  A stochastic birth-death competition model for particles with excluded volume
is proposed. The particles move, reproduce, and die on a regular lattice. While
the death rate is constant, the birth rate is spatially nonlocal and implements
inter-particle competition by a dependence on the number of particles within a
finite distance. The finite volume of particles is accounted for by fixing an
upper value to the number of particles that can occupy a lattice node,
compromising births and movements. We derive closed macroscopic equations for
the density of particles and spatial correlation at two adjacent sites. Under
different conditions, the description is further reduced to a single equation
for the particle density that contains three terms: diffusion, a linear death,
and a highly nonlinear and nonlocal birth term. Steady-state homogeneous
solutions, their stability which reveals spatial pattern formation, and the
dynamics of time-dependent homogeneous solutions are discussed and compared, in
the one-dimensional case, with numerical simulations of the particle system.
