  We consider the dynamics of spatially-distributed, diffusing populations of
organisms with antagonistic interactions. These interactions are found on many
length scales, ranging from kilometer-scale animal range dynamics with
selection against hybrids to micron-scale interactions between poison-secreting
microbial populations. We find that the dynamical line tension at the interface
between antagonistic organisms suppresses survival probabilities of small
clonal clusters: the line tension introduces a critical cluster size that an
organism with a selective advantage must achieve before deterministically
spreading through the population. We calculate the survival probability as a
function of selective advantage $\delta$ and antagonistic interaction strength
$\sigma$. Unlike a simple Darwinian selective advantage, the survival
probability depends strongly on the spatial diffusion constant $D_s$ of the
strains when $\sigma>0$, with suppressed survival when both species are more
motile. Finally, we study the survival probability of a single mutant cell at
the frontier of a growing spherical cluster of cells, such as the surface of an
avascular spherical tumor. Both the inflation and curvature of the frontier
significantly enhance the survival probability by changing the critical size of
the nucleating cell cluster.
