## 2 Materials and Methods
### Mathematical Analysis
#### 2.3.2 Boundedness  
For model system (4), let \(\hat{P}=\max\{P(0),K\}\) with \(\eta=\frac{K(r^{*}(l)+d+\mu)^{2}}{4r^{*}(l)}\). Let \(\hat{L}=\max\big{\{}cP(0)+L(0),\frac{\eta}{d+\mu}\big{\}}\). Then, for any positive \(t>0\), we have \(P(t)\leq\hat{P}\) and \(L(t)\leq\hat{L}\). Proof. : The proof of Lemma 4 closely resembles that of Lemma 2.2 in [9], where a similar argument is presented. Notably, modifications to the growth rate function \(r(t)\) do not alter the essence of the proof. The set on which the system (4) is defined, \(\Omega=\big{\{}(P,L)\in\mathbb{R}_{2}:P\geq 0,\,L\geq 0,P\leq\hat{P},L\leq\hat{L} \big{\}}\) is positively invariant, where \(\hat{P}=\max\big{\{}P(0),K\big{\}}\) and \(\hat{L}=\max\big{\{}cP(0)+L(0),\frac{K(r^{*}(l)+d+\mu)^{2}}{4r^{*}(l)(d+\mu)} \big{\}}\). Proof. : From the first equation of the model system (4), it is clear that if \(P=0\) then \(\frac{dP}{dt}=0\). Moreover, from the second equation when \(L=0\) and \(P\geq 0\), we have \(\frac{dt}{dt}=0\). Therefore, we can conclude that the trajectory of the solution of the system is directed inward on the boundary points of the positive quadrant, that is, on the set \(\partial\mathbb{R}_{2}^{+}=\big{\{}(P,L)\in\mathbb{R}_{2}^{+}|P=0,L=0\big{\}}\). Hence, the conclusion of the theorem follows immediately from Lemma 1 and 3 above. The above permanence of the system verifies that, for the model system (4), with the given non-negative initial conditions, there exists a \(\omega\)-periodic solution to it that remains in the domain for all \(t\geq 0\), and the solution is globally attractive. Thus, the system is mathematically and biologically well posed. Coexistence Solution and Livestock-Free Solution  
When the livestock population is always positive, that is, \(L>0\), from Equation (4) the term  
\[c\bigg{(}\frac{aP(t)}{b+P(t)}\bigg{)}\]  
represents the average rate of emergence of livestock and is an increasing function of \(P\). Similarly, the term \(d+\mu\) represents the rate of removal of livestock. If \(\frac{caP(t)}{b+P(t)}<d+\mu\), then the rate of livestock mortality will always be higher than its emergence. Hence, the livestock population eventually will decay to extinction. Therefore, for the non-extinction of livestock, we must always have a condition that  
\[\frac{caP(t)}{b+P(t)}>d+\mu. \tag{12}\]  
Thus, define \(\mathcal{R}_{0}(t)=\frac{caP(t)}{(b+P(t))(d+\mu)}\) to be a threshold function value for the system. Since \(r^{*}(t)\) is periodic, the solution \(P^{*}(t)\) of (4) also becomes periodic and continuous, and there exist \(P^{*}_{\min}\) and \(P^{*}_{\max}\) such that \(P^{*}_{\min}\leq P^{*}(t)\leq P^{*}_{\max}\), where \(P^{*}_{\max}\) and \(P^{*}_{\min}\) are solutions of Equation (9) corresponding to \(r_{\min}\) and \(r_{\max}\), which are constant values. Therefore, the livestock emergence rates in each case are related as \(\frac{aP^{*}_{\min}}{b+P^{*}_{\min}}<\frac{aP^{*}(t)}{b+P^{*}(t)}\leq\frac{aP^{* }_{\max}}{b+P^{*}_{\max}}\). On the other hand, Equation (12) becomes  
\[\frac{caP^{*}(t)}{(b+P^{*}(t))(d+\mu)}>1, \tag{13}\]  
using the solution \(P^{*}(t)\) of (9), and we can find a critical threshold value that ensures the stability of coexistence. Considering the provided estimates, our focus will be on examining both the upper and lower bounds of \(\mathcal{R}_{0}(t)\). We start with the upper system which, usingthe fact that, for all \(t\geq 0\), \(\frac{aP^{*}(t)}{b+P^{*}(t)}\leq\frac{aP_{\max}^{*}}{b+P_{\max}^{*}}\). Then, the upper bound reproduction ratio is obtained as follows:  
\[\mathcal{R}_{0}^{M}=\frac{caP_{\max}^{*}}{(b+P_{\max}^{*})(d+\mu)} \tag{14}\]  
Our next step is to determine the lower bound for the threshold, which is obtained by using the fact that, for all \(t\geq 0\), \(\frac{aP_{\min}^{*}}{b+P_{\min}^{*}}\leq\frac{aP^{*}(t)}{b+P^{*}(t)}\).