Interplay of sources of stochastic noise in a resource-based model

We introduce a stochastic resource-based model in which the agents acquire resources from the environment. The model dynamics is described by two coupled stochastic differential equations with distinct sources of noise, one acting on the population size and the other affecting the availability of resources. Random fluctuations from the environment and the population dynamics itself are simulated as stochastic effects causing variations in the resource influx and death rates. We derive analytical results for the probability density of the population size and amount of available resource when a single source exists. The analytical expressions and numerical results, that uses the Milstein approach, show a remarkable agreement. In particular, the presence of stochasticity in the death rate leads to an effectively higher death rate in comparison to the deterministic counterpart of the model. On the other hand, when the two sources of stochastic fluctuations are considered, interesting antagonistic effects are revealed by the Pearson coefficient of the correlation between the population size and resource availability. Indeed, small fluctuations in resource availability reveal strong negative correlations. In contrast, when the noise in the death rate is too low the equilibrium population size and available resource become nearly uncorrelated. We also generalize the model to handle an arbitrary number of species and resources. This allows us to probe the underlying conditions for coexistence among species. We find a stable coexistence of many species over a wide range of parameter values. In the most general conditions, the system does not violate the competitive exclusion principle.