Summary: For random measure-valued stochastic partial differential equations for biological processes, growth represented by scalar partial differential equations at each point of the support and spread being a diffusion on \(\mathbb{R}^d\), solutions are constructed by smearing the growth processes at each spatial point and composing the resulting generator with the generator for the spread. If these solutions are unique, the equation is called solvable. We find conditions for the noise term of a solvable equation to have trivial effect and we identify some non-solvable equations, for example the diffusion-free bilinear equation. The search led to an investigation of explosion and the effect of point barriers for scalar stochastic differential equations with linear drift; this is used to explain the clustering effect in the usual superprocess.