# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 2 The Competition Equation and Its Main Properties  
### The Single-Species Impulsive Beverton-Holt Equation  
The subsequent impulsive discrete Beverton-Holt equation for a single species, [26; 27; 33] extends the standard one, which has been described in the introduction:  
\[x^{-}(k+1)=\frac{\mu(k)K(k)x(k)}{K(k)+(\mu(k)-1)x(k)},\]  
\[x(k+1)=x^{+}(k+1)=b(k+1)x^{-}(k+1)-d(k+1);\forall k\in\mathbf{N}_{0} \tag{1}\]with given finite initial conditions \(x(0)\geq 0\), where \(\mu(k)\in R_{0+}:=R_{+}\cup\{0\}\) is the so-called _intrinsic growth rate_ of the population, determined by life cycle and demographic properties (such as, for instance, species growth rate, survivorship rate, etc.) and \(K(k)\in R_{+}\) is the so-called _carrying capacity_, a characteristic of the habitat dependent on resource availability, temperature, humidity, etc. Typically, the constraints \(\mu(k)>1\) and \(K(k)\in R_{+}\) for all \(k\in N_{0}\) are assumed for the Beverton-Holt equation, since if, otherwise, the carrying capacity is zero, then the habitat is incompatible with the species evolution within it; if \(\mu(k)\equiv 1\), then the species stock remains constant through time; if \(0<\mu(k)<1\), then \(x^{-}(k+1)/x(k)=\mu(k)/[1-(1-\mu(k))x(k)/K(k)]\), so that two special atypical situations can happen, namely, if \(1>\mu(k)>1-(1-\mu(k))x(k)/K(k)\) for the initial samples and there is a sufficiently small population stock, which keeps the right-hand-side positivity, then the species stock grows over time, but it can tend to negative values, making the model to be not well-posed for a sufficiently extended time. If, on the contrary, \(\mu(k)<1-(1-\mu(k))x(k)/K(k)\) for all samples, then the population asymptotically extinguishes. On the other hand, the impulsive parameters, which can modify the stocks of population, to the right of the sampling instants related to their left-values are _the harvesting quota_\((1-b(k))\in R_{+}\); \(\forall k\in N_{0}\), which depends on the authorities policy depending on the spawning stock and foreseen recruitments and the _independent consumption_\(d(k)(\leq b(k)x(k))\in R_{i}\); \(\forall k\in N_{0}\) that implies recruitment variations depending on unforeseen disturbances, artificial re-population actions and local migrations \"towards\" or \"from\" the studied habitat. For easy presentation of some of the technical results, the _complementary harvesting quota_\(b(k)\in R_{0+}\); \(\forall k\in N_{0}\) is used. Some particular results in this paper are given for the case of intrinsic growth rates under unity and for negative harvesting quotas, or, equivalently, for complementary harvesting quotas exceeding unity. The physical interpretation of this situation is that a re-population with a number of adults exceeding to population stock is performed with a ratio larger than unity. **Remark 1**. : _Note that the public regulations on fishing or hunting taking into account the stocks of population imply that \(b(k)\leq 1\); \(\forall k\in N_{0}\). However, if a repopulation with adults is intended, which is sometimes the case, then it can be decided to increase the above coefficient exceeding unity in the case that the population stock is low. Therefore, we admit, at the level of a general formulation, that the complementary harvesting quota is non-negative and bounded. It turns out that the independent consumption is positive, if there is a net flux of individuals from the studied habitat to outside of it, so that the population stock decreases, and it is negative, if there is a net flux entering the habitat, so that the population stock increases. Finally, it is zero if there is no interchange of population of the habitat with the environment. The independent consumption can also take into account repopulation actions that are not linked to a proportion on the existing population stock._  
Note that, if \(x(0)=0\), or if \(K(0)=0\) and \(\{d(k)\}_{k=0}^{\infty}\equiv 0\), then \(\{x(k)\}_{k=0}^{\infty}\equiv 0\).