Unifying Models of Trophic Exploitation: A Mathematical Framework for Understanding the Paradox of Enrichment

The rapid increase in the world’s human population has largely been attributed to efforts aimed at enhancing primary productivity and enriching food resources. However, an intriguing proposition of M. Rosenzweig, known as the paradox of enrichment, challenged the notion that such enrichment schemes always lead to sustained population growth. Instead, they can disrupt the delicate equilibrium of predator–prey systems, potentially driving one or both species to extinction. In this study, we develop a comprehensive mathematical framework that unifies Rosenzweig’s six analytical models of trophic exploitation through the Richards growth model, which can be viewed as a Box–Cox transformation of one species’ abundance relative to carrying capacity. Our analysis not only elucidates the connections and similarities between each model but also presents a generalized framework that unveils the underlying relationships between the proposed functions. Using the generalized logarithm and exponential functions of nonextensive statistical mechanics, we offer a fresh perspective and highlight the importance of a cautious approach when enriching ecosystems. This unification clarifies how the parameters that govern growth dynamics and predator–prey interactions determine system stability in diverse ecological contexts. Through numerical simulations and isoclinic analysis, we demonstrate that our generalized model accurately reproduces the classic paradox of enrichment while providing new insights into the mechanisms driving population fluctuations after environmental enrichment. This mathematical synthesis advances both theoretical ecology and practical conservation efforts by enabling a more accurate assessment of enrichment risks in managed ecosystems.