Pattern formation with jump discontinuity in a macroalgae-herbivore model with strong Allee effect in macroalgae

Abstract

Abstract This paper is concerned with the existence, uniqueness and stability of nonconstant steady states of a reaction-diffusion-ODE system modeling macroalgae-herbivore interaction with strong Allee effect in macroalgae. By applying a generalized mountain pass lemma, we prove the existence of steady states with jump discontinuity. We explore also the structure of stationary solutions on one-dimensional domain and construct various types of steady states, which may be monotone, symmetric or irregular. Moreover, the asymptotic behavior of steady states is considered as the diffusion coefficient tends to infinity. Finally, around a constant steady state, we construct spatially heterogeneous steady states with the aid of the bifurcation theory and study their stability.