Continuous-Time Model with Long-Range Dispersal

As p 0, w - oo, and as p - oo, w grows linearly with p. This ensures the existence of a minimum velocity but does not guarantee a maximum velocity. In fact, in this case, no maximum velocity exists, and the front velocity grows without bound in the fast reaction limit, as for the RD equation. 

2 Continuous-Time Model with Long-Range Dispersal 

Consider particles that follow a CTRW, such that the random time T between jumps is exponentially distributed with rate X, f(T t) = exp(—A.t). The mean-field equation for the particle density is the Master equation for the compound Poisson process with logistic growth (5.2), Hyperbolic scaling yields 

Both Hamiltonians (5.11) and (5.21) involve the maximum velocities ajx and aX, respectively. The front velocities tend to infinity in the fast reaction limit when the diffusion approximation is considered. This means that H p) depends quadratically on ap for ap small. The two models (5.11) and (5.20) differ fundamentally with respect to propagating fronts discreteness in time leads to a finite propagation rate, while the continuous-in-time model leads to an infinite velocity of propagation in the limit of fast reaction, r - oo. The front velocity for model (5.21) is 

5 Reaction-Transport Fronts Propagating into Unstable States A, In rcosh(flp) - ll + r 

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Continuous-Time Model with Long-Range Dispersal then the Hamiltonian takes the form... 

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