Nearly exact discretization of single species population models

Abstract: We present a nonstandard discretization method related to the methods of Mickens and Elaydi and Sacker for converting single species population models from continuous time to discrete time. The method falls in the category of the so-called nonstandard discretization schemes, that is more advantageous than the classical discretization methods (such as adaptive step-size) since it allows large step sizes. For instance, a large step size could better represent a generation time or a time interval between empirical measurements. Examples of single-species models with and without negative density dependence, with an Allee effect, and with an alternative positive stable equilibrium (predator pit) are studied. Comparative analyses of bifurcations of ordinary differential equations and difference equations show how the new discretization proposed here preserves the dynamical properties of the continuous-time models. Recommendations for Resource Managers: The discretization method we propose preserves the original dynamic properties of the continuous model, in the sense of equilibria, their stability, and bifurcation characteristics. Unlike the traditional numerical methods that are widely used in ecology, the dynamical consistency of our method does not depend on the size of the step size used. The discretization method we propose produces solution trajectories in a remarkable agreement with those of the corresponding continuous models irrespective of the size of the time interval used. Results presented here will be important to future ecological studies that seek to evaluate the pervasiveness and strength of negative density dependence as well as Allee effects, along with the prospects of alternative stable states, in natural populations.