The dynamics of a nonlocal non-monotone reaction-diffusion model with free boundary

This paper is concerned with the dynamics for a class of non-local reaction-diffusion equations with non-quasi-monotonicity and double front free boundaries. We mainly obtained the existence and uniqueness of the global positive solution, as well as its corresponding asymptotical behaviors. We also illustrated the traveling wave solutions of the equations involved with finite length by phase plane analysis. Moreover, we obtained the estimate for the spreading speed when spreading happens.