# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 2 The Competition Equation and Its Main Properties
### The Competition Impulsive Beverton-Holt Equation  
The class of impulsive discrete Beverton-Holt Equation of \(n\) competing species proposed for further analysis is a generalization of the discrete Beverton-Holt Equation (1) for the particular case of a single species, i.e., for \(n=1\), with a constant parameterization of intrinsic growth rates and carrying capacities through time being expressed for \(n\) competing species as  
\[x_{i}^{-}(k+1)=\frac{\mu_{i}K_{i}x_{i}(k)}{K_{i}+\sum_{j=1}^{n}c_{ji}(\mu_{i}- 1)x_{j}(k)} \tag{2}\]  
\[x_{i}(k+1)=b_{i}(k+1)x_{i}^{-}(k+1)-d_{i}(k+1);\,\,\forall i\in\overline{n}=\{ 1,2,\ldots,n\},\forall k\in\mathbb{N}_{0} \tag{3}\]  
with given finite initial conditions \(x_{i}(0)\geq 0\). For some coupling coefficients \(c_{ij}\geq 0\); \(\forall i,j\in\overline{n}\) with \(c_{ii}=1\); \(\forall i\in\overline{n}\). The subscripts stand for the populations and model parameters of the various competing species, while each current sample is denoted in parenthesis. The superscript \"-\" stands for the population values to the left of a sampling instant, while the absence of superscript indicates its value to the right of sampling instant. Note that it was assumed, for the sake of exposition simplicity, that the intrinsic growth rate, carrying capacity and inter-species coupling dynamics coefficients are constant through time for each one of the species. The impulsive control parameters are the members of the real sequences \(\{b_{i}(k)\}_{k=1}^{\infty}\) and \(\{d_{i}(k)\}_{k=1}^{\infty}\); \(\forall i\in\overline{n}\). Those sequences may modify the levels of populations to the right of the sampling instants. If those sequences are identically zero for any population at a sampling instant, then its value after sampling remains unaltered. The combination of (2) and (3) yields  
\[x_{i}(k+1)=\frac{[b_{i}(k+1)\mu_{i}K_{i}-d_{i}(k+1)(\mu_{i}-1)]x_{i}(k)-d_{i}(k +1)\Big{(}K_{i}(k)+\sum_{j(i)=1}^{n}c_{ji}(k)(\mu_{i}-1)x_{j}(k)\Big{)}}{K_{i} ()+\sum_{j=1}^{n}c_{ji}(\mu_{i}-1)x_{j}(k)};\forall k\in\mathbb{N}_{0},\,\, \forall i\in\overline{n}. \tag{4}\]  
It is of interest to summarize, first, some basic properties of the impulsive time-invariant single Beverton-Holt equation for only one species to then interpret some parallel properties associated with a tandem of competing species. It is apparent that the equilibrium point exists also if the left and right values at the sampling time instants are non-negative and finite, even if they are distinct. In the case that they are distinct, there is an oscillatory equilibrium solution switching among those left and right values at the sampling time instants. In the case that the four parameters are potentially time-varying,the mentioned impulsive Beverton-Holt equation for a single species (that is, \(n=1\)) is the following one:  
\[x^{-}(k+1)=\frac{\mu(k)K(k)x(k)}{K(k)+(\mu(k)-1)x(k)};\]  
\[x(k+1)=b(k+1)x^{-}(k+1)-d(k+1)\]  
that, combined, yield  
\[\begin{array}{l}x(k+1)=b(k+1)x^{-}(k+1)-d(k+1)\\ =\frac{[\mu(k)b(k+1)K(k)-d(k+1)(\mu(k)-1)]x(k)-d(k+1)K(k)}{K(k)+(\mu(k)-1)x(k)} \end{array}\]