# 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}&\left[\begin{array}{c}x_{1}\\ x_{2}\end{array}\right]=\left[\begin{array}{cc}1&c_{21}\\ c_{12}&1\end{array}\right]^{-1}\left[\begin{array}{c}\frac{K_{1}(b_{1}\mu_{1} -1)}{\mu_{1}-1}\\ \frac{K_{3}(b_{2}\mu_{2}-1)}{\mu_{2}-1}\\ \end{array}\right]=\frac{1}{1-c_{12}c_{21}}\left[\begin{array}{cc}1&-c_{21} \\ -c_{12}&1\end{array}\right]\left[\begin{array}{c}\frac{K_{1}(b_{1}\mu_{1}-1)}{ \mu_{1}-1}\\ \frac{K_{3}(b_{2}\mu_{2}-1)}{\mu_{2}-1}\end{array}\right]\\ =\frac{1}{1-c_{12}c_{21}}\left[\begin{array}{c}\frac{K_{1}(b_{1}\mu_{1}-1)}{ \mu_{1}-1}-\frac{c_{12}K_{1}(b_{1}\mu_{1}-1)}{\mu_{1}-1}\\ \frac{K_{3}(b_{2}\mu_{2}-1)}{\mu_{2}-1}-\frac{c_{12}K_{1}(b_{1}\mu_{1}-1)}{\mu _{1}-1}\end{array}\right]\succ\succ 0\end{split}  \tag{26}\]  
_provided that \(K_{i}>0\) for \(i=1,2\) and either \(\mu_{i}>\text{max}\Big{(}1,\,b_{i}^{-1}\Big{)}\), or \(\mu_{i}<\text{min}\Big{(}1,\,b_{i}^{-1}\Big{)}\), for \(i=1,2\) and \(c_{12}\) and \(c_{21}\) are sufficiently small according to the relations_  
\[c_{ij}<\frac{K_{j}(\mu_{i}-1)\big{(}b_{j}\mu_{j}-1\big{)}}{K_{i}(\mu_{j}-1)(b_ {i}\mu_{i}-1)};\,i,j(\
eq i)=1,2 \tag{27}\]  
The equilibrium consensus, that is, the equilibrium points of both species are identical, \(x=x_{1}=x_{2}=\frac{a}{1-c_{12}c_{21}}>0\) for some prefixed \(a>0\) is achieved for the coupling coefficients satisfying  
\[c_{ij}=\left(\frac{K_{i}(b_{i}\mu_{i}-1)}{\mu_{i}-1}-a\right)\frac{\mu_{j}-1 }{K_{j}\big{(}b_{j}\mu_{j}-1\big{)}}\in[0,\,1);\,i,j(\
eq i)=1,2 \tag{28}\]  
so that the parameter \(a>0\) has to satisfy, furthermore,  
\[\frac{K_{i}(b_{i}\mu_{i}-1)}{\mu_{i}-1}-a<\frac{K_{j}\big{(}b_{j}\mu_{j}-1 \big{)}}{\mu_{j}-1};\,i,j(\
eq i)=1,2 \tag{29}\]  
or  
\[ \begin{split} a>\text{max}\Bigg{(}0,\underset{1\leq i,j(\
eq i )\leq 2}{\max}\Bigg{(}\frac{K_{j}\big{(}b_{j}\mu_{i}-1\big{)}}{\mu_{j}-1}-\frac{K_ {i}(b_{j}\mu_{i}-1)}{\mu_{i}-1},\,\frac{K_{i}(b_{j}\mu_{i}-1)}{\mu_{i}-1}- \frac{K_{i}(b_{j}\mu_{i}-1)}{\mu_{j}-1}\Bigg{)}\Bigg{)}\\ =\text{max}\Big{(}0,\Big{|}\frac{K_{2}(b_{2}\mu_{2}-1)}{\mu_{2}-1}-\frac{K_ {1}(b_{1}\mu_{1}-1)}{\mu_{1}-1}\Big{|}\Big{)}\end{split}  \tag{30}\]  
The generalization to \(n\geq 3\) species, if \(d_{i}=0;\,\forall i\in\mathbb{N}_{0}\), leads to a vector of equilibrium points given by  
\[ \begin{split} x=\left[\begin{array}{c}x_{1}\\ x_{2}\\ \cdots\\ x_{n}\end{array}\right]&=\left[\begin{array}{cccc}1&c_{21}&\cdots&c_{n1} \\ c_{12}&1&c_{32}&\cdots&c_{n2}\\ \cdots&\cdots&\cdots&\cdots\\ c_{1n}&c_{2n}&\cdots&1\end{array}\right]^{-1}\left[\begin{array}{c}\frac{K_ {1}(b_{1}\mu_{1}-1)}{\mu_{1}-1}\\ \vdots\\ \vdots\\ \frac{K_{n}(b_{n}\mu_{n}-1)}{\mu_{n}-1}\end{array}\right]\\ =\left(I_{n}+o(\varepsilon)\right)^{-1}\left[\begin{array}{c}\frac{K_{1}(b_{1} \mu_{1}-1)}{\mu_{1}-1}\\ \vdots\\ \frac{K_{n}(b_{n}\mu_{n}-1)}{\mu_{n}-1}\end{array}\right]\end{split}  \tag{31}\]  
provided that \(K_{i}>0\), \(\mu_{i}>\text{max}\Big{(}1,\,b_{i}^{-1}\Big{)}\); \(\forall i\in\overline{n}\) with \(c_{ij}\geq 0\); \(\forall i,j(\
eq i)\in\overline{3}\) and \(\underset{1\leq i,j(\
eq i)\leq n}{\max}c_{ij}\leq\varepsilon\) for some \(\varepsilon\in\mathbb{R}_{0+}\) so that, if \(\varepsilon\) is sufficiently small, then the inverse of \((I_{n}+o(\varepsilon))\) exists and \(x\succ\succ 0\), i.e., all its components, that is, the equilibrium states of all the species, are positive. The following technical result, which establishes equilibrium point upper-bounds when one of the species in the competing set is dominant to the left and right samples, holds. **Proposition 3. **_Let us define the population inverses \(y_{i}^{-}=1/x_{i}^{-}\) and \(y_{i}=1/x_{i}\) at the equilibrium points and let us assume that the \(i-th\) species is the dominant one in the set of \(n\) competing species in the set that has the best fitness.