Hot-spot threshold of the Short-Brantingham reaction-diffusion model
Source:R/ledger_1_4_3.R
morie_hotspot_threshold.RdThe spatially uniform Short-Brantingham system
\(B' = -\omega B + \epsilon D \rho (A_0 + B)\), \(\rho' = -\rho (A_0 + B) + \gamma\)
has exactly one homogeneous steady state, \(\bar B = \epsilon D \gamma / \omega\),
\(\bar\rho = \gamma / \bar A\) with \(\bar A = A_0 + \bar B\)
(Research.P4.Hotspot.steady_iff). Linearising at wavenumber-squared \(k\) gives a
\(2 \times 2\) matrix whose determinant is \(P k^2 - Q k + R\) with
\(P = \eta D^2 / z\), \(Q = D (3 \epsilon D \bar\rho - \omega - \eta \bar A / z)\) and
\(R = \omega \bar A > 0\) (det_eq, P_eq, Q_eq, R_eq); its trace is
negative at every \(k \ge 0\) (tr_neg), so a mode grows iff the determinant is
negative (unstable_iff). Hence the threshold: with
\(3 \epsilon D \bar\rho \le \omega + \eta \bar A / z\) the uniform state is stable at every
wavelength (no_hotspots_below_threshold; in particular without repeat reinforcement,
no_hotspots_without_reinforcement), and with \(Q > 0\) and \(Q^2 > 4 P R\) the
critical mode \(k^* = Q / (2P)\) grows (hotspots_above_threshold).
Arguments
- A0
Baseline attractiveness (positive).
- omega
Decay rate of the dynamic attractiveness (positive).
- eps
Repeat-victimisation reinforcement (non-negative).
- D
Diffusion rate of the criminal density (positive).
- gamma
Generation rate of criminals (positive).
- eta
Relative diffusion of attractiveness (non-negative).
- z
Lattice coordination number (positive; 4 on a square lattice, 1 in the continuum form).
- k
Optional wavenumbers-squared at which the trace, determinant and growth rate are tabulated.
Value
A list with steady (\(\bar B\), \(\bar A\), \(\bar\rho\)), the coefficients
P, Q, R, threshold (the two sides of
\(3 \epsilon D \bar\rho \lessgtr \omega + \eta \bar A / z\)), unstable, the critical
wavenumber-squared k_crit with det_crit and growth_crit, an optional
dispersion table and theorems.