferior gang in the proportion of sites occupied, which happens around 173 time-steps into the simulation. At equilibrium, the superior gang holds a proportion \(\widehat{p}_{1} = 0.38\) of the sites and the inferior gang a proportion \(\widehat{p}_{2} = 0.21\) of the sites. The inferior competitor survives in spite of the absolute competitive superiority they face at each site.

The competitively inferior gang may also persist if it maintains a lower activity cessation rate relative to the superior gang. However, this strategy is more limited. Rearranging Eq. (9) to solve for \(m_{2}\), and assuming that the two gangs have the same activity spread rate \(c_{1} = c_{2} = c\), yields (Tilman 1994):

$$m_{2} < m_{1} \left( {1 - \frac{{\widehat{p}_{1} }}{{1 - \widehat{p}_{1} }}} \right).$$

(11)

The inferior gang can persist only if it ceases activities at a rate slower than the superior gang. The activity cessation rate for the inferior gang can be very close to that of the superior gang when the superior gang occupies very few sites at equilibrium (i.e., when \(\widehat{p}_{1} \approx 0\)). However, the activity cessation rate for the inferior gang must quickly approach zero as the equilibrium proportion of sites occupied by the superior gang approaches \(\widehat{p}_{1} = 0.5\). As the superior gang increases its hold on space, the inferior gang is put under more displacement pressure and therefore *must* hold on to any sites that it *does* occupy for as long as it can. If the superior gang occupies more than a proportion \(\widehat{p}_{1} > 0.5\) of sites at equilibrium, the inferior gang cannot rely on reducing activity cessation rates to persist (Tilman 1994). To see why, notice that the term \(\widehat{p}_{1} /\left( {1 - \widehat{p}_{1} } \right)\) in Eq. (11) is analogous to the odds that any given site is occupied by the superior gang. The odds are greater than 1 when \(\widehat{p}_{1} > 0.5\), meaning that the inferior gang cannot be guaranteed to find any open space.

Persistence of the inferior gang as a result of lower activity cessation rates is illustrated in Fig. 2b. Here the two gangs have the same rate of activity spread, but different rates of activity cessation. Qualitatively the trajectory towards equilibrium looks similar to the case of differential activity spread rates, even though the mechanism is very different. Starting at the same low initial abundances, the inferior gang early on comes to occupy a large fraction of the environment, holding approximately \(p_{2} = 0.70\) of the sites only 14 time steps into the simulation. Eventually, however, the superior competitor occupies a sufficient proportion of sites that it starts to competitively exclude the inferior competitor, driving down its abundance. At equilibrium, the superior competitor occupies a proportion \(\widehat{p}_{1} = 0.33\) of the site, while the inferior competitor occupies a proportion \(\widehat{p}_{2} = 0.25\) of sites.

The outcomes shown Fig. 2a and b are not the only ones possible. The inferior gang can be driven to extinction under a wide range of conditions (not shown). It is also possible for the inferior gang to persist with a lower activity spread rate than the superior gang as long as it has an activity cessation rate sufficiently below that of its competitor (Fig. 2c). It is also possible for the inferior gang to exist at a greater abundance than the superior gang. This seems counter intuitive, but is possible if the inferior gang has an activity spread rate that is higher than the superior competitor *and* an activity cessation rate that is lower than the superior competitor. Such a case is shown in Fig. 2d. Here the equilibrium proportion of the inferior competitor is \(\widehat{p}_{2} = 0.21\), while the proportion of the superior competitor is \(\widehat{p}_{1} = 0.17\).

### Competition in a community of asymmetric gangs

Tilman (1994) illustrates how the two gang model can be extended to a community of street gangs. We start with the same environment consisting of *s* different discrete sites or settings in which gangs might be active. There is community of *n* total gangs present in the environment and they can be ranked into a strict competitive hierarchy \(i = 1,2, \ldots , n\). The most competitive gang is positioned at the top (\(i = 1\)) and the least competitive gang at the bottom (\(i = 1\)) of the hierarchy.^{Footnote 1} Gangs positioned higher in the hierarchy can displace all gangs lower in the hierarchy. Conversely, gangs positioned lower in the hierarchy are never able to displace gangs higher up. This is a strict “pecking order” consistent with the analyses in Papachristos (2009) and Randle and Bichler (2017).

To model the dynamics of this competitive hierarchy, let \(p_{i}\) be the proportion of sites registering activity by the gang \(i\). Let \(c_{i}\) be the rate at which gang \(i\)’s activity spreads to other sites in the environment. Let \(m_{i}\) represent the rate at which gang \(i\)’s activity ceases at sites with activity. The change in the fraction of sites occupied by gang *i* (Tilman 1994) is:

$$\frac{{dp_{i} }}{dt} = c_{i} p_{i} \left( {1 - p_{i} - \mathop \sum \limits_{j = 1}^{i - 1} p_{j} } \right) - m_{i} p_{i} - \mathop \sum \limits_{j = 1}^{i - 1} c_{j} p_{j} p_{i} .$$

(12)

The term in Eq. (12) states that gang *i* cannot spread to any site that is currently held by any gang higher in the competitive hierarchy. That is, the available space is reduce by sites held by the highest ranked gang \(j = 1\), the second highest ranked gang \(j = 2\), and so on, through to the sites held by gang \(j = i - 1\), the gang immediately above \(i\) in the hierarchy. Gang \(i\) also interferes with its own spread. Gang \(i\) ceases activity at a rate \(m_{i} p_{i}\), a density dependent effect. Gang \(i\) is also displaced by all higher ranked gangs. This can be seen in the second summation, which takes into account the spread of the highest ranked gang \(j = 1\), the second highest ranked gang \(j = 2\), and so on, through to gang \(j = i - 1\), the gang immediately superior to gang \(i\). Setting Eq. (12) to zero and solving for \(p_{i}\) gives the equilibrium frequency of gang \(i\) within the strict competitive hierarchy (Tilman 1994):

$$\widehat{p}_{i} = 1 - \frac{{m_{i} }}{{c_{i} }} - \mathop \sum \limits_{j = 1}^{i - 1} \widehat{p}_{j} \left( {1 + \frac{{c_{j} }}{{c_{i} }}} \right).$$

(13)

The first two terms on the right-hand side of Eq. 13 together reflect the how the activities of gang \(i\) influence its own equilibrium proportion, independent of competitive effects. As in the two-gang case, gang \(i\) can only hold territory if \(m_{i} < c_{i}\). The second term on the right-hand side reflects the additional impact of competitive displacement by higher-ranked gangs. In general, the equilibrium proportion of gang \(i\) is reduced by the total proportion of space occupied by superior gangs, scaled by the ratio of activity spread rates for each superior gang relative to gang \(i\).

The conditions under which any inferior gang *i* can survive in the face of competition from any number of superior gangs can be established by solving Eq. (13) for \(\widehat{p}_{i}\) > 0 and isolating either \(c_{i}\) or \(m_{i}\). The mathematical results are conceptually the same as for the two-gang asymmetrical case so we do not detail them here (see Tilman 1994, p. 7). In general, a gang \(i\) must have an activity spread rate \(c_{i}\) that is faster than that of the next higher ranked gang \(c_{i - 1}\), scaled by the proportion of sites left open by higher ranked gangs. Alternatively, the activity cessation rate for gang \(i\) must be less than the cessation rate for the immediately superior gang \(i - 1\) scaled by the proportion of sites left open by superior gangs. Inferior competitors up and down the hierarchy can also mix different activity spread and cessation rates to ensure survival.

Equation (12) leads to a range of outcomes (Fig. 3). Gangs may occupy space in proportions that are positively rank-order correlated with their competitive abilities, although this outcome can result from different mechanisms. In Fig. 3a, for example, gangs ranked 1–4 in the competitive hierarchy achieve equilibrium proportions \(\widehat{p}_{i} = \left\{ {0.2, 0.16,0.08,0.05} \right\}\), respectively, as a result of differing activity spread rates. In Fig. 3b, they occupy exactly the same proportions of space at equilibrium, but this time because of differing activity cessation rates. Most importantly, gangs may occupy space in proportions that do not at all track their relative competitive abilities. In Fig. 3c, for example, gangs ranked 1–4 in the competitive hierarchy occupy proportions \(\widehat{p}_{i} = \left\{ {0.05, 0.08, 0.16,0.2} \right\}\), respectively, a perfect inversion of the actual competitive ranking. This is achieved by gangs deploying a mixture of activity spread and cessation rates. The key observation is that the proportion of space occupied by a gang (i.e., territory size) is alone not sufficient to infer competitive dominance.