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Front sight, front sight, front sight, squeeze subtly not pull, front sight, front sight — oooo, it went off!" ON the Top Gun range, with the target three yards away — most gunfights occur at close quarters in conditions of darkness and confusion — I kept hitting low and to the left.
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Figure 8 Plot of variance in position of a stochastic pulled front.
Such scaling is a consequence of the asymptotic relaxation of the leading edge of the deterministic pulled front.
We begin by constructing a homogeneous neural field equation that supports a propagating pulled front, and calculate its wave speed.
For sufficiently localized initial conditions, the asymptotic velocity of the resulting pulled front is given by the linear spreading velocity, which is determined by linearizing about the unstable state within the leading edge of the front.
Assuming that the full nonlinear system supports a pulled front, then a sufficiently localized initial perturbation (one that decays faster than e - λ 0 x ) will asymptotically approach the traveling front solution with the minimum wave speed c0 = c λ0).
Since pulled front solutions of the neural field equation (Equation 2.1) exhibit similar dynamics, it suggests that there will also be subdiffusive wandering of these fronts in the presence of multiplicative noise.
(a) Velocity dispersion curves c = c for a pulled front solution of the neural field equation (Equation 2.1) with piecewise linear firing rate function (Equation 2.3) and a Gaussian weight distribution (Equation 2.6) with amplitude W0.
In Figure 5a, we show snapshots of a pulled front in the case of a homogeneous network with Gaussian weights (Equation 2.6) and piecewise linear firing rate function (Equation 2.3).
Plot of variance σ X 2 ( t ) of the position of a stochastic pulled front as a function of time for noise amplitude ε = 0.005 and g ( A ) = A L. Other parameters are W0 = 1.2, κ = 0.8, and σ = 1.
The rate at which a population can expand in a landscape without a climate window (invasion speed) is, as expected for a pulled front, linearly correlated to the implemented dispersal distance parameter δ*.
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