This thesis studies the related problems of modelling highly non-linear Ordinary and Stochastic equations whose solutions remain positive but either converge to an equilibrium point or blow-up. Neither a metric nor rigorous results in continuous time to characterise these solutions asymptotic behaviour exist. Direct discretisations of the equation using fixed-step numerical schemes fail to reproduce important qualitative properties such as positivity. The thesis develops a suitable metric, a generalised Liapunov exponent, to describe the asymptotic convergence and reliable adaptive numerical schemes that are optimal for both ODEs and SDEs. The schemes are optimal in the sense of minimising computational effort by taking the largest step-size possible whilst preserving the qualitative properties and correct asymptotic behaviour of the continuous-time solution. The schemes recover the qualitative properties and asymptotic rates of convergence under assumptions of monotonicity and regular variation. The critical rate of decay for the step-size is identified. The work shows the resulting error in the convergence rate is insensitive to the assumption of regular variation. Transforming the co-ordinate system is essential to preserving positivity in the case of SDEs. We determine the class of suitable transforms to use and identify that a logarithmic pre-transformation is optimal for ODEs. The class of suitable transformation shows that the problems of hitting an equilibrium and explosion in solutions for ODEs are not equivalent problems in terms of numeric schemes. We develop a quasi-adaptive scheme that can revert to a fixed-step when less computational effort is needed for SDEs. This quasi-adaptive scheme is universal: the scheme works on the highly non-linear problems covered by the thesis and on more standard problems with non-positive solutions, exponential or sub-exponential convergence. The Implicit, Explicit and Transformed schemes can be ranked as measured by the error in convergence rates. No scheme is superior in all circumstances but a ranking can always be achieved.