Analysis of a Mathematical Model of Protocell

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- centration u r, t. of nutrient, the density ¨ r, t. of building material, and the radius R t. of the organism which is assumed to be spherically symmetric. The functions u r, t., ¨ r, t. satisfy a system of reaction]diffusion equations in the region 0Fr-R t., t)0, and ¨ satisfies a Stefan condition on the free-boundary rsR t.. We give precise conditions for existence of one stationary solution, two 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 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 stationary configuration with the larger radius; and c. in the last case the transient protocell disappears in finite time.