We model, analyze, and simulate a novel run-and-tumble model with autochemotaxis, biologically inspired by the phytoplankton, Heterosigma akashiwo. Developing a fundamental understanding of planktonic movements and interactions through phototaxis and chemotaxis is vital to comprehending why harmful algal blooms (HABs) start to form and how they can be prevented. We develop a one- and two-dimensional mathematical and computational model reflecting the movement of an ecology of plankton, incorporating both run-and-tumble motion and autochemotaxis. We present a succession of complex and biologically meaningful models combined with a sequence of laboratory and computational experiments that inform the ideas underlying the model. By analyzing the dynamics and pattern formation which are similar to experimental observations, we identify parameters that are significant in plankton's pattern formation in the absence of bulk fluid flow. We find that the precise form of chemical deposition and plankton sensitivity to small chemical gradients are crucial parameters that drive nonlinear pattern formation in the plankton density.