**5. Numerical Simulation**

In this section, we present some numerical simulations to show the occurrence of the paradox of enrichment of systems (1) and (2) with Holling type II when increasing the carrying capacity of prey. We use time series and phase space graphs to present the dynamic behavior, and present bifurcation diagrams to explain a map of the dynamic behaviors of systems (1) and (2). The values of the parameters for both systems (1) and (2) were selected to satisfy Theorems 3 and 4, in which the dynamic behavior is stable, but different values of carrying capacity are used.

The values of system (1) are as follows:

$$a = 16.0, \ e\_1 = 0.7, \ h\_1 = 0.5, \ u = 0.65, \ x(0) = 0.5, \ y(0) = 0.2 \tag{43}$$

The values of system (2) are as follows:

$$\begin{aligned} \text{If } u = 10.0, \; \beta = 7.0, \; c\_1 = 1.00, \; c\_2 = 0.4, \; h\_1 = 1.5, \; h\_2 = 1.7, \; c\_1 = 0.03, \; c\_2 = 0.02, \\\ u = 0.05, \; w = 0.1, \; x(0) = 0.5, \; y(0) = 0.2, \; z(0) = 0.2 \end{aligned} \tag{44}$$

When taking the value of k = 1 of system (1), the dynamic behavior in the first case is stable, as shown in Figure 1. However, when increasing the carrying capacity to k = 4 in the second case, the dynamic behavior oscillates for a period of time and then ends, finally stabilizing, as shown in Figure 2. In the third case, when k = 7, the dynamic behavior oscillates to become a limit cycle, as shown in Figure 3. Consequently, the probability of extinction in the third case would be higher than in the first and second cases. Figure 4 shows the changes of the dynamic behavior of system (1) with Holling type II, from stable to periodic cases. The points in Figure 4 appear because oscillation exists in the dynamic behavior.

**Figure 1.** Dynamic behavior of system (1) when *k* = 1: (**a**) time series of two trophics *x* and *y*; (**b**) phase space of two trophics.

**Figure 2.** Dynamic behavior of system (1) when *k* = 4: (**a**) time series of two trophics *x* and *y*; (**b**) phase space of two trophics.

**Figure 3.** Dynamic behavior of system (1) when *k* = 7: (**a**) time series of two trophics *x* and *y*; (**b**) phase space of two trophics.

**Figure 4.** Bifurcation diagram for system (1) with Holling type II, using carrying capacity (*k*) as the bifurcation parameter.

Following the same process, we used different values of k from system (2). As shown in Figure 5, the dynamic behavior is stable when k =1 in the first case. However, when the carrying capacity is increased to k = 2 in the second case, the dynamic behavior oscillates for a period of time and then ends, finally stabilizing, as shown in Figure 6. Whereas in the third case, when k = 3, the dynamic behavior oscillates to create a limit cycle, as shown in Figure 7. Therefore, the probability of extinction in the third case would be greater than in the first and second cases. Figure 8 shows the changes in the dynamic behavior of system (2) with Holling type II, from stable to periodic, quasi-periodic, or chaos cases. The points in Figure 8 appear because oscillation occurs in the dynamic behavior. The numerical simulations show the occurrence of the paradox of enrichment in systems (1) and (2) with Holling type II.

**Figure 5.** Dynamic behavior of system (2) when *k* = 1: (**a**) time series of three trophics *x, y* and *z*; (**b**) phase space of three trophics.

**Figure 6.** Dynamic behavior of system (2) when *k* = 2: (**a**) time series of three trophics *x, y* and *z*; (**b**) phase space of three trophics.

**Figure 7.** Dynamic behavior of system (2) when *k* = 3: (**a**) time series of three trophics *x, y* and *z*; (**b**) phase space of three trophics.

**Figure 8.** Bifurcation diagram for system (2) with Holling type II, using carrying capacity (*k*) as the bifurcation parameter.
