  In the present paper we study a lattice model of two species competing for
the same resources. Monte Carlo simulations for d=1, 2, and 3 show that when
resources are easily available both species coexist. However, when the supply
of resources is on an intermediate level, the species with slower metabolism
becomes extinct. On the other hand, when resources are scarce it is the species
with faster metabolism that becomes extinct. The range of coexistence of the
two species increases with dimension. We suggest that our model might describe
some aspects of the competition between normal and tumor cells. With such an
interpretation, examples of tumor remission, recurrence and of different
morphologies are presented. In the d=1 and d=2 models, we analyse the nature of
phase transitions: they are either discontinuous or belong to the
directed-percolation universality class, and in some cases they have an active
subcritical phase. In the d=2 case, one of the transitions seems to be
characterized by critical exponents different than directed-percolation ones,
but this transition could be also weakly discontinuous. In the d=3 version,
Monte Carlo simulations are in a good agreement with the solution of the
mean-field approximation. This approximation predicts that oscillatory
behaviour occurs in the present model, but only for d>2. For d>=2, a steady
state depends on the initial configuration in some cases.
