# 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
It turns out that, in the general case, the zeros of (17), namely,  
\[0\leq\frac{x_{i}(k+1)}{x_{i}^{-}(k+1)}=b_{i}(k+1)\;-\frac{d_{i}(k+1)}{x_{i}^{ -}(k+1)}\leq\frac{M_{i}}{x_{i}^{-}(k+1)}<+\infty \tag{12}\]8  
are both non-negative and real if \(\hat{b}_{i}>0\) and \(\hat{b}_{i}^{2}\geq 4(\mu_{i}-1)\hat{c}_{i}\) which hold for each \(i\in\overline{n}\) if \(\overline{d}_{i}\) and \(\overline{c}_{ji}\); \(\forall j(\
eq i)\) are sufficiently small in view of (18) and (19), such that \(\hat{c}_{i}\) is sufficiently small, compared to \(K_{i}(b_{i}\mu_{i}-1)\) (that is, sufficiently small related to the first additive term of \(\left|\hat{b}_{i}\right|\)) from the property of the continuity of the zeros of a polynomial with respect to its coefficients. Property (i) has been proved. Property (ii) is now proved by guaranteeing the validity of \(\hat{b}_{i}^{2}\geq 4(\mu_{i}-1)\hat{c}_{i}\) depending on the coupling equilibrium dynamics from the remaining species \(j(\
eq i)\in\overline{n}\) to the \(i-th\) one. First, let us define the auxiliary amount \(\Delta_{i}=(\mu_{i}-1){\sum}_{j(\
eq i)=1}^{n}c_{ji}x_{j}\) so that, from (18) and (19),  
\[0\leq\frac{x_{i}(k+1)}{x_{i}^{-}(k+1)}=b_{i}(k+1)\;-\frac{d_{i}(k+1)}{x_{i}^{ -}(k+1)}\leq\frac{M_{i}}{x_{i}^{-}(k+1)}<+\infty \tag{12}\]9  
\[B_{i}=[K_{i}(b_{i}\mu_{i}-1)-(\mu_{i}-1)d_{i}]\geq 0\]0  
then  
\[ \begin{array}{l}g(\Delta_{i})=\hat{b}_{i}^{2}-4(\mu_{i}-1)\hat{c}_{i}=(K_{i }(b_{i}\mu_{i}-1)-(\mu_{i}-1)d_{i}-\Delta_{i})^{2}-4(\mu_{i}-1)(d_{i}(K_{i}+ \Delta_{i}))\\ =\Delta_{i}^{2}-2[K_{i}(b_{i}\mu_{i}-1)-(\mu_{i}-1)d_{i}]\Delta_{i}+(K_{i}(b_{ i}\mu_{i}-1)-(\mu_{i}-1)d_{i})^{2}-4(\mu_{i}-1)d_{i}K_{i}-4(\mu_{i}-1)d_{i} \Delta_{i}\\ =\Delta_{i}^{2}-2[K_{i}(b_{i}\mu_{i}-1)-(\mu_{i}-1)d_{i}]\Delta_{i}+(K_{i}(b_{ i}\mu_{i}-1)-(\mu_{i}-1)d_{i})^{2}-4(\mu_{i}-1)d_{i}K_{i}\\ =\Delta_{i}^{2}-2B_{i}\Delta_{i}+C_{i}\geq 0\end{array}  \tag{21}\]  
holds, where  
\[B_{i}=[K_{i}(b_{i}\mu_{i}-1)-(\mu_{i}-1)d_{i}] \tag{22}\]  
\[C_{i}=(K_{i}(b_{i}\mu_{i}-1)-(\mu_{i}-1)d_{i})^{2}-4(\mu_{i}-1)d_{i}K_{i}=B_{i }^{2}-4(\mu_{i}-1)d_{i}K_{i} \tag{23}\]  
Let us consider the subsequent two cases:  
_Case 1_ (\(g(\Delta_{i})=0\)_has real zeros_\(\Delta_{i1,i2}\)). For the above constraint (21) to be well-posed with real zeros \(\Delta_{i1,i2}\) of \(g(\Delta_{i})=0\), with \(\Delta_{i2}=B_{i}+\sqrt{B_{i}^{2}-C_{i}}\geq\Delta_{i1}=B_{i}-\sqrt{B_{i}^{2} -C_{i}}\) (according to the definition of \(\Delta_{i}\)), so that \(g(\Delta_{i})\geq 0\) if and only if \({\sum}_{j(\
eq i)=1}^{n}c_{ji}x_{j}\in\left[max\left(0,\frac{B_{i}+\sqrt{B_{i}^ {2}-C_{i}}}{\mu_{i}-1}\right),\infty\right)\), a necessary condition being that \(B_{i}^{2}\geq C_{i}\), equivalently, \((\mu_{i}-1)d_{i}\geq 0\). Provided that such a necessary condition holds, sufficient conditions for \(\Delta_{i2}=\geq 0\) are as follows:  
(a) \(B_{i}\geq 0\), equivalently, \(0\leq d_{i}\leq\frac{K_{i}(b_{i}\mu_{i}-1)}{\mu_{i}-1}\), which, combined with the above necessary condition, also implies that \(b_{i}\geq\mu_{i}^{-1}\);  
(b) \(B_{i}<0\) together with \(C_{i}\leq 0\). These two joint sufficiency-type conditions are respectively equivalent to \(d_{i}>\frac{K_{i}(b_{i}\mu_{i}-1)}{\mu_{i}-1}\) and the implicit constraint \(d_{i}\geq\underline{d}_{i}=\frac{B_{i}^{2}}{4(\mu_{i}-1)K_{i}}=\frac{(\lambda_ {i}-\mu_{i}+1)^{2}d_{i}^{2}}{4(\mu_{i}-1)K_{i}}\), obtained after defining the auxiliary parameter \(\lambda_{i}=\frac{K_{i}(b_{i}\mu_{i}-1)}{d_{i}}\) for \(d_{i}\
eq 0\). Note that the particular case \(d_{i}=0\) requires, for feasibility, that \(B_{i}\geq 0\) (from the necessary condition \(B_{i}^{2}\geq C_{i}\)), so that it is included in the sufficient condition \(B_{i}\geq 0\) (or \(d_{i}\leq\frac{K_{i}(b_{i}\mu_{i}-1)}{\mu_{i}-1}\)), that is, it is included in (a).