# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 3 Stability Results Obtained by Lyapunov Method  
It is now assumed that the Beverton-Holt equation is time-varying, in general, rather than parameterized by constant parameters. A Lyapunov sequence candidate is proposed for stability analysis around eventual non-extinction equilibrium points. From Assumption 2, an artificial auxiliary sequence \(\{\
u_{i}(k)\}_{k\in\mathbf{N}}\subset\mathbf{R}\) is defined by \(\
u_{i}(k)=\frac{d_{i}(k)}{x_{i}^{-}(k)}\in(-\infty,\ b_{i}(k)]\);\(\forall i\in\overline{n}\), \(\forall k\in\mathbf{N}\) (note that \(d_{i}(k)\) is not defined for \(k=0\)). That definition, with the above constraint, guarantees that  
\[d_{i}(k+1)=\
u_{i}(k+1)x_{i}^{-}(k+1)\leq b_{i}(k+1)x_{i}^{-}(k+1);\ \forall i\in \overline{n},\ \forall k\in\mathbf{N} \tag{60}\]  
**Remark 3**. : _The above artificial sequence allows an easy integration of the eventual independent consumption contribution to the stability of non-extinction equilibrium points._  
The above artificial sequence is not adequate to describe finite-time extinction \(x_{i}(k)=x_{i}^{-}(k+1)=0\) with \(d_{i}(k+1)<0\) (individuals of the species \(i-th\) enter the habitat from outside) implying that the extinguished \(i-th\) species to the left of the \((k+1)\) sampling instants recovers from previous extinction to the right of the \((k+1)\) sampling instant. We can now describe the model to the right of the sampling instants as follows, if \(\big{\{}x_{i}^{-}(k)\big{\}}_{k=0}^{\infty}\subset\mathbf{R}_{+}\), \(\forall i\in\overline{n}\),  
\[x_{i}(k+1)=\frac{\mu_{i}(k)K_{i}(k)(b_{i}(k+1)-\
u_{i}(k+1))}{K_{i}(k)+(\mu_{i }(k)-1)\sum_{j=1}^{n}c_{ji}(k)x_{j}(k)}x_{i}(k);\forall i\in\overline{n},\ \forall k\in\mathbf{N} \tag{61}\]  
for given initial finite conditions \(x_{i}(0)>0\); \(\forall i\in\overline{n}\), where \(\
u_{i}(k)=\frac{d_{i}(k)}{x_{i}^{-}(k)}\in(-\infty,\ b_{i}(k)]\). Let us define the \(i-th\) population inverse \(y_{i}(k)=x_{i}^{-1}(k)\); \(\forall i\in\overline{n}\), \(\forall k\in\mathbf{N}\) so that  
\[ \begin{split}& y_{i}(k+1)=(\mu_{i}(k)(b_{i}(k+1)-\
u_{i}(k+1)))^{- 1}y_{i}(k)\\ &+\frac{\mu_{i}(k)-1}{\mu_{i}(k)K_{i}(k)(b_{i}(k+1)-\
u_{i}(k+1)) }\Big{(}\sum_{j=1}^{n}c_{ji}(k)x_{j}(k)\Big{)}y_{i}(k)\end{split}  \tag{62}\]  
which can be rewritten as  
\[ \begin{split} y_{i}(k+1)&=(a_{i}(k)+h_{i}(k))y_{i} (k)=a_{i}(k)\Big{(}1+\frac{\mu_{i}(k)-1}{K_{i}(k)}\Big{(}\sum_{j=1}^{n}c_{ji}( k)x_{j}(k)\Big{)}\Big{)}y_{i}(k)\\ &=a_{i}(k)\Big{(}1+\frac{\mu_{i}(k)-1}{K_{i}(k)}+\frac{\mu_{i}(k) -1}{K_{i}(k)}\Big{(}\sum_{j=1}^{n}c_{ji}(k)x_{j}(k)\Big{)}\Big{)}y_{i}(k)\end{split}  \tag{63}\]  
where  
\[a_{i}(k)=(\mu_{i}(k)(b_{i}(k+1)-\
u_{i}(k+1)))^{-1} \tag{64}\]  
\[ \begin{split} h_{i}(k)&=\frac{\mu_{i}(k)-1}{\mu_{i }(k)K_{i}(b)(b_{i}(k+1)-\
u_{i}(k+1))}\Big{(}\sum_{j=1}^{n}c_{ji}(k)x_{j}(k) \Big{)}\\ &=\frac{(\mu_{i}(k)-1)a_{i}(k)}{K_{i}(k)}\Big{(}\sum_{j=1}^{n}c_ {ji}(k)x_{j}(k)\Big{)}\end{split}  \tag{65}\]  
The following result relies on the asymptotic stability of the non-extinction equilibrium point of the competition impulsive Beverton-Holt equation. **Theorem 5**. : _Let us assume that all the competing species have non-zero equilibrium points \(x_{i}\in\mathbf{R}_{+}\). Those equilibrium points are asymptotically stable, provided that the following constraints hold:_  
\[ \begin{split}&\text{(a) }a_{i}(k)+h_{i}(k)=a_{i}(k)\Big{(}1+\frac{\mu_{i}(k)-1}{K_{i}(k)}\Big{(} \sum_{j=1}^{n}c_{ji}(k)x_{j}(k)\Big{)}\Big{)}<1;\\ &\forall k\in\mathbf{N}_{0}\\ &\text{(b) }\limsup_{k\to\infty}\Big{[}a_{i}(k)\Big{(}1+\frac{\mu_{i}(k)-1}{K_{i }(k)}\Big{(}\sum_{j=1}^{n}c_{ji}(k)x_{j}(k)\Big{)}\Big{)}\Big{]}\leq 1\end{split} \]and that, furthermore, at least one of the subsequent four constraints hold for some set of real parameters \(\theta_{i}>0\); \(\forall i\in\overline{n}\):_  
\[