In this paper we study a mathematical model of growth of protocell proposed by Tarumi and Schwegler. The model comprises three unknown functions: the con-Ž . Ž . centration u r, t of nutrient, the density ¨r, t of building material, and the Ž . radius R t of the organism which is assumed to be spherically symmetric. The Ž . Ž . functions u r, t , ¨r, t satisfy a system of reaction᎐diffusion equations in the Ž . region 0 F r -R t , t ) 0, and ¨satisfies a Stefan condition on the free-boundary Ž . r s R t . We give precise conditions for existence of one stationary solution, two Ž . stationary solutions, or none. We then prove that a in the first case the stationary solution is unstable so that the transient protocell either disappears in finite time Ž . or expands unboundedly; b in the second case the stationary solution with the larger radius is stable whereas the one with the smaller radius is unstable, so that the transient protocell generally either disappears in finite time or converges to the Ž . stationary configuration with the larger radius; and c in the last case the transient protocell disappears in finite time.