In this paper, we prove the existence and uniqueness of a positive solution for a nonlocal logistic equation arising from the birth-jump processes. For this, we establish a sub-super solution method for nonlocal elliptic equations, we perform a study of the eigenvalue problems associated with these equations and we apply these results to the nonlocal logistic equation.

Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviourthan for the usual Fisher equation. A striking numerical observation is that a travelingwave with minimal speed can connect a dynamically unstable steady state 0 to a Turingunstable steady state 1, see [12]. This is provedin [1, 6] inthe case where the speed is far from minimal, where we expect the wave to be monotone.

Here we introduce a simplified nonlocal Fisher equation for which we can build simpleanalytical traveling wave solutions that exhibit various behaviours. These travelingwaves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connectthese two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. Thelatter exist in a regime where time dynamics converges to another object observed in[3, 8]: awave that connects 0 to a pulsating wave around 1.