A triangle model of criminality

This paper is concerned with a quantitative model describing the interaction of three sociological species, termed as owners, criminals and security guards, and denoted by XX, YY and ZZ respectively. In our model, YY is a predator of the species XX, and so is ZZ with respect to YY. Moreover, ZZ can also be thought of as a predator of XX, since this last population is required to bear the costs of maintaining ZZ.We propose a system of three ordinary differential equations to account for the time evolution of X(t)X(t), Y(t)Y(t) and Z(t)Z(t) according to our previous assumptions. Out of the various parameters that appear in that system, we select two of them, denoted by HH, and hh, which are related with the efficiency of the security forces as a control parameter in our discussion. To begin with, we consider the case of large and constant owners population, which allows us to reduce ,  and  to a bidimensional system for Y(t)Y(t) and Z(t)Z(t). As a preliminary step, this situation is first discussed under the additional assumption that Y(t)+Z(t)Y(t)+Z(t) is constant. A bifurcation study is then performed in terms of HH and hh, which shows the key role played by the rate of casualties in YY and ZZ, that results particularly in a possible onset of bistability. When the previous restriction is dropped, we observe the appearance of oscillatory behaviours in the full two-dimensional system. We finally provide a exploratory study of the complete model ,  and , where a number of bifurcations appear as parameter HH changes, and the corresponding solutions behaviours are described.