# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 5 Numerical Experiments
Thus, we are in the condition of applying Propositions 1 and 2 and Theorem 2(i) to guarantee that all populations are non-negative and, furthermore, \(x_{1}(k)>x_{2}(k)>x_{3}(k)\), as Proposition 2 ensures. These results can be observed in Figure 6, where the evolution of the three populations is displayed. In addition, Figure 7 displays the evolution of the three species when we consider the same parameters as before, but \(b_{1}=0.98\), \(b_{2}=0.96\) and \(b_{3}=0.95\) with \(d_{1}=0.01\), \(d_{2}=0.02\) and \(d_{3}=0.03\) (i.e., \(b_{1}>b_{2}>b_{3}\) while \(d_{3}>d_{2}>d_{1}\)). It is checked, in this case, that both Propositions 1 and 2 hold when a harvesting quota, along with an external consumption, is included in the system. It can be also observed, in Figure 7, that species 3 extinguishes. This situation is also predicted by Theorem 2(ii), since \(b_{3}=0.95>1/\mu_{3}=0.55\), making \(x_{3}(k)\) converge to zero asymptotically. Finally, Theorem 2(iii) is also verified, since, as it can be observed in Figures 6 and 7, all populations are bounded for all time. **Example 6**. : _Let us consider the multi-species Beverton-Holt Equations (2) and (3) parameterized by the constant values \(b_{i}(k)=1\), \(d_{i}(k)=0\), \(K_{i}(k)=200\) and \(\mu_{1}(k)=2.4\), \(\mu_{2}(k)=2\) and \(\mu_{3}(k)=1.8\) with \(x_{1}(0)=100\), \(x_{2}(0)=50\) and \(x_{3}(k)=30\). The coupling matrix is given by_  
\[C=\left( \begin{array}{ccc}1&0.004&0.005\\ 0.0025&1&0.004\\ 0.005&0.0035&1\end{array} \right)\]  
The evolution of the three species under this parameterization is displayed in Figure 9. It can be observed, in Figure 9, that there exists a consensus equilibrium point in which the equilibrium points of all species converge to the same unique value. The numerical location of the equilibrium point is given by \(x=198.50\), which is close to the theoretically calculated one of 198.70 predicted by Proposition 4 in (43). Figure 7: Evolution of \(x_{1}(k)\), \(x_{2}(k)\) and \(x_{3}(k)\) for constant parameters in Example 4. Species 1 dominates the other two and harvesting quota and consumption are included. Figure 8: Evolution of \(x_{1}(k)\), \(x_{2}(k)\) and \(x_{3}(k)\) for constant parameters in Example 5 when harvesting quota is included and coupling matrix \(\mathbb{C}\) has small non-diagonal components. Figure 9: Evolution of \(x_{1}(k)\), \(x_{2}(k)\) and \(x_{3}(k)\) for constant parameters in Example 6 when no harvesting quota nor external consumption are included so that left-hand side values and right-hand side values are coincident. The coupling matrix \(\mathbb{C}\) has small non-diagonal entries and the three species converge to the same equilibrium point. **Example 7**. : _Let us consider the multi-species Beverton-Holt Equations (2) and (3) parameterized by the constant values \(b_{i}(k)=1\), \(d_{i}(k)=0\), \(K_{1}(k)=300\), \(K_{2}(k)=K_{3}(k)=200\) and \(\mu_{1}(k)=1.5+0.4^{k}\), \(\mu_{2}(k)=1.45+0.35^{k}\) and \(\mu_{3}(k)=1.35+0.3^{k}\) with \(x_{1}(0)=100\), \(x_{2}(0)=75\) and \(x_{3}(k)=50\). The coupling matrix is given by_  
\[C=\left( \begin{array}{ccc}1&0.4&0.005\\ 0.25&1&0.4\\ 0.05&0.035&1\end{array} \right)\]  
The evolution of the three species under this parameterization is displayed in Figure 10. Since we are in the condition of applying Theorem 4(i), because \(0=d_{i}<b_{i}x_{i}\) and \(\prod_{j=k_{\ell}}^{j=k_{\ell+1}}[b_{i}(k_{\ell+1}+1-j)\mu_{i}(k_{\ell+1}-j)] \geq\underline{\rho}>1\), then there is no asymptotic extinction of the populations, as Figure 10 shows. Moreover, all the populations converge to an equilibrium point which is asymptotically stable, as established by Theorem 5. In addition, Figure 11 displays the populations of the three species when \(\mu_{1}(k)=0.5+0.4^{k}<1\), \(\mu_{2}(k)=0.45+0.35^{k}<1\) and \(\mu_{3}(k)=0.35+0.3^{k}<1\). As it is claimed in Theorem 4(ii), the three species extinguish asymptotically, since \((1-\mu_{i}(0))\sum_{j=1}^{n}c_{ji}x_{i}(0)<1\).