We consider a bistable reaction-diffusion-advection system as a population model describing the growth of biological individuals which move by diffusion and chemotaxis. Assuming that the diffusion rate and the chemotactic rate are both very small with the growth rate, we introduce the limiting system by using the singular limit procedure and study the dynamics of growth patterns arising in this system. It is shown that in two dimensional domain planar traveling front solutions are transversally stable when the chemotaxis effect is weak and, when it becomes stronger, they are destabilized. But, these results depend on the sign of the second derivative of the sensitive function. Numerical simulations reveal that the destabilized solution evolves into complex patterns with dynamic spot structures and so on. To explain the appearance of these complex patterns, we obtain the estimate of the attractor dimension from below and it tends to infinity as the chemotaxis effect is increasing.