# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 3 Stability Results Obtained by Lyapunov Method
 \tag{69}\]  
although it is not as easy to test as the conditions given in the theorem statement. Note that stability around the zero equilibrium implies the asymptotic extinction of all the species, This also implies that the inverse populations diverge asymptotically to \(+\infty\). Note that  
\[ \begin{split} y_{i}(k+1)=\frac{1}{\mu_{i}(k)K_{i}(k)\left(b_{i}(k +1)-d_{i}(k+1)y_{i}^{-}(k+1)\right)}(K_{i}(k)y_{i}(k)+\mu_{i}(k)-1)\\ +\frac{\mu_{i}(k)-1}{\mu_{i}(k)K_{i}(k)\left(b_{i}(k+1)-d_{i}(k+1) y_{i}^{-}(k+1)\right)}\left(\sum_{j(\
eq i)=1}^{n}c_{ji}(k)\frac{y_{i}(k)}{y_{j}(k) }\right)\end{split}  \tag{70}\]  
The asymptotic extinction is formalized through the subsequent result. **Theorem 6**. : _Let us assume that the following conditions hold:_  
_(a)_ \(\{K_{i}(k)\}_{k=0}^{\infty}\subset\left(0\,,\overline{K}\right)\)_,_ \(\{\mu_{i}(k)\}_{k=0}^{\infty}\subset(1,\,\overline{n})\)_,_ \(\{b_{i}(k)\}_{k=0}^{\infty}\subset\left[0,\,\overline{b}\right]\)_,_ \(c_{ii}(k)=1\)_,_ \(\{c_{ji}(k)\}_{k=0}^{\infty}\subset[0,1]\) _and_ \(d_{i}(k)\in\left(-\infty,\ b_{i}(k)x_{i}^{-}(k)\right];\forall i\in\overline{ n},\,\forall k\in\mathbf{N}_{0}\)_._  
_(b) If there is some finite_ \(k\in\mathbf{N}_{0}\) _such that_ \(x_{i}(k)=0\)_, then_ \(x_{i}(k+1)=-d_{i}(k+1)=0\)__  
_(c)_ \(\underset{k\to\infty}{lim}\left(\prod_{j=0}^{\infty}[\alpha_{i}(j)]\right)=+\infty\)__  
Then, \(\{x_{i}(k)\}_{k=0}^{\infty}\to 0\); \(\forall i\in\overline{n}\) if  
\[\underset{k\to\infty}{lim}\inf\frac{K_{i}(k)y_{i}(k)+\mu_{i}(k)-1}{\mu_{i}(k) K_{i}(k)\left(b_{i}(k+1)-d_{i}(k+1)y_{i}^{-}(k+1)\right)}>1\]  
**Proof. ** Note that the assumption \(d_{i}(k)\leq b_{i}(k)x_{i}^{-}(k)\) is equivalent to \(b_{i}(k)-d_{i}(k)y_{i}^{-}(k)\geq 0\); \(\forall i\in\overline{n},\,\forall k\in\mathbf{N}_{0+}\) and implies that \(x_{i}(k)\geq 0\) and \(y_{i}(k)\geq 0\); \(\forall i\in\overline{n},\,\forall k\in\mathbf{N}_{0}\). Let us define the auxiliary variables for \(i\in\overline{n},\,k\in\mathbf{N}_{0}\) as  
\[\alpha_{i}(k)=\frac{K_{i}(k)y_{i}(k)+\mu_{i}(k)-1}{\mu_{i}(k)K_{i}(k)\left(b_{ i}(k+1)-d_{i}(k+1)y_{i}^{-}(k+1)\right)} \tag{71}\]  
\[\beta_{i}(k)=\frac{\mu_{i}(k)-1}{\mu_{i}(k)K_{i}(k)\left(b_{i}(k+1)-d_{i}(k+1) y_{i}^{-}(k+1)\right)} \tag{72}\]  
\[\gamma_{i}(k)=\sum_{j(\
eq i)=1}^{n}c_{ji}(k)\frac{y_{i}(k)}{y_{j}(k)} \tag{73}\]  
so that  
\[y_{i}(k+1)=\alpha_{i}(k)y_{i}(k)+\beta_{i}(k)\gamma_{i}(k);\ \forall i\in \overline{n},\ \forall k\in\mathbf{N}_{0} \tag{74}\]  
One obtains, via recursive calculations with the above relation, that  
\[ \begin{split} y_{i}(k+1)-\sum_{j=0}^{k}\left(\prod_{j=0}^{k} \left[\alpha_{i}(\ell)\right]^{k-j}\beta_{i}(j)\gamma_{i}(j)\right)=\left(\prod _{j=0}^{k}[\alpha_{i}(j)]\right)y_{i}(0);\\ \forall i\in\overline{n},\,\forall k\in\mathbf{N}_{0}\end{split}  \tag{75}\]  
And, if \(y_{i}(0)>0\) and finite (equivalently, \(0<x_{i}(0)<+\infty\)) for all \(i\in\overline{n}\), then  
\[\underset{k\to\infty}{lim}\left(y_{i}(k+1)-\sum_{j=0}^{k}\left(\prod_{j=0}^{k} \left[\alpha_{i}(\ell)\right]^{k-j}\beta_{i}(j)\gamma_{i}(j)\right)\right)=+ \infty;\ \forall i\in\overline{n} \tag{76}\]  
The following cases can arise:_Case 1_: \(\left\{y_{i}(k)\right\}_{k=0}^{\infty}\to+\infty\); \(\forall i\in\overline{n}\), equivalently, \(\left\{x_{i}(k)\right\}_{k=0}^{\infty}\to 0\); \(\forall i\in\overline{n}\). The proof follows directly. _Case 2_: \(\left\{y_{i}(k)\right\}_{k=0}^{\infty}\) is bounded and \(\lim\limits_{k\to\infty}\left(\sum_{j=0}^{k}\left(\prod_{j=0}^{k}\left[\alpha_ {i}(\ell)\right]^{k-j}\beta_{i}(j)\gamma_{i}(j)\right)\right)=+\infty\) for some \(i\in\overline{n}\). Case 2 can hold through several sub-cases as follows:  
a) _Sub-case 2a_: Either \(\left\{\alpha_{i}(k)\right\}_{k=0}^{\infty}\to+\infty\) or \(\left\{\beta_{i}(k)\right\}_{k=0}^{\infty}\to+\infty\).