In this paper, we construct a simple economic dynamics model linking population dynamics and renewable resource dynamics and apply the model to provide a plausible explanation of sustainable or non-sustainable growth of a small island. The "small island" means that it is isolated from other parts of the world and its national product strongly depends on the limited stock of renewable resource and the size of population. In the model human being behaves as the predator and the resource base as the prey and thus the structure of our model is the same as that of the Lotka-Volterra predator-prey model. We demonstrate, with a help of numerical simulations, that the model can have a wide spectrum of growth patterns ranging from monotonic convergence to a steady state to complex dynamics involving erratic (chaotic) behavior depending on the configuration of parameter values. This study indicates that the growth of the small island is very sensitive to a choice of parameters' values reflecting changes in technology, the environment, the economic policy or human behavior.