Abstract
In this thesis, the researcher introduced the aggregation-diffusion equation in two-dimensional space and investigated the existence of steady-states, which contain several states, some of which are trivial and some of which are not. The categories studied numerically using the finite-volume method, then followed by the use of the third-order SSP-RK method, and finally used MATLAB to clarify the state of stability in each case. The categories are divided into three main categories depending on the value of theta, one of which diffusion prevails, while aggregation dominates, and the last one, in which the balance between diffusion and aggregation wins.