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

We study mild solutions u to the semilinear Cauchy problem ∂∂t ut(x)=12 x(1−x) ∂2∂x2 ut(x)+γut(x)(1−ut(x)) (t⩾0),u0(x)=f(x)with x∈[0,1], f a nonnegative measurable function and γ a positive constant. Solutions to this equation are given by ut=Utf, where (Ut)t⩾0 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 γ⩽1, the mass in the interior (0,1) dies out after a finite random time, while for γ>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 γ−1 and is approximately uniformly distributed over (0,1). We apply these results to show that (Ut)t⩾0 has precisely four fixed points when γ⩽1 and five fixed points when γ>1, and determine their domains of attraction.