# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 2 The Competition Equation and Its Main Properties
### Equilibrium Points  
The subsequent result holds for the case of one single species. **Theorem 1**. : _Let us assume that either the intrinsic growth rates, carrying capacities, harvesting quotas and independent consumptions are constant parameters, or that the related parameterizing sequences converge asymptotically to constant parameters. Then, the following properties hold:_  
_(i) There exist equilibrium points (or, more properly, left and right limits of equilibrium points)_  
\[x_{1,2}=\frac{\mu bK-d(\mu-1)-K\pm\sqrt{\left(\mu bK-d(\mu-1)-K\right)^{2}-4( \mu-1)dK}}{2(\mu-1)}\]  
\[x_{1,2}^{-}\geq\frac{x_{1,2}+d}{b}\]  
_with physical sense, or feasible, (i.e., being real non-negative) of the single-species impulsive Beverton-Holt equation if and only if_  
\[d\in\left[-x_{1},\ \min\left(\frac{\left(\sqrt{\mu b}-1\right)^{2}K}{\mu-1},\ \frac{(\mu b-1)K}{\mu-1}\right)\right]\cup\left[\frac{\left(\sqrt{\mu b}+1 \right)^{2}K}{\mu-1},\infty\right)\]  
_If \(\mu\geq\max\bigl{(}1,\ b^{-1}\bigr{)}\), then the above constraint becomes_  
\[d\in\left[-x_{1},\ \frac{\left(\sqrt{\mu b}-1\right)^{2}K}{\mu-1}\right]\cup \left[\frac{\left(\sqrt{\mu b}+1\right)^{2}K}{\mu-1},\ \infty\right)\]  
_(ii) If \(d=0\) and \(b=1\), then there are no impulses at the equilibrium steady-state, so that the equilibrium solution is not oscillatory between its left and right values at the sampling instants, that is, \(x_{1}=x_{1}^{-}=0\) (extinction) and \(x_{2}=x_{2}^{-}=K\)._  
_If \(d=0\), then the equilibrium points become \(x_{1}=0\) (extinction) and \(x_{2}=\frac{(\mu b-1)K}{\mu-1}>0\), provided that either \(\mu>\max\bigl{(}1,\ b^{-1}\bigr{)}\) or \(\mu<\min\bigl{(}1,\ b^{-1}\bigr{)}\). If, in addition, \(\mu>1\) and \(b=\mu^{-1}\), then \(x_{1}=x_{2}=0\)._  
_(iii) If \(d=\frac{(\mu b-1)K}{\mu-1}<0\), then there is only a non-negative equilibrium point \(x_{1}=\sqrt{\left|\frac{d}{\mu-1}\right|}K>0\) with either \(\mu>1\) and \(d<0\) or \(\mu<1\) and \(d<0\)._  
_(iv) If \(d=\frac{\mu bK}{\mu-1}\), then there is only a non-negative equilibrium point \(x_{1}=\frac{\sqrt{1+4(1-\mu b)}-1}{2(\mu-1)}K>0\) with \(\mu>1\) and \(b\in\bigl{[}0,\mu^{-1}\bigr{)}\). Let us assume that negative harvesting quotas are allowed, then \(x_{1}=\frac{1+\sqrt{4\mu b-3}}{2(1-\mu)}>0\), provided that \(b>\max\bigl{(}\mu^{-1},\ \frac{3}{4}\mu\bigr{)}\)._  
**Proof. ** The equilibrium point to the left and right of the sampling instants satisfies \(x^{-}=\frac{x+d}{b}\) subject to the constraint  
\[x=\frac{[\mu b\ K-d(\mu-1)]x-dK}{K+(\mu-1)x}\]  
The above constraint may be rewritten as  
\[(\mu-1)x^{2}-[\mu bK-d(\mu-1)-K]x+dk=0\]  
whose zeros \(x_{1},x_{2}\geq x_{1}\) are  
\[x_{1,2}=\frac{\mu bK-d(\mu-1)-K\pm\sqrt{(\mu bK-d(\mu-1)-K)^{2}-4(\mu-1)dK}}{2( \mu-1)},\]  
which have to be real and non-negative for the equilibrium point to have a physical sense, so that two joint necessary conditions are  
\[d\leq d_{0}=\frac{(\mu b-1)K}{\mu-1};(\mu bK-d(\mu-1)-K)^{2}\geq 4(\mu-1)dK\]  
The second constraint can be re-arranged in the form  
\[(\mu-1)^{2}d^{2}-2[(\mu b-1)+2]K(\mu-1)d+(\mu b-1)^{2}K^{2}\geq 0\]  
Defining the auxiliary variable \(f=(\mu-1)d\), the above inequality takes the form  
\[g(f)=f^{2}-2(\mu b+1)Kf+(\mu b-1)^{2}K^{2}\geq 0\]  
One obtains that the zeros of the left-hand-side of the above constraint are  
\[x_{1,2}^{-}\geq\frac{x_{1,2}+d}{b}\]0  
and the constraint \(g(f)\geq 0\) holds, since \(g(f)=0\) is a convex parabola of zeros \(f_{1},\ f_{2}\geq f_{1}\), if any of the two constraints below holds  
\[x_{1,2}^{-}\geq\frac{x_{1,2}+d}{b}\]1  
In order that, in addition, \(x_{1,2}^{-}\geq\frac{x_{1,2}+d}{b}\geq 0\) (in order for the equilibrium points to the left of the sampling instants to be non-negative as well), \(d\geq-x_{1}\) since \(0\leq x_{1}\leq x_{2}\). Furthermore, \(d\in[-x_{1},\ \min(d_{0},d_{1})]\cup[d_{2}+\infty)\). If \(\mu\geq\max(1,\ b^{-1})\), then \(\big{(}\sqrt{\mu b}-1\big{)}^{2}\leq\mu b-1\). Property (i) follows directly from the above constraints.