**6. Conclusions**

We studied the occurrence of the paradox of enrichment in prey–predator models with Holling types I and II functional and numerical responses. We proved through Theorems 1 and 2 that the paradox of enrichment does not occur with Holling type I in two or three dimensions. However, the paradox of enrichment occurs with Holling type II in two and three dimensions, respectively, as shown through Corollary 1, Theorem 5, and the numerical simulations. The numerical simulations explain the occurrence of the paradox of enrichment in systems (1) and (2) with Holling type II when the carrying capacity of prey increases and a map of changes of the dynamic behaviors is given for stable to periodic, quasi-periodic, or chaos cases. We conclude that the linearity and nonlinearity of functional and numerical responses plays important roles in the occurrence of the enrichment paradox. We introduce a new approach connecting the enrichment paradox phenomenon and persistence and extinction dynamics by deriving the persistence and extinction conditions based on the carrying capacity parameter (*k*). We used different analytical techniques to derive the persistence and extinction conditions. We introduce several theorems and corollaries to present our results. We introduce some biological explanations to support our results.

**Acknowledgments:** This work is funded by the Basic Science Research Unit, Scientific Research Deanship at Majmaah University under the research project No. 31/37. The author is extremely grateful to Majmaah University, Deanship of Scientific Research and Basic Science Research Unit, Majmaah University.

**Conflicts of Interest:** The author declares no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.
