Morphogenetic processes such as neurulation and gastrulation involve coordinated movements of cells. The processes under consideration happen due to long-range signaling, although the detailed mechanisms are not completely understood. The biological model-systems along with self-organization of cells and the mechanisms of signaling are of great interest. A major question is whether or not short-range signaling or local interaction of cells can also be the cause of coordinated movement and morphogenetic processes. As a model problem authors analyze ripple formation of myxobacteria considered due to purely local interaction, a hypothesis which is discussed in the biological literature. These ripples can be observed before the final aggregation of the bacteria and fruiting body formation take place. The mathematical model described here is the hyperbolic system of Goldstein-Kac type in one dimension with density-dependent coefficients, \[ u_t^++\gamma u_x^+=-(\mu +\lambda ^+)u^++(\mu +\lambda ^-)u^-, \]
\[ u_t^--\gamma u_x^-=(\mu +\lambda ^+)u^+-(\mu +\lambda ^-)u^-, \] where \(\mu > 0 \) is the rate of autonomous turning and \(\lambda ^{\pm }=\lambda^{\pm }(u^+,u^-)\geq 0\) are the turning rates due to interaction. The Neumann boundary conditions on the interval \([0,l]\) are imposed, i.e., \(u^+(t,x)=u^-(t,x)\) for \(x\in [0,l]\). Here the rates depend pointwise on the cell densities. It is assumed that the turning rates are continuously differentiable with locally Lipschitz continuous first partial derivatives. The main result is that for all initial data \(u^{\pm }\in C^1([0,l])\) which satisfy the compatibility conditions there exists a unique solution \((u^+,u^-)\) of the considered problem in the class \(C^1([0,T]\times [0,l])^2\) for some time \(T>0\). Conditions for the existence of travelling waves are discussed by methods of linear analysis and the construction of invariant domains.