# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 2 The Competition Equation and Its Main Properties
### Non-Negativity and Boundedness of the Solutions
Now, note, from (49), that, for any given \(i\in\overline{n}\),  
\[\frac{1}{\rho_{i}(k+1)\mu_{i}(k)}=\frac{1}{\mu_{i}(k)\big{(}b_{i}(k+1)-d_{i}(k +1)y_{i}^{-}(k+1)\big{)}}<1 \tag{54}\]  
if \(\mu_{i}(k)\big{(}b_{i}(k+1)-d_{i}(k+1)y_{i}^{-}(k+1)\big{)}>1\), equivalently, if \(x_{i}^{-}(k+1)>\frac{d_{i}(k+1)\mu_{i}(k)}{\mu_{i}(k)b_{i}(k+1)-1}\) and, in a similar way, one has  
\[\frac{1}{\rho_{i}(k+1)\mu_{i}(k)}>1\text{ if }x_{i}^{-}(k+1)<\frac{d_{i}(k+1)\mu_ {i}(k)}{\mu_{i}(k)b_{i}(k+1)-1} \tag{55}\]  
\[\frac{1}{\rho_{i}(k+1)\mu_{i}(k)}=1\text{ if }x_{i}^{-}(k+1)=\frac{d_{i}(k+1) \mu_{i}(k)}{\mu_{i}(k)b_{i}(k+1)-1} \tag{56}\]  
Extinction at the limit as time tends to infinity needs some extra constraints, as follows, since \(x_{i}(k+1)<\frac{d_{i}(k+1)\mu_{i}(k)}{\mu_{i}(k)b_{i}(k+1)-1}\) does not hold at the limit with the left-hand side and right-hand side both converging to zero. So, the convergence to extinction of \(x_{i}(k)\) as \(k\to\infty\) implies that \(x_{i}(k)=b_{i}(k)x_{i}^{-}(k)-d_{i}(k)\to 0\) as \(k\to\infty\), then, equivalently, \(x_{i}^{-1}(k)=y_{i}(k)\to\infty\) as \(k\to\infty\) and we convey that also \(x_{i}^{-}(k)\to 0\) (basically, the asymptotic convergences to the right and left of the sampling instants jointly hold), \(b_{i}(k)\to 1\) and \(d_{i}(k)\to 0\) as \(k\to\infty\) with \(\frac{d_{i}(k)}{x_{i}(k)}\to 0\) as \(k\to\infty\) (so that \(d_{i}(k)=o\Big{(}x_{i}^{-1}(k)\Big{)}\) as \(k\to\infty\)) and then \(\rho_{i}(k)=b_{i}(k)-\frac{d_{i}(k)}{x_{i}(k)}\to 1\) as \(k\to\infty\) from (49). The following result, whose proof involves the use of the above considerations, is concerned with non-asymptotic and asymptotic extinction conditions. **Theorem 4**. : _The following properties hold:_  
1. _Let us assume that the_ \(i-th\) _species has the best fitness,_ \(x_{i}(0)>0\) _is finite, that the_ \(i-th\) _independent consumption fulfills_ \(d_{i}(k+1)<b_{i}(k+1)x_{i}(k+1)\)_;_ \(\forall k\in\mathbf{N}_{0}\) _and that there is a strict sequence of non-negative integer numbers \(\{k_{\ell}\}_{\ell=0}^{\infty}\subset\mathbf{N}_{0}\) with \(k_{\ell+1}-k_{\ell}\leq\overline{k}<+\infty\); \(\forall\ell\in\mathbf{N}_{0}\) such that \(\prod_{j=k_{\ell}}^{j=k_{\ell+1}}[\rho_{i}(k_{\ell+1}+1-j)\mu_{i}(k_{\ell+1}-j) ]\geq\underline{\rho}>1\). Then, there is no asymptotic extinction of the \(i-th\) species with the best fitness and its solution sequence is bounded for all time._
2. _Let us assume that \((1-\mu_{i})\sum_{j=1}^{n}c_{ji}x_{i}(0)<1\), \(b_{i}(k)\to 1\), \(d_{i}(k)\to 0\) and \(\frac{d_{i}(k)}{x_{i}(k)}\to 0\) as \(k\to\infty\) and that \(\liminf\limits_{k\to\infty}\mu_{i}(k)\leq 1\) with \(\sum_{k=1}^{\infty}(\mu_{i}(k)-1)=+\infty\). Then, there is asymptotic extinction of the_ i-th _species with the best fitness and then that of all the remaining ones._  
Proof. : From recursive calculations by using (50), the inverse of the \(i-th\) population is given by  
\[ \begin{array}{c}y_{i}(k+1)=\prod_{j=0}^{k}\Big{[}(\rho_{i}(k+1-j)\mu_{i}(k- j))^{-1}\Big{]}y_{i}(0)\\ +\sum_{j=0}^{k}\prod_{\ell=j}^{k-1}\big{[}(\rho_{i}(\ell+2)\mu_{i}(\ell+1))^{-1 }\big{]}\frac{\mu_{i}(j)-1}{k_{i}(j)\mu_{i}(j)\rho_{i}(j+1)}(\sum_{\ell=1}^{n} c_{\ell i}(j)\lambda_{\ell i}(j))\end{array}  \tag{57}\]  
with the notation convention \(\prod_{\ell=j}^{k-1}\big{[}(\rho_{i}(\ell+2)\mu_{i}(\ell+1))^{-1}\big{]}=1\) for \(j=k\), \(\lambda_{ii}(j)=1\), \(\lambda_{\ell i}(j)\leq 1\); \(\forall\ell(\
eq i)\in\overline{n}\) since the \(i-th\) species has the best fitness. Since there exists a strictly increasing sequence \(\{k_{\ell}\}_{\ell=0}^{\infty}\subset\mathbf{N}_{0}\) with \(k_{\ell+1}-k_{\ell}\leq\overline{k}<+\infty\); \(\forall\ell\in\mathbf{N}_{0}\) such that  
\[ \begin{array}{c}\prod_{j=k_{\ell}}^{j=k_{\ell+1}}[\rho_{i}(k_{\ell+1}+1-j) \mu_{i}(k_{\ell+1}-j)]\geq\underline{\rho}>1\text{ then }\\ \lim\limits_{k\to\infty}\prod_{j=0}^{k}\big{[}\rho_{i}(k+1-j)\mu_{i}(k-j)^{-1 }\big{]}\end{array}  \tag{58}\]  
and  
\[