**1. Introduction**

Prey–predator interactions are important in applied mathematics and mathematical biology, receiving considerable attention from many researchers [1–8]. The Lotka-Volterra model is considered the basis for formulating prey–predator interaction models; it was proposed independently by Lotka and Volterra, so it is known as the Lotka-Volterra model. In the literature, predation and competition relationships are two main relationship types used for modeling any prey–predator system [9,10]. Mathematically, prey–predator interactions are described by nonlinear differential equations.

Counterintuitive observations have generally attracted more attention than observations that confirm intuition. These observations are called paradoxes that unexpectedly challenge normal intuition [11]. One of these observations, the paradox of enrichment, states that increasing the carrying capacity of prey in a stable prey–predator system leads to the destabilization of the system, which can be mathematically represented by limit cycles. Destabilization might lead to extinction, which is interpreted when the limit cycle is sufficiently large for one of the species or all species, so that the limit cycle is approximately close to zero. This phenomenon was discovered by Rosenzweig in 1971 [12].

Several experimental studies rejected the hypothesis that the enrichment phenomenon would destabilize community dynamics [13–16]. The studies that rejected enrichment paradox phenomenon explained that the paradox was actually caused by a difference between the mathematical construction and real prey–predator interactions. However, recent experimental studies showed the occurrence of the paradox of enrichment. Fussmann et al. [17] showed that enrichment led to the predator's extinction in their experiment on rotifer algae. Cottingham et al. [18] showed that in some lakes, anthropogenic eutrophication of ecosystems destabilized lakes. The process of lake eutrophication has been suggested to be an example of the paradox of enrichment [11]. Recently, Meyer et al. [19] predicted the occurrence of the paradox of enrichment for communities with multiple aboveground and belowground trophic levels and suggested that extinction and destabilization are more likely in fertilized agroecosystems than in natural communities.

One of the most important dynamics in prey–predator systems is stability, which is the first property usually studied in these systems. Some models show that the predator equilibrium density increases when the carrying capacity raises, but the prey equilibrium density would not increase as shown in the Rosenzweig–MacArthur model [20]. Notably, increasing the carrying capacity affects the prey and predator equilibrium densities. Losing stability transitions the dynamic behavior to cycle dynamics, which are relevant to persistence and extinction dynamics. The persistence and extinction of prey-predator systems have been studied by many researchers [21–41] due to their importance. Some methodologies have been used to find the conditions of persistence and extinction in two and three dimensions (trophics). Hutson and Vickers [23] determined the main criteria of a two prey, one predator model that depends on the Lyapunov function or average Lyapunov function. Freedman and Waltman [24] introduced a definition of persistence and determined the general criteria for three interacting populations. Freedman [21], in his book, summarized Kolmogorov conditions of prey–predator systems, which have been applied to derive the persistence and extinction conditions of two dimensions (trophics).

Persistence is defined analytically as follows: For a population *<sup>x</sup>*(*t*), *if x*(0) > 0 and lim*t*→ ∞ inf*x*(*t*) > 0, *x*(*t*) persists: geometrically, defined each trajectory of differential equations is defined as eventually bounding away from the coordinate planes [24]. Extinction is defined analytically as follows: if *x*(0) > 0 and lim*t*→ ∞ inf*x*(*t*) = 0, then *x*(*t*) becomes extinct: geometrically, the trajectory of differential equations is defined as touching the coordinate planes.

Dubey and Upadhyay [30] studied persistence and extinction according to the Hutson and Vickers method. They explained that the conditions of persistence and extinction depend on the equilibrium levels of prey and predators and food conversion coefficients, capturing the rates and comparing them with the mortality rates of predators. Gakkhar et al. [31] studied persistence and extinction in their proposed model based on the Freedman and Waltman method. They proved that persistence is not possible for two predators competing for one prey species when any one of the boundary prey–predator planes has a stable equilibrium point. They presented numerical simulations of persistence in the case of periodic solution. They concluded that the principle of competitive exclusion holds in this case. Alebraheem and Abu Hassan [38–41] studied different scenarios of persistence and extinction in their modified model. However, the carrying capacity of the systems was widely excluded to study the dynamic behavior.

In this paper, we introduce a new approach that involves a mathematical connection between the occurrence of the enrichment paradox and the persistence and extinction dynamics. The question that we aimed to answer here is if enrichment of prey affects the persistence and extinction of predators. Therefore, we derived the persistence and extinction conditions and completed numerical simulations based on the carrying capacity that affects the occurrence of the paradox of enrichment. To study this idea, we used the same systems that were used by Alebraheem and Abu Hassan [38–42], but considered the carrying capacity. Two systems were examined: a prey–predator model that represents two dimensions (trophics), and a one prey, two predators system that describes three dimensions (trophics). Kolmogorov analysis and Freedman and Waltman methods were used to study the persistence and extinction dynamics.

The remainder of this paper is structured as follows. In Section 2, we introduce the mathematical systems of prey–predator used to study the relationship between the paradox of enrichment and the dynamics of persistence and extinction. In Section 3, we study the occurrence of the paradox of enrichment phenomenon. In Section 4, we study a theoretical approach to persistence and extinction. In Section 5, we present some numerical simulations. In Section 6, we draw our conclusions.
