Extinction versus exponential growth in a supercritical super-Wright-Fisher diffusion

We study mild solutions u to the semilinear Cauchy problem with x[set membership, variant][0,1], f a nonnegative measurable function and [gamma] a positive constant. Solutions to this equation are given by , where is the log-Laplace semigroup of a supercritical superprocess taking values in the finite measures on [0,1], whose underlying motion is the Wright-Fisher diffusion. We establish a dichotomy in the long-time behavior of this superprocess. For [gamma][less-than-or-equals, slant]1, the mass in the interior (0,1) dies out after a finite random time, while for [gamma]>1, the mass in (0,1) grows exponentially as time tends to infinity with positive probability. In the case of exponential growth, the mass in (0,1) grows exponentially with rate [gamma]-1 and is approximately uniformly distributed over (0,1). We apply these results to show that has precisely four fixed points when [gamma][less-than-or-equals, slant]1 and five fixed points when [gamma]>1, and determine their domains of attraction.