  In this paper we study invasion probabilities and invasion times of
cooperative parasites spreading in spatially structured host populations. The
spatial structure of the host population is given by a random geometric graph
on $[0,1]^n$, $n\in \mathbb{N}$, with a Poisson($N$)-distributed number of
vertices and in which vertices are connected over an edge when they have a
distance of at most $r_N\in \Theta\left(N^{\frac{\beta-1}{n}}\right)$ for some
$0<\beta<1$ and $N\rightarrow \infty$. At a host infection many parasites are
generated and parasites move along edges to neighbouring hosts. We assume that
parasites have to cooperate to infect hosts, in the sense that at least two
parasites need to attack a host simultaneously. We find lower and upper bounds
on the invasion probability of the parasites in terms of survival probabilities
of branching processes with cooperation. Furthermore, we characterize the
asymptotic invasion time.
  An important ingredient of the proofs is a comparison with infection dynamics
of cooperative parasites in host populations structured according to a complete
graph, i.e. in well-mixed host populations. For these infection processes we
can show that invasion probabilities are asymptotically equal to survival
probabilities of branching processes with cooperation.
  Furthermore, we build in the proofs on techniques developed in [BP22], where
an analogous invasion process has been studied for host populations structured
according to a configuration model.
  We substantiate our results with simulations.
