# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 3 Stability Results Obtained by Lyapunov Method
\begin{array}{c}\mbox{(c) }x_{i}(k)<(1+a_{i}(k)+h_{i}(k))\frac{\gamma_{i}}{2};\ \forall k\in\mathbf{N}_{0}\\ \mbox{(d) }\sum_{i=1}^{n}\frac{\theta_{i}}{x_{i}(k)}(1+a_{i}(k)+h_{i}(k))>2 \sum_{i=1}^{n}\frac{\theta_{i}}{x_{i}};\ \forall k\in\mathbf{N}_{0}\\ \mbox{(e) }\frac{\max\limits_{i\leq j\in\overline{n}}x_{i}^{2}(k)}{\min \limits_{i\leq j\in\overline{n}}x_{i}(k)}<\frac{\sum_{i=1}^{n}\theta_{i}\left(1 -(a_{i}(k)+h_{i}(k))^{2}\right)}{2\sum_{i=1}^{n}\theta_{i}\left[1-a_{i}(k)-h _{i}(k)\right]}x_{i}\\ \mbox{(f) }\frac{\sum_{i=1}^{n}\theta_{i}x_{i}^{2}(k)}{\sum_{i=1}^{n}\theta_{i}x_{i} (k)}<\frac{\min\limits_{i\leq j\in\overline{n}}\left(1-(a_{i}(k)+h_{i}(k))^{2 }\right)}{2\sum_{i\leq j\in\overline{n}}\left(1-a_{i}(k)-h_{i}(k)\right)}x_{i} \end{array} \]  
**Proof. ** Let us define \(y_{i}=x_{i}^{-1}\); \(\forall i\in\overline{n}\) with \(x_{i}\) being a nonzero equilibrium point of the \(i\)-th species and \(\widetilde{y_{i}}(k)=y_{i}(k)-y_{i}\); \(\forall i\in\overline{n}\), \(\forall k\in\mathbf{N}_{0}\). Let us consider the Lyapunov sequence candidate \(V(k)=\sum_{i=1}^{n}\theta_{i}(y_{i}(k)-y_{i})^{2}\); \(\forall k\in\mathbf{N}_{0}\) for some set of design scalar coefficients \(\theta_{i}>0\); \(\forall i\in\overline{n}\). Then, the one-step incremental Lyapunov sequence candidate verifies the subsequent chain of relations:  
\[ \begin{array}{c}\Delta V(k)=V(k+1)-V(k)=\sum_{i=1}^{n}\theta_{i}\left( \widetilde{y}_{i}^{2}(k+1)-\widetilde{y}_{i}^{2}(k)\right)\\ \mbox{}\hskip 14.226378 pt=\sum_{i=1}^{n}\theta_{i}(\widetilde{y}_{i}(k+1)+ \widetilde{y}_{i}(k))\ (\widetilde{y}_{i}(k+1)-\widetilde{y}_{i}(k))\\ \mbox{}\hskip 14.226378 pt=\sum_{i=1}^{n}\theta_{i}([a_{i}(k)+h_{i}(k)]y_{i}(k )-y_{i}+y_{i}(k)-y_{i})\\ \mbox{}\hskip 14.226378 pt\times([a_{i}(k)+h_{i}(k)]y_{i}(k)-y_{i}-y_{i}(k)+y_ {i})\\ \mbox{}\hskip 14.226378 pt=\sum_{i=1}^{n}\theta_{i}([a_{i}(k)+h_{i}(k)+1]y_{i}( k)-2y_{i})\ ([a_{i}(k)+h_{i}(k)-1]y_{i}(k))\\ \mbox{}\hskip 14.226378 pt=\sum_{i=1}^{n}\theta_{i}([a_{i}(k)+h_{i}(k)+1]\ [a_{i}(k)+h_{i}(k)-1]y_{i}^{2}(k)\\ \mbox{}\hskip 14.226378 pt-2\sum_{i=1}^{n}\theta_{i}y_{i}\ [a_{i}(k)+h_{i}(k)-1]y_{i}(k)\\ \mbox{}\hskip 14.226378 pt=-\sum_{i=1}^{n}\theta_{i}((1+a_{i}(k)+h_{i}(k))y_{i} (k)-2y_{i})(1-a_{i}(k)-h_{i}(k))y_{i}(k)\\ \mbox{}\hskip 14.226378 pt\leq-\sum_{i=1}^{n}\left((1+a_{i}(k)+h_{i}(k))y_{i} (k)-2y_{i}\right)\min\limits_{1\leq i\leq n}[\theta_{i}(1-a_{i}(k)-h_{i}(k))y_{ i}(k)];\\ \mbox{}\hskip 14.226378 pt\forall k\in\mathbf{N}_{0}\end{array}  \tag{66}\]  
Other upper-bounds of \(\Delta V(k)\) are clearly the following ones:  
\[ \begin{array}{c}-\left(\sum_{i=1}^{n}\theta_{i}\Big{[}1-(a_{i}(k)+h_{i}(k))^ {2}\Big{]}\right)\min\limits_{1\leq i\leq n}y_{i}^{2}(k)\\ \mbox{}\hskip 14.226378 pt+2(\sum_{i=1}^{n}\theta_{i}y_{i}\ [1-a_{i}(k)-h_{i}(k)])\max \limits_{1\leq i\leq n}y_{i}(k)\end{array}  \tag{67}\]  
and  
\[ \begin{array}{c}-\min\limits_{1\leq i\leq n}\big{(}1-(a_{i}(k)+h_{i}(k))^{2} \big{)}\big{(}\sum_{i=1}^{n}\theta_{i}y_{i}^{2}(k)\big{)}\\ \mbox{}\hskip 14.226378 pt+2\max\limits_{1\leq i\leq n}\ (1-a_{i}(k)-h_{i}(k))(\sum_{i=1}^{n} \theta_{i}y_{i}(k)y_{i})\end{array}  \tag{68}\]  
Thus, if the constraints (a) and (b) and one of the constraints from (c) to (f) hold, then \(\left\{-\Delta V_{k}\right\}_{k=0}^{\infty}\subset\mathbf{R}_{+}\), so that \(\left\{V_{k}\right\}_{k=0}^{\infty}\) is bounded and strictly decreasing, so that \(\left\{V_{k}\right\}_{k=0}^{\infty}\to 0\) so that \(y_{i}(k)\to y_{i}=x_{i}^{-1}\), equivalently, \(x_{i}(k)\to x_{i}\,;\forall i\in\overline{n}\); \(\forall k\in\mathbf{N}_{0}\). \(\Box\)  
**Remark 4. **_Note that the constraint (a) of Theorem 5 requires, as a necessary condition, \(\mu_{i}(k)(b_{i}(k+1)-v_{i}(k+1))>1\); \(\forall i\in\overline{n}\); \(\forall k\in\mathbf{N}_{0+}\)._Note also that a sufficient condition for Theorem 5 to hold is  
\[ \begin{split}\sum_{i=1}^{n}\theta_{i}((1+a_{i}(k)+h_{i}(k))y_{i}(k)- 2y_{i})(1-a_{i}(k)-h_{i}(k))>0;\\ \forall i\in\overline{n},\ \forall k\in\mathbf{N}_{0}\end{split}