# 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  
The subsequent assumptions are eventually made in the sequel in some of the technical results. **Assumption 1**. : _The carrying capacities, intrinsic growth rates and inter-species coupling dynamics are constant, i.e., \(K_{i}>0\), \(\mu_{i}>1\), \(c_{ii}=1\), \(c_{ij}\geq 0\); \(\forall i,j(\
eq i)\in\overline{n}\)._  
**Assumption 2**. : _The complementary harvesting quotas and independent consumption sequences satisfy the upper-bounding constraints \(0\leq b_{i}(k)\leq\overline{b}_{i}<+\infty\); \(-\infty<d_{i}(k)\leq b_{i}(k)x_{i}^{-}(k)\); \(\forall i\in\overline{n}\), \(\forall k\in\mathbf{N}_{0}\)._  
It can be pointed out that Assumption 1 can be extended directly to the case that the constant parameters are sequences depending on time. It can be also generalized for \(\mu_{i}\leq 1\) under the additional constraint \(\frac{K_{i}}{1-\mu_{i}}\geq\sum_{j=1}^{n}c_{ji}x_{j}(k);\forall i,j(\
eq i)\in \overline{n}\) or for its extension to the case when carrying capacities, intrinsic growth rates and inter-species coupling dynamics are sequences depending on time. The interpretation is that, if the intrinsic growth rate is under unity, then the coupling sequences from the remaining populations to the \(i-th\) have to be small enough related to the ratio \(K_{i}(k)/(1-\mu_{i}(k))\). Note that the populations stocks to the left and right limits of the sampling instants have to be non-negative for coherency of the model evolution. The constraints of Assumption 2 guarantee that the populations to the left limits of the sampling instants are non-negative if those to the right limits are non-negative. The following result on the non-negativity of the solution is directly obtained from (2)-(3) and Assumptions 1-2. **Proposition 1**. : _Under Assumptions 1-2, for all \(i\in\overline{n}\) and any given \(k\in\mathbf{N}_{0}\), if \(x_{i}(k)\geq 0\), then \(x_{i}^{-}(k+1)\geq 0\) and \(x_{i}(k+1)\geq 0\). If \(x_{i}(k)>0\), then \(x_{i}^{-}(k+1)>0\) and, also, \(x_{i}(k+1)>0\) if, furthermore, \(d_{i}(k+1)<b_{i}(k+1)x_{i}^{-}(k+1)\)._Assumptions 1-2 also lead to the subsequent result on the dominance (or best fitness) of a certain population on another one for all samples. Let us assume that, for any given \(i,j(\
eq i)\in\overline{n}\), \(\mu_{i}\geq\mu_{j}\), \(K_{i}\geq\overline{K}_{j}=\max\limits_{\ell\in n}\left(\frac{c_{\ell i}}{c_{\ell j }}\right)\frac{\mu_{i}(\mu_{i}-1)}{\mu_{j}(\mu_{j}-1)}K_{j}\), \(\left\{b_{i}(k)-b_{j}(k)\right\}_{k=1}^{\infty}\subset\R_{0+}\) and \(\left\{d_{j}(k)-d_{i}(k)\right\}_{k=1}^{\infty}\subset\R_{0+}\) and that Assumptions 1-2 hold. Thus, if \(x_{i}(0)\geq x_{j}(0)\), then \(\left\{x_{i}^{-}(k)-x_{j}^{-}(k)\right\}_{k=0}^{\infty}\subset\R_{0+}\) and \(\left\{x_{i}(k)-x_{j}(k)\right\}_{k=0}^{\infty}\subset\R_{0+}\). The above conditions may be relaxed as follows:  
If \(K_{i}\geq\overline{K}_{j}\), \(\mu_{i}\geq\mu_{j}\) and \(x_{i}(0)\geq\frac{\mu_{i}}{\mu_{i}}x_{j}(0)\), then \(\left\{x_{i}^{-}(k)-x_{j}^{-}(k)\right\}_{k=1}^{\infty}\subset\R_{0+}\) and \(\left\{x_{i}^{-}(k)-x_{j}^{-}(k)\right\}_{k=1}^{\infty}\subset\R_{0+}\),provided that \(d_{j}(k)-d_{i}(k)\geq b_{j}(k)x_{j}^{-}(k)-b_{i}(k)x_{i}^{-}(k)\); \(\forall k\in\N\). Note that  
\[ \begin{array}{l}x_{i}^{-}(1)=\frac{\mu_{i}x_{i}(0)}{1+\sum_{\ell=1}^{\infty }c_{\ell i}(\mu_{i}-1)K_{i}^{-1}x_{\ell}(0)}\\ \geq x_{j}^{-}(1)=\frac{\mu_{i}x_{j}(0)}{1+\sum_{\ell=1}^{\infty}c_{\ell j}( \mu_{j}-1)K_{j}^{-1}x_{\ell}(0)}\end{array}  \tag{5}\]  
if \(x_{i}(0)\geq x_{j}(0)\), \(\mu_{i}\geq\mu_{j}\) and  
\[0\leq\sum_{\ell=1}^{n}\left(c_{\ell i}(\mu_{i}-1)K_{i}^{-1}-c_{\ell j}(\mu_{j }-1)K_{j}^{-1}\right)x_{\ell}(0)\leq 0\]  
which holds if \(K_{i}\geq\overline{K}_{j}\) and \(\mu_{i}x_{i}(0)\geq\mu_{j}x_{j}(0)\), which leads to \(x_{i}^{-}(1)\geq x_{j}^{-}(1)\). Furthermore, \(x_{i}(1)\geq x_{j}(1)\) since \(x_{i}(0)\geq x_{j}(0)\), \(b_{i}(1)-b_{j}(1)\geq 0\) and \(d_{j}(1)-d_{i}(1)\geq 0\).