# 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
Thus, there is a subsequence \(\left\{x_{i}(k_{R})\right\}_{k=0}^{\infty}\subset\left\{x_{i}(k)\right\}\), with \(n_{k+1}-n_{k}<+\infty\), which is strictly increasing, so that \(\left\{x_{i}(n_{k})\right\}_{k=0}^{\infty}\rightarrow+\infty\) and then, from (7),  
\[0\leq\frac{x_{i}^{-}(k+1)}{x_{i}(k)}\leq\frac{\mu_{i}K_{ie}(k)}{(\mu_{i}-1)x_{ i}(k)};\;\forall k\in\mathbf{N}_{0}\]  
then \(\lim\limits_{k\rightarrow\infty}\frac{x_{i}^{-}(n_{k}+1)}{x_{i}(n_{k})}=0\). Furthermore,  
\[ \begin{array}{c}\frac{x_{i}^{-}(n_{k+1})}{x_{i}(n_{k})}=\left(\frac{x_{i}^{- }(n_{k+1})}{x_{i}(n_{k+1}-1)}\right)\left(\frac{x_{i}(n_{k+1}-1)}{x_{i}^{-}(n_ {k+1}-1)}\right)\\ \times\left(\frac{x_{i}^{-}(n_{k+1}-1)}{x_{i}(n_{k+1}-2)}\right)\left(\frac{x_ {i}(n_{k+1}-2)}{x_{i}^{-}(n_{k+1}-2)}\right)\ldots\left(\frac{x_{i}^{-}(n_{k} +1)}{x_{i}(n_{k})}\right)\\ \leq C(n_{k},j)\frac{x_{i}^{-}(n_{k}+1)}{x_{i}(n_{k})}\end{array}  \tag{11}\]  
where \(C(n_{k},j)\) is a positive real constant which is finite, since it is bounded to a finite number of iterations between \(n_{k}\) and \(n_{k+1}\), since \(n_{k+1}-n_{k}<+\infty\) and since one has, from Assumption 2, that  
\[x_{i}(k+1)=b_{i}(k+1)x_{i}^{-}(k+1)-d_{i}(k+1)\leq M_{i}<+\infty;\;\forall i \in\overline{n};\forall k\in\mathbf{N}_{0}\]  
and  
\[0\leq\frac{x_{i}(k+1)}{x_{i}^{-}(k+1)}=b_{i}(k+1)\;-\frac{d_{i}(k+1)}{x_{i}^{ -}(k+1)}\leq\frac{M_{i}}{x_{i}^{-}(k+1)}<+\infty \tag{12}\]  
if \(x_{i}^{-}(k+1)\
eq 0\) and \(\frac{x_{i}(k+1)}{x_{i}^{-}(k+1)}=\emptyset=+1<+\infty\) if \(x_{i}^{-}(k+1)=0\), still from Assumption 2. Then, from (11) and (12), one obtains \(\lim\limits_{k\rightarrow\infty}\frac{x_{i}(n_{k+1})}{x_{i}(n_{k})}=0\), since \(\left\{\frac{x_{i}^{-}(n_{k}+1)}{x_{i}(n_{k})}\right\}_{k=0}^{\infty}\) is bounded and \(\left\{x_{i}(n_{k})\right\}_{k=0}^{\infty}\) is strictly increasing from the contradiction argument hypothesis. Thus, there is a subsequence \(\left\{x_{i}\big{(}m_{n_{k}}\big{)}\right\}_{k=0}^{\infty}\subset\left\{x_ {i}(n_{k})\right\}\) such that \(\frac{x_{i}\big{(}m_{n_{k+1}}\big{)}}{x_{i}\big{(}m_{n_{k}}\big{)}}<1\), so that one concludes, from D'Alembert criterion for convergence of sequences, that \(\left\{x_{i}\big{(}m_{n_{k}}\big{)}\right\}_{k=0}^{\infty}\to 0\). As a result, \(\left\{x_{i}(k)\right\}_{k=0}^{\infty}\) is bounded. Property (iii) has been proved. \(\square\)  
Note that Theorem 2 (i) guarantees the boundedness of all the populations under Assumption 1, provided that the harvesting quota and the independent consumption are zero (so that there are no impulsive effects at sampling instants), while Theorem 2(iii) guarantees such a boundedness under Assumptions 1 and 2. In the following, the limits of the convergent sequences are denoted with the same notation as the sequences of elements by suppressing the sample argument \"k\" for the sake of notation simplicity. Theorem 3: Let us assume that all the populations converge to equilibrium points, that is, \(\left\{x_{i}(k)\right\}_{0}^{\infty}\to x_{i}\). Then, the following properties hold:  
(i) Let us assume that Assumptions 1 and 2 hold and that \(\left\{b_{i}(k)\right\}_{0}^{\infty}\to b_{i}\left(>\mu_{i}^{-1}\right)\), \(\left\{d_{i}(k)\right\}_{0}^{\infty}\to d_{i}\in\left[0,\overline{a}_{i} \right]\), \(\left\{c_{ji}(k)\right\}_{0}^{\infty}\to c_{ji}\in\left[0,\overline{c}_{ji} \right]\) for some \(\overline{a}_{i}\in\mathbf{R}_{0+}\), \(\overline{c}_{ji}\in\mathbf{R}_{0+}\);\(\forall j(\
eq i)\), \(i\in\overline{n}\). If \(\overline{d}_{i}\) and \(\overline{c}_{ji}\); \(\forall j(\
eq i)\), \(i\in\overline{n}\) are sufficiently small compared to \(K_{i}(b_{i}\mu_{i}-1)\), then there are two equilibrium points \(x_{i}\geq 0\); \(\forall i\in\overline{n}\).