# On the Properties of a Class of Impulsive Competition Beverton-Holt Equations
## 3 Stability Results Obtained by Lyapunov Method
Since the numerators of their respective general terms are finite, then either the denominators converge to zero or they have some element which is zero, that is, either \(b_{i}(k)-d_{i}(k)y_{i}^{-}(k)(\geq 0)\to 0\) as \(k\to\infty\) for some \(i\in\overline{n}\) or there are some \(k\in N_{0}\) and some \(i\in\overline{n}\), such that \(b_{i}(k)-d_{i}(k)y_{i}^{-}(k)=0\). However, these two sub-cases imply, respectively, \(\left\{x_{i}(k)\right\}_{k=0}^{\infty}\to 0\) and \(x_{i}(k)=0\), which implies that \(x_{i}^{-}(k+1)=0\) and \(x_{i}(k+1)=-d_{i}(k+1)=0\) from the assumption (b). In both cases, extinction is achieved either asymptotically or in a finite time and the claim of Case 2 that \(\left\{y_{i}(k)\right\}_{k=0}^{\infty}\) is bounded cannot hold for any \(i\in\overline{n}\) so that \(\left\{y_{i}(k)\right\}_{k=0}^{\infty}\to+\infty\); \(\forall i\in\overline{n}\). Thus, Case 2, supported by Sub-case 2a, cannot hold. b) _Sub-case 2b_: Either \(\left\{\gamma_{i}(k)\right\}_{k=0}^{\infty}\to+\infty\) or \(\left\{\gamma_{i}(k)\right\}_{k=0}^{\infty}\to+\infty\) for some given \(i\in\overline{n}\) and some \(j(\
eq i)\in\overline{n}\). Since the sequences \(\left\{c_{ji}(k)\right\}_{k=0}^{\infty}\subset\left[0,\;1\right]\) then either \(\left\{\frac{y_{i}(k)}{y_{j}(k)}\right\}_{k=0}^{\infty}\to\infty\), or \(\left\{\frac{y_{i}(k)}{y_{j}(k)}\right\}_{k=0}^{\infty}=+\infty\), for \(j(\
eq i)\in\overline{n}\). Since \(\left\{y_{i}(k)\right\}_{k=0}^{\infty}\) is claimed to be bounded, then the denominators of the quotients defining the general terms of the above sequence either converge to zero as \(k\to\infty\) or are zero for some finite \(k\in N_{0}\). However, this would imply the unboundedness of some of the subpopulations contradicting the boundedness property of Theorem 2, which still holds for the extended Beverton-Holt evolving under the time-varying parameterizing sequences \(\left\{K_{i}(k)\right\}_{k=0}^{\infty}\left\{\mu_{i}(k)\right\}_{k=0}^{\infty} \left\{b_{i}(k)\right\}_{k=0}^{\infty}\) and \(\left\{d_{i}(k)\right\}_{k=0}^{\infty}\), fulfilling the assumption (a) of this theorem. Therefore, Case 2 cannot hold when being supported by Sub-case 2b. c) _Sub-case 2c_: \(\lim\limits_{k\to\infty}\left(\sum_{j=0}^{k}\left(\prod_{j=0}^{k}\left[\alpha_ {i}(\ell)\right]^{k-j}\beta_{i}(j)\gamma_{i}(j)\right)\right)=+\infty\) with \(\lim\limits_{k\to\infty}\left(\sum_{j=0}^{k}\left(\prod_{j=0}^{k}\left[\alpha _{i}(\ell)\right]^{k-j}\right)\right)=+\infty\) and \(\left\{\beta_{i}(j)\gamma_{i}(j)\right\}_{j=0}^{\infty}\) being bounded for any \(i\in\overline{n}\). Since \(\lim\limits_{k\to\infty}\left(\prod_{j=0}^{k}\left[\alpha_{i}(j)\right]\right)=+\infty\) and since the above partial sum diverges to \(+\infty\) as \(k\to\infty\), one also concludes, from the assumed condition (c), i.e., \(\lim\limits_{k\to\infty}\left(\prod_{j=0}^{k}\left[\alpha_{i}(j)\right]\right)=+\infty\), that the Sub-case 2c reduces to Case 1. As a conclusion, only Case 1, including Sub-Case 2c, is possible, which implies the asymptotic extinction of all the subspecies. \(\square\)  
**Remark 5**: _As it could be expected from intuition, a large hunting/fishing quota or a certain level of positive independent consumption might lead to extinction. Such conclusions arise from inspecting either the zeroing in finite time or the asymptotic convergence to zero of the amount \(b_{i}(k+1)-d_{i}(k+1)y_{i}^{-}(k+1)\). In a parallel way, sufficiently small hunting/fishing quotas or small positive independent consumption can keep the populations free of extinction._