  The well-known Solow growth model is the workhorse model of the theory of
economic growth, which studies capital accumulation in a model economy as a
function of time with capital stock, labour and technology efficiency as the
basic ingredients. The capital is assumed to be in the form of manufacturing
equipments and materials. Two important parameters of the model are: the saving
fraction $s$ of the output of a production function and the technology
efficiency parameter $A$, appearing in the production function. The saved
fraction of the output is fully invested in the generation of new capital and
the rest is consumed. The capital stock also depreciates as a function of time
due to the wearing out of old capital and the increase in the size of the
labour population. We propose a stochastic Solow growth model assuming the
saving fraction to be a sigmoidal function of the per capita capital $k_p$. We
derive analytically the steady state probability distribution $P(k_p)$ and
demonstrate the existence of a poverty trap, of central concern in development
economics. In a parameter regime, $P(k_p)$ is bimodal with the twin peaks
corresponding to states of poverty and well-being respectively. The associated
potential landscape has two valleys with fluctuation-driven transitions between
them. The mean exit times from the valleys are computed and one finds that the
escape from a poverty trap is more favourable at higher values of $A$. We
identify a critical value of $A_c$ below (above) which the state of poverty
(well-being) dominates and propose two early signatures of the regime shift
occurring at $A_c$. The economic model, with conceptual foundation in nonlinear
dynamics and statistical mechanics, shares universal features with dynamical
models from diverse disciplines like ecology and cell biology.
