The continuum model of sdbsim() has one uniform steady state,
\(\bar b = \epsilon D \gamma / \omega\), \(\bar A = a_0 + \bar b\),
\(\bar\rho = \gamma / \bar A\). Linearised about it, a disturbance of
wavenumber-squared \(k\) evolves by a 2 by 2 matrix whose trace is
negative at every \(k\) and whose determinant is \(P k^2 - Q k +
R\), \(R = \omega \bar A > 0\): a mode grows exactly when the
determinant is negative. So hot spots form when \(Q > 0\), that is
\(3 \epsilon D \bar\rho > \omega + \eta \bar A\) (repeat
victimisation outweighs decay and spreading), and \(Q^2 > 4 P R\);
the fastest-growing mode is near \(k^* = Q / (2 P)\). Without
reinforcement (eps = 0) the uniform state is stable at every
wavelength (proof("sdbthr")). z places the lattice's neighbour
count on both diffusion terms, as sdbsim() does (z = 1 is the
continuum form).
Arguments
- a0, omega, eps, dif, gam, eta, z
The model's constants, as for
sdbsim(): one number each,epsandetafrom 0, the others positive.- k
NULLor wavenumbers-squared (numbers from 0) at which to tabulate the trace, the determinant and the growth rate.
Value
An object of class sdbthr, a list: steady (b, a,
rho); p, q, r, the determinant's coefficients; thr, the two
sides of the threshold (lhs \(3 \epsilon D \bar\rho\), rhs
\(\omega + \eta \bar A\)); unstable; kcrit, detcrit,
grcrit, the critical mode, its determinant and growth rate (NA
unless p and q are positive); and disp, NULL or a data frame
(k, trace, det, growth).
References
Short, M. B., Bertozzi, A. L. and Brantingham, P. J. (2010). Nonlinear patterns in urban crime: hotspots, bifurcations, and suppression. SIAM Journal on Applied Dynamical Systems, 9(2), 462-483. doi:10.1137/090759069