Proliferating L\'evy Walkers and Front Propagation

We develop non-linear integro-differential kinetic equations for proliferating L'{e}vy walkers with birth and death processes. A hyperbolic scaling is applied directly to the general equations to get the Hamilton-Jacobi equations that will allow to determine the rate of front propagation. We found the conditions for switching, birth and death rates under which the propagation velocity reaches the maximum value, i.e. the walker’s speed. In the strong anomalous case the death rate was found to influence the velocity of propagation to fall below the walker’s maximum speed.