Complex dynamics of a family Cubic-Logistic map

0,xR}. In this work we study the complex dynamics of this family i.e. L= {L_ (z)=z^2 (1-z): >0,zC}. That is we study the Fatou and Julia sets of this maps. In fact we give a whole description for these sets These two new types of logistics maps can address some of life's problems as shown in the introduction.We prove for any R, L_L is preservers R, critically finite, maps the negative x-axis into positive real line and has no any complex periodic point. Fatou set of these maps has no Siegel disk, Baker domain, and has no Wandering domain so they consist of parabolic domains and basins of attraction. Finally, we use escaping algorithm to construct the Fatou and Julia sets of our maps for various values of .]]>