DYNAMICS OF STREET GANG GROWTH 5
because of various types of economic externalities and social spill- over effects. He used systems of partial differential equations
to model interactions between rates of violent death, housing abandonment, homelessness, fires, substance abuse, and AIDS.
Empirical testing confirms the validity of these models. Rowe and his colleagues have done the most extensive empirical work.
Their Markov models consistently yield reasonably accurate pre- dictions of the incidences of various social problems Rogers and
Rowe, 1993; Rowe and Gulley, 1992; Rowe et al., 1989, 1992a, 1992b; Rowe and Rodgers, 1991a. Crane 1989 found that distri-
butions of the rates of various types of crimes in the neighborhoods of Chicago and Los Angeles were consistent with his model. Crane
1991 found that particular nonlinear patterns of neighborhood effects on dropping out and teenage childbearing were also consis-
tent with that same model; and Wallace 1990 found significant geographic correlations among the rates of different social prob-
lems, as predicted by his model.
There is no empirical work on gang populations that would enable us to address the issues raised by these models directly.
But Hutson et al. 1995 found that, over the last 16 years, there has been an epidemic of gang-related homicides. This pattern of
growth is similar to that of a number of the social problems mod- eled in the papers described above.
The model developed in this paper is unique in two ways. First, it deals with the problem of gangs specifically, rather than with
related phenomena. Second, it is more general than those de- scribed above in that it is not explicitly predicated on an assump-
tion of contagion. Nevertheless, it is completely consistent with such an assumption, and given the literature reviewed above, we
believe that contagion probably is an important factor in the gang growth process.
3. THE MODEL
On a very general level, the relationship between gang popula- tion growth and the social control response can be represented by
the classic predator–prey model in ecology Lotka, 1925; Volterra, 1925. This model assumes that two factors determine the prey
population, in this case the number of gang members in a commu- nity. One is the natural growth trajectory of the prey in the absence
of the predator, which defines the relationship between the size of the population and the growth rate. The other is the magnitude
6 J. Crane et al.
of the response of the predator to the level of the population. In this case, the “predator” takes the form of public programs aimed
at reducing the size of the gang population either by rehabilitating or incarcerating gang members. The only other assumption of the
general model is that the relationship between the two factors is additive, which is simplest and seems to be accurate for predator–
prey relationships in nature. More formally, we will assume that the growth rate of the gang population is given by 1:
dxdt 5 g x 2 px
1
where x is the size of the gang population at time t, gx is the intrinsic growth function, and px is the public response function
that describes the amount of resources the public devotes to the problem at each level of the population.
To glean some insights into the dynamics of the gang population, we must specify functional forms for gx and px. As discussed
above, there are several reasons why the growth rate of gangs will tend to increase with the size of the population and then level
off at some point. This is an extremely common pattern for many different types of populations. It can be modeled in a very general
way with the logistic function Pearl and Reed, 1920; Verhulst, 1838. We will use a slight variant of this function by adding an
initial condition. Suppose that, in the absence of exogenous forces i.e., px 5 0, the growth of gang membership over time can be
described in this way 2:
dx dt 5 gx 5 rx1x
12xk 2
where r, k, and x the initial condition
2
are positive constants. Figure 1a shows how the growth rate varies with the size of the
gang population, and Figure 1b shows the time path of the popula- tion. The gang grows first at an increasing rate to some point.
Then the rate of growth levels off, and the population approaches some maximum level k asymptotically, as t approaches infinity.
For the response function, assume that as gang membership in a community grows, the public will devote more resources to the
problem. So px is monotone increasing. But because resources are limited, the investment will ultimately reach or approach some
2
Note that the initial condition implies that a gang population of zero is not an equilib- rium, since at t 5 0 and x 5 0, dxdt 5 rX
. If the initial condition is omitted, the details of the analysis are a little different, but the basic results are the same.
DYNAMICS OF STREET GANG GROWTH 7
Figure 1. a Natural growth of a street gang: gx 5 rx 1 x
1 2 xk; r 5 0.5, x
5 0.5, k 5 10. b Natural growth of a street gang: dxdt 5 gx 5 rx 1 x
1 2 xk; r 5 0.5, x 5
0.5, k 5 10.
maximum level. This pattern of response can be modeled in a very general way, as follows 3:
p x 5 ax
a
b 1x
a
3
where a, b, and a are each constants greater than 0. Note that p
x approaches a asymptotically as x approaches infinity, so a is the maximum response level. The parameters b and a are inversely
related to px. Thus, if b andor a are relatively small, px approaches the maximum relatively quickly as x rises i.e., the
public responds more aggressively to a growing gang population.
8 J. Crane et al.
4. ANALYSIS
