We consider a spatial branching particle system with locally regulated mean of the offspring distribution. The regulation at one site is of logistic type and depends on a weighted average of the population density in the neighbourhood of that site. Due to the local regulation particles reproduce sub-critically in overcrowded regions and super-critically in sparsely populated regions. We choose Poisson distribution as the offspring distribution. The model describes in discrete time the evolution of a population on $\Z^d$ in which individuals compete for some resources. The generations are non-overlapping, i.e. each individual dies after one unit of time and leaves behind a random number of offspring according to the offspring distribution. The offspring move instantaneously to a new location chosen according to a random walk kernel. After that individuals do not move until the next reproduction step. The local regulation is not only plausible from the ecological point of view. It also endows the system with stability, which is lacking in the case of branching random walks in dimension smaller than or equal to two. It is known that in these dimensions branching random walks die out locally or grow beyond all bounds. We show that for certain parameters the locally regulated system has positive survival probability starting from any non-trivial initial conditions. Here we call initial conditions trivial if, starting with these conditions the population dies out almost surely within one step. Furthermore, using a coupling argument we prove existence and uniqueness of a non-trivial extremal invariant distribution. As the zero-configuration is absorbing, the Dirac measure in that configuration is trivially invariant. The proofs use comparison of suitable events with oriented percolation. For the non-trivial invariant distributions of the oriented percolation and of the locally regulated model we prove exponential decay of the space and time correlations. If in the locally regulated model we replace Poisson random variables by their means, then we obtain deterministic spatial dynamical system. As a by-product of the proof of the coupling result we classify the equilibria and their domain of attraction of that deterministic system.