# 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
To prove Property (iii), note that, since \(b_{i}=1\) and \(d_{i}=0\); \(\forall i\in\overline{n}\), one has, at the equilibrium points,  
\[x^{-}=x=\frac{\mu_{i}K_{i}x}{K_{i}+\sum_{j=1}^{n}c_{ji}(\mu_{i}-1)x} \tag{45}\]  
whose solutions are \(x=0\) and, after direct calculation for \(x\
eq 0\), one obtains  
\[x^{-}=x=\frac{K_{i}}{\sum_{j=1}^{n}c_{ji}}=K_{ie}=K_{e};\ \forall i\in \overline{n}. \tag{46}\]  
\(\square\)  
Let us assume, for generalization purposes, that the intrinsic growth rates, the carrying capacities and inter-species coupling coefficients might also be time-varying, in general. Thus, one obtains, from the resulting generalized (2) and (3) and, provided that \(x_{i}^{-}(k+1)\
eq 0\) and \(\lambda_{ji}(k)=x_{j}(k)/x_{i}(k)\), that  
\[ \begin{array}{c}y_{i}^{-}(k+1)=\frac{1}{x_{i}^{-}(k+1)}\\ =\mu_{i}^{-1}(k)y_{i}(k)+\frac{\mu_{i}(k)-1}{K_{i}(k)\mu_{i}(k)}+\frac{\mu_{i} (k)-1}{K_{i}(k)\mu_{i}(k)}\sum_{j(\
eq)=1}^{n}c_{ji}(k)\lambda_{ji}(k)\\ \\ y_{i}(k+1)=\rho_{i}^{-1}(k+1)y_{i}^{-}(k+1)\end{array}  \tag{47}\]  
for any \(i,j(\
eq i)\in\overline{n}\), where  
\[\rho_{i}(k+1)=x_{i}(k+1)/x_{i}^{-}(k+1)=b_{i}(k+1)-d_{i}(k+1)y_{i}^{-}(k+1)>0 \tag{49}\]  
for non-negativity and non-extinction in finite-time of the solution sequence, so that \(d_{i}(k+1)<b_{i}(k+1)x_{i}^{-}(k+1)\). Then, it follows, from (47) and (48), that  
\[ \begin{array}{c}y_{i}(k+1)=\left(\rho_{i}(k+1)\mu_{i}(k)\right)^{-1}\!\!y_ {i}(k)+\frac{\mu_{i}(k)-1}{K_{i}(k)\mu_{i}(k)\rho_{i}(k+1)}\\ +\frac{\mu_{i}(k)-1}{K_{i}(k)\mu_{i}(k)\rho_{i}(k+1)}\sum_{j(\
eq)=1}^{n}c_{ji }(k)\lambda_{ji}(k)\end{array}  \tag{50}\]  
for any \(i\in\overline{n}\). It can be proved that  
\[ \begin{array}{c}x_{i}(k+1)>x_{i}^{-}(k+1)\\ \Leftrightarrow d_{i}(k+1)<(b_{i}(k+1)-1)x_{i}^{-}(k+1)\Leftrightarrow\rho_{i }(k+1)>1\end{array}  \tag{51}\]  
\[ \begin{array}{c}x_{i}(k+1)<x_{i}^{-}(k+1)\\ \Leftrightarrow d_{i}(k+1)>(b_{i}(k+1)-1)x_{i}^{-}(k+1)\Leftrightarrow\rho_{i }(k+1)<1\end{array}  \tag{52}\]  
\[ \begin{array}{c}x_{i}(k+1)=x_{i}^{-}(k+1)\\ \Leftrightarrow d_{i}(k+1)=(b_{i}(k+1)-1)x_{i}^{-}(k+1)\Leftrightarrow\rho_{i }(k+1)=1\end{array}  \tag{53}\]for any \(i\in\overline{n}\). We first prove (51) in detail by using contradiction arguments and logics of propositions rules as follows. Let us assume that \(\rho_{i}(k+1)\leq 1\) for some given \(i\in\overline{n}\). Then, one obtains, from (49), that  
\[b_{i}(k+1)x_{i}^{-}(k+1)>d_{i}(k+1)\geq(b_{i}(k+1)-1)x_{i}^{-}(k+1)\]  
so that  
\[\rho_{i}(k+1)\leq 1\Rightarrow d_{i}(k+1)\geq(b_{i}(k+1)-1)x_{i}^{-}(k+1)\]  
which is the contrapositive logic proposition and then equivalent, logic proposition to the logic implication  
\[d_{i}(k+1)<(b_{i}(k+1)-1)x_{i}^{-}(k+1)\Rightarrow\rho_{i}(k+1)>1\]  
and the part \"\(\Rightarrow\)\" is proved in (51) for any \(i\in\overline{n}\) if \(x_{i}(k+1)>x_{i}^{-}(k+1)\). Now, let us assume that \(\rho_{i}(k+1)>1\) and \(d_{i}(k+1)\geq(b_{i}(k+1)-1)x_{i}^{-}(k+1)\). Then,  
\[ \begin{array}{c}\rho_{i}(k+1)=b_{i}(k+1)-d_{i}(k+1)y_{i}^{-}(k+1)\\ \leq b_{i}(k+1)-(b_{i}(k+1)-1)x_{i}^{-}(k+1)y_{i}^{-}(k+1)\\ =b_{i}(k+1)-(b_{i}(k+1)-1)x_{i}^{-}(k+1)y_{i}^{-}(k+1)=1\end{array} \]  
which contradicts that \(\rho_{i}(k+1)>1\). Then,  
\[\rho_{i}(k+1)>1\Rightarrow d_{i}(k+1)<(b_{i}(k+1)-1)x_{i}^{-}(k+1)\]  
which proves the part \"\(\Leftarrow\)\" of (51) for any \(i\in\overline{n}\). As a result, (51) has been proved for any \(i\in\overline{n}\). The proofs of (52) and (53) are very close to the one above and were therefore omitted.