A mathematical model for the impact of noise on population dynamics of a single species experiencing Lombard effect

Noise is a form of pollution resulting from the undeniable increase in industrialization worldwide. Consequently, it is becoming increasingly important to understand the underlying mechanisms and potential effects of noise on ecosystems. In this study, we propose a deterministic mathematical model that uses a system of nonlinear, non-autonomous differential equations to describe the population dynamics of a single species exposed to noise. The Lombard effect is a phenomenon that involves increasing the intensity of acoustic signals in response to noise, which can mask and degrade acoustic signals and prevent them from being recognized or discriminated by their target receivers. However, when the anthropogenic noise is chronic and critical (i.e., that by its long duration and high intensity positively affects the mortality rate), the increase in the intensity of acoustic signals (due to the Lombard effect) only increases the chronic critical anthropogenic noise and also increase energetic, behavioral and predation costs. Therefore, the critical noise generated by the use of higher intensity acoustic signals (due to the Lombard effect) together with the chronic critical anthropogenic noise, negatively affect population survival. We analyzed the persistence of the population and found that our results are consistent with the observed ecological data as they suggest that, the maximum intensity level of critical chronic anthropogenic noise, consequently, by the Lombard effect, the maximum intensity of self generated acoustic signals, must decrease to ensure population persistence. However, when the maximum intensity level of critical chronic anthropogenic noise is uncontrollable, it is sufficient to reduce its mean intensity level to ensure persistence in the population mean. Furthermore, decreasing the degree to which noise affects the population favors the survival of the species. Finally, to validate our results, we performed numerical simulations.