Sustainability of nonlinear consumption schemes in resource dynamics with Allee and crowding effects

Abstract Mathematical models of production and consumption are useful in devising policies for sustainable management of renewable resources. Analysis of simple but not too naive models of resource dynamics can provide initial or supplementary insights in formulating comprehensive policies. In these models, density-dependence of resources, nonlinearities in consumption schemes, and uncertainties in parameters are essential to be considered to account for the temporal complex dynamics of the resource. Here, we study minimal ordinary differential equation models which are compartmentalized into production and consumption functions. We propose density-dependent production functions with Allee and crowding effects, and nonlinear consumption functions that are hyperbolic and sigmoidal. We identify scenarios where our results differ from the existing classical logistic models with linear consumption scheme, and determine quantitative conditions leading to sustainable consumption. Our results propose conditions in which consumption efforts will not cause eventual depletion of the resource under specific assumptions. Furthermore, we analyze how delay and stochasticity affect the derived sustainability conditions. Our results can be used as initial input in formulating strategies to properly manage renewable resources, especially in the absence of models requiring extensive data availability.