  In this paper we introduce a critical curve separating the asymptotic
behavior of the moments of the symbiotic branching model, introduced by
Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127--160] into
two regimes. Using arguments based on two different dualities and a classical
result of Spitzer [Trans. Amer. Math. Soc. 87 (1958) 187--197] on the exit-time
of a planar Brownian motion from a wedge, we prove that the parameter governing
the model provides regimes of bounded and exponentially growing moments
separated by subexponential growth. The moments turn out to be closely linked
to the limiting distribution as time tends to infinity. The limiting
distribution can be derived by a self-duality argument extending a result of
Dawson and Perkins [Ann. Probab. 26 (1998) 1088--1138] for the mutually
catalytic branching model. As an application, we show how a bound on the 35th
moment improves the result of Etheridge and Fleischmann [Stochastic Process.
Appl. 114 (2004) 127--160] on the speed of the propagation of the interface of
the symbiotic branching model.
