A two-type two-sex branching process is introduced with the aim of describing the interaction of predator and prey populations with sexual reproduction and promiscuous mating. In each generation and in each species the total number of individuals which mate and produce offspring is controlled by a binomial distribution with size given by this number of individuals and probability of success depending on the density of preys per predator. The resulting model enables us to depict the typical cyclic behaviour of predator-prey systems under some mild assumptions on the shape of the function that characterises the probability of survival of the previous binomial distribution. We present some basic results about fixation and extinction of both species as well as conditions for the coexistence of both of them. We also analyse the suitability of the process to model real ecosystems comparing our model with a real dataset.