  We consider the use of adaptive timestepping to allow a strong explicit
Euler-Maruyama discretisation to reproduce dynamical properties of a class of
nonlinear stochastic differential equations with a unique equilibrium solution
and non-negative, non-globally Lipschitz coefficients. Solutions of such
equations may display a tendency towards explosive growth, countered by a
sufficiently intense and nonlinear diffusion.
  We construct an adaptive timestepping strategy which closely reproduces the
a.s. asymptotic stability and instability of the equilibrium, and which can
ensure the positivity of solutions with arbitrarily high probability. Our
analysis adapts the derivation of a discrete form of the It\^o formula from
Appleby et al (2009) in order to deal with the lack of independence of the
Wiener increments introduced by the adaptivity of the mesh. We also use results
on the convergence of certain martingales and semi-martingales which influence
the construction of our adaptive timestepping scheme in a way proposed by Liu &
Mao (2017).
