In this paper we study chemical oscillations and wave formation in a cubic chemical reaction model (a modified Selkov model) by using hyperbolic partial differential equations, i.e., hyperbolic reaction-diffusion equations, instead of the usual parabolic reaction-diffusion equations. We have carried out a comparative investigation for the hyperbolic and parabolic reaction-diffusion equations in order to help determine which system of partial differential equations is more suitable for describing chemical oscillations and wave phenomena. It is shown that the hyperbolic system approaches the parabolic system as a parameter called the reaction-diffusion number increases to a large value. In contrast to the case of parabolic differential equations, the travelling wave speed has no singular behavior if the hyperbolic system is used. The hyperbolic system predicts that there is no necessity to take different diffusion coefficients to have the Turing instability unlike the parabolic system. Numerical solutions are presented for a one-dimensional case to show that the patterns change as the reaction-diffusion number is changed from the hyperbolic regime to the parabolic regime. The entropy productions are also calculated which show that, given the same boundary and initial conditions, different patterns have the same global entropy production, but the system gets organized to local structures of some particular frequencies and wave numbers of high entropy productions at the expense of energy and matter on the part of the rest of the frequency and wave number modes. Some modes appropriate to themselves most of energy and matter available to the system as a whole and get organized to local structures. It is also found that as the trajectories become chaotic the entropy production becomes relatively smaller than the ordered structures. The second law of thermodynamics, however, does not have direct control over such local organizations except for providing thermodynamically consistent evolution equations. In the case of a bistable system, the entropy production is found to make a rather abrupt change as the system makes transition from one stable state to another.