The aim of this paper is to apply the newly trending Atangana-Baluanu derivative operator to model some symbiosis systems describing commmensalism and predator-prey processes. The choice of using this derivative is due to the fact that it combines nonlocal and nonsingular properties in its formulation, which are the essential ingredients when dealing with models of real-life applications. In addition, it is only the Atangana-Baleanu derivative that has both Markovian and non-Markovian properties. Also, its waiting time takes into account the power, exponential, and Mittag-Leffler laws in its formulation. Mathematical analysis of these dynamical models is considered to guide in the correct use of parameters therein, with chaotic and spatiotemporal results reported for some instances of fractional power α.