# 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
Then, one obtains recursively that \(\left\{x_{i}^{-}(k)-x_{j}^{-}(k)\right\}_{k=0}^{\infty}\subset\R_{0+}\) and \(\left\{x_{i}(k)-x_{j}(k)\right\}_{k=0}^{\infty}\subset\R_{0+}\), as claimed in the first part of the proposition. The second part is direct, since, under the assumptions that \(K_{i}\geq K_{j}\) and \(x_{i}(0)\geq\frac{\mu_{i}}{\mu_{i}}x_{j}(0)\), one obtains \(\mu_{i}K_{i}x_{i}(0)\geq\mu_{j}K_{j}x_{j}(0)\) (not necessarily under the joint constraints \(x_{i}(0)\geq x_{j}(0)\), \(\mu_{i}\geq\mu_{j}\) and \(K_{i}\geq K_{j}\)). This implies that \(x_{i}^{-}(1)\geq x_{j}^{-}(1)\), which, together with \(d_{j}(1)-d_{i}(1)\geq b_{j}(1)x_{j}^{-}(1)-b_{i}(1)x_{i}^{-}(1)\) (not necessarily with \(d_{j}(1)\geq d_{1}(1)\)), implies that \(x_{i}(1)\geq x_{j}(1)\). The result for \(k\geq 1\) follows recursively. \(\square\)  
It follows, from Proposition 2, that a population \(i\in\overline{n}\) is dominant on the remaining ones, or, in other words, it has the best fitness, for all \(j(\
eq i)\in\overline{n}\) and the respective left and right sampled values if \(K_{i}\geq\max\limits_{(j\
eq i)\in\overline{n}}\overline{K}_{j}\), \(x_{i}(0)\geq\frac{1}{\mu_{i}}\max\limits_{(j\
eq i)\in\overline{n}}\left(\mu_ {i}x_{j}(0)\right)\) and \(\min\limits_{(j\
eq i)\in\overline{n}}\left(d_{j}(k)-b_{j}(k)x_{j}^{-}(k) \right)\geq d_{i}(k)-b_{i}(k)x_{i}^{-}(k)\); \(\forall k\in\N\). It turns out, furthermore, that an increase in an individual carrying capacity facilitates the increase in the corresponding species numbers, if all the remaining parameters remaining unchanged. The above constraints generalize Proposition 2, although at the expense of involving conditions on the left amounts of the population stocks. It can be pointed out that, eventually, there could exist more than one species with an identical best fitness within the set of competing species. The following result is now proved on non-negativity, boundedness and eventual extinction of the species populations. The following properties hold:  
(i): _If_ \(\infty>x_{i}(0)>0\); \(\forall i\in\overline{n}\)_, Assumption 1 holds,_ \(\left\{b_{i}(k)\right\}_{0}^{\infty}\equiv 1\) _and_ \(\left\{d_{i}(k)\right\}_{0}^{\infty}\equiv 0\)_;_ \(\forall i\in\overline{n}\)_; then,_ \(\left\{x_{i}(k)\right\}_{k=0}^{\infty}\) _is non-negative and bounded;_ \(\forall i\in\overline{n}\)_. Furthermore, if_ \(\left\{x_{i}(k)\right\}_{0}^{\infty}\to 0\) _(i.e., asymptotic extinction of the_ \(i\)_th species), then_ \(\liminf\limits_{k\to\infty}\inf\limits_{0<k<\infty}K_{i\ell}(k)=0\)_._
2. _If_ \(\infty>x_{i}(0)>0\)_;_ \(\forall i\in\overline{n}\) _and Assumptions_ 1 _and_ 2 _hold with_ \(\{d_{i}(k)\}_{k=0}^{\infty}\subset\mathbf{R}_{0+}\)_, then and it is a bounded non-negative sequence if_ \(\prod_{k=0}^{\infty}[{\mu_{i}b_{i}(k)}]=0\)_;_ \(\forall i\in\overline{n}\)_. This property holds, in particular, if_ \(b_{i}(k)>\mu_{i}^{-1}\)_;_ \(\forall k\in\mathbf{N}_{0}\)_. If, furthermore,_ \(0\leq d_{i}(k)<b_{i}(k)x_{i}^{-}(k)\)_;_ \(\forall k\in\mathbf{N}_{0}\)_, then extinction in finite time does not occur for any_ \(i\in\overline{n}\)_, so that extinction only happens asymptotically and_ \(\{x_{i}(k)\}_{k=0}^{\infty}\) _is a bounded non-negative sequence if_ \(\prod_{k=0}^{\infty}[{\mu_{i}b_{i}(k)}]\leq A<\infty\)_. In particular, if_ \(\prod_{k=j}^{i+a-1}[{b_{i}(k)}]\leq\mu_{i}^{-a}\)_;_ \(\forall k(\geq j_{0})\in\mathbf{Z}_{0+}\) _and some finite_ \(j_{0}\in N_{0}\) _and_ \(a\in\mathbf{N}\)_, then_ \[x_{i}^{-}(k+1)=\frac{\mu_{i}K_{i}x_{i}(k)}{K_{i}+\sum_{j=1}^{n}c_{ji}(\mu_{i}- 1)x_{j}(k)}\] (6) _If Assumptions_ 1 _and_ 2 _hold with_ \(\{d_{i}(k)\}_{k=0}^{\infty}\subset R\)_, then_ \(\{x_{i}(k)\}_{k=0}^{\infty}\to 0\) _if_ \(\prod_{k=0}^{\infty}[{\mu_{i}b_{i}(k)}-{d_{i}(k+1)}]=0\)_._
3. _If Assumptions_ 1 _and_ 2 _hold, then_ \(\{x_{i}(k)\}_{k=0}^{\infty}\) _is bounded;_ \(\forall i\in\overline{n}\)_._  
**Proof.