# 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
\begin{split}\frac{\mu_{i}b_{i}-1}{\mu_{i}b_{i}}y_{i}^{-}& \leq\frac{\mu_{i}-1}{\mu_{i}K_{i}}+\frac{\mu_{i}-1}{\mu_{i}K_{i} }\Sigma_{j(\
eq i)=1}^{n}\,c_{ji}\left(b_{i}^{-1}+\frac{d_{i}}{b_{i}\left(b_{ i}-d_{i}y_{i}^{-}\right)}\right)+\frac{\mu_{i}^{-1}d_{i}}{b_{i}\left(b_{i}-d_{i}y_{i}^{ -}\right)}y_{i}^{-}\\ &\leq\frac{\mu_{i}-1}{\mu_{i}K_{i}}+\frac{\mu_{i}-1}{\mu_{i}K_{i }}\Sigma_{j(\
eq i)=1}^{n}\,c_{ji}\left(\frac{b_{i}-d_{i}y_{i}^{-}+d_{i}}{b_{i }\left(b_{i}-d_{i}y_{i}^{-}\right)}\right)+\frac{1}{\mu_{i}\left(b_{i}-d_{i}y_ {i}^{-}\right)}\end{split}  \tag{41}\]  
which is identical to (32); then, (33) is direct for the equilibrium values at the right of the sampling instants. The further two upper-bounds if \(d_{i}\leq 0\) inequalities follow directly from (32)-(33) and the proof is completed. \(\square\)  
The next definition relies on the concept of equilibrium consensus among the species in the sense that all of them reach the same equilibrium points. **Definition 1**. : _It is said that the \(n(\geq 2)\) competing species set has a left/right equilibrium consensus if there exist at least two, non-necessarily distinct, non-negative real numbers \(x^{-}\) and \(x\) such that \(x_{i}^{-}=x^{-}\) and \(x_{i}=x\); \(\forall i\in\overline{n}\), that is, the equilibrium point is common to all the species._  
Some sufficient conditions for equilibrium consensus are found in the subsequent result. **Proposition 4**. : _The following parametric constraints hold:_  
1. _A right equilibrium consensus for a given common population_ \(x\) _is reached under the independent consumptions_ \[d_{i}=\frac{(b_{i}\mu_{i}-1)K_{i}-(\mu_{i}-1)\left(\sum_{j=1}^{n}c_{ji}\right) x}{K_{i}+(\mu_{i}-1)\left(\sum_{j=1}^{n}c_{ji}\right)x}x;\,\,\forall i\in \overline{n}\] (42)
2. _A right equilibrium consensus for a common equilibrium population of value_ \[x=\frac{(b_{i}\mu_{i}-1)K_{i}}{(\mu_{i}-1)\left(\sum_{j=1}^{n}c_{ji}\right)}\] (43) _exists if_ \(d_{i}=0\)_;_ \(\forall i\in\overline{n}\) _with either_ \(\mu_{i}>max\left(1,\,\,b_{i}^{-1}\right)\) _or_ \(\mu_{i}<\min\left(1,\,\,b_{i}^{-1}\right)\) _for each_ \(i\in\overline{n}\) _for_ \(x>0\)_, or, jointly,_ \(\mu_{i}=b_{i}^{-1}\
eq 1;\,\forall i\in\overline{n}\) _for_ \(x=0\) _(that is, \"extinction\" consensus)._
3. _Let us assume that the competing species set has an equilibrium consensus and that there are no impulses at the equilibrium points, that is,_ \(b_{i}=1\) _and_ \(d_{i}=0\)_;_ \(\forall i\in\overline{n}\)_. Then, the equivalent carrying capacities_ \(K_{i\epsilon}=K_{\epsilon}\)_;_ \(\forall i\in\overline{n}\) _are identical for all the species, the carrying capacities are_ \(K_{i}=K_{\epsilon}\sum_{j=1}^{n}c_{ji}\)_;_ \(\forall i\in\overline{n}\) _and the nonzero common equilibrium point coincides with such a common equivalent carrying capacity, i.e.,_ \[x=x^{-}=\frac{K_{i}}{\sum_{j=1}^{n}c_{ji}}=K_{i\epsilon}=K_{\epsilon};\forall i \in\overline{n}.\]  
**Proof. ** One obtains, from (4), for \(x_{i}=x\); \(\forall i\in\overline{n}\) that  
\[ \begin{array}{c}(\mu_{i}-1)\left(\sum_{j=1}^{n}c_{ji}\right)x^{2}\\ =\left[(b_{i}\mu_{i}-1)K_{i}-d_{i}(\mu_{i}-1)\left(\sum_{j=1}^{n}c_{ji}\right) \right]x-d_{i}K_{i};\ \forall i\in\overline{n}\end{array}  \tag{44}\]  
which proves Property (i). Property (ii) follows directly from Property (i), by making \(d_{i}=0\); \(\forall i\in\overline{n}\), which leads to \(x=\frac{(b_{i}\mu_{i}-1)K_{i}}{(\mu_{i}-1)\left(\sum_{j=1}^{n}c_{ji}\right)}\); \(\forall i\in\overline{n}\). Since \(c_{ji}\geq 0\); \(\forall i,j\in\overline{n}\) implies that \(\frac{K_{i}}{\sum_{j=1}^{n}c_{ji}}=\frac{\mu_{i}-1}{b_{i}\mu_{i}-1}x\geq 0\), it follows that a necessary condition of fulfillment of those constraints for each \(i\in\overline{n}\) is that either \(\mu_{i}\geq max\Big{(}1,\ b_{i}^{-1}\Big{)}\) or \(\mu_{i}\leq\min\Big{(}1,\ b_{i}^{-1}\Big{)}\), or, jointly, \(\mu_{i}=b_{i}^{-1}\
eq 1\); \(\forall i\in\overline{n}\).