## 2 Materials and Methods
### Mathematical Analysis
#### 2.3.2 Boundedness
Then, we define  
\[\mathcal{R}_{0}^{m}=\frac{caP_{\min}^{*}}{(b+P_{\min}^{*})(d+\mu)}. \tag{15}\]  
It should be noted that the interesting qualitative results obtained in this section are due to the estimates of the complex threshold value. In this regard, the time-average basic reproduction number, \(\mathcal{R}_{0}\), of the non-autonomous system (9) can be obtained by setting the time-dependent \(\omega\)-periodic function \(P^{*}(t)\) to its time average within the period \(\omega\), as  
\[\langle P^{*}\rangle=\frac{1}{\omega}\int_{0}^{\omega}P^{*}(t)dt. \tag{16}\]  
In doing this, we have  
\[\mathcal{R}_{0}=\bigg{(}\frac{1}{d+\mu}\bigg{)}\bigg{(}\frac{ca \langle P^{*}\rangle}{b+\langle P^{*}\rangle}\bigg{)}. \tag{17}\]  
Hence, we have the following theorem. Theorem 3.2: _The livestock basic reproduction number \(\mathcal{R}_{0}\) of system (4) is bounded from below by \(\mathcal{R}_{0}^{m}\) and from above by \(\mathcal{R}_{0}^{M}\). That is \(\mathcal{R}_{0}^{m}\leq\mathcal{R}_{0}\leq\mathcal{R}_{0}^{M}\)._  
The reproduction ratio in Theorem 3.2 can be illustrated graphically, showing that each of our four site areas exhibits distinct basic reproduction ratio values due to variations in rainfall across these locations (Figure 3). The analysis of livestock population growth rates as described by the threshold value, \(\mathcal{R}_{0}\), in these four site areas from 2020 to 2023 reveals a concerning downward trend (Figure 3), with critical periods observed around day 250 and day 650, which correspond to September 2020 and September 2021, respectively. During these times, \(\mathcal{R}_{0}\) values approached 1, indicating a heightened risk of livestock population decline that is potentially linked to seasonal rainfall patterns [40]. This observation aligns with findings by [PERSON] et al. [41], who documented similar declines in livestock data trends in the region. Forecasts made using Neural Network Autoregression (NNAR) models, as presented by [PERSON] et al. [41], and supported by historical data [42], confirmed the persistence of this decline. The uniformity of this trend across all studied areas highlights an urgent need for further investigation into contributing factors, such as forage availability and environmental stressors, to develop adaptive strategies for sustaining livestock populations. Theorem 3.3: _Suppose the model system (4) is given with a positive initial plant population \((P(0)>0)\). If \(\mathcal{R}_{0}\leq 1\), then as time goes on, the livestock population decays to zero, whereas the plant population grows to the carrying capacity \(K\) of the environment._  
Proof: Let us bound the values of \(\frac{dL}{dt}\) from above. To do this, we bound the livestock birth rate from above. Since the maximum livestock emergence rate is \(\frac{caK}{b+K}\), to bound this from above for \(t\geq 0\), we have \(P(t)\) bounded above by a function \(P_{1}(t)\), which when \(P(0)>0\) has the property that \(P_{1}(t)\longrightarrow K\) as \(t\longrightarrow\infty\). Hence, for any \(\epsilon_{1}>0\ \ \exists\ \ T_{1}\geq 0\) such that \(P(t)\leq P_{1}(t)\leq K+\epsilon\), for \(t\geq T_{1}\). Using Lemma 3, we have the relation \[\frac{caP}{b+P}\leq\frac{ca(K+\epsilon_{1})}{b+(K+\epsilon_{1})}\;\;\text{for}\;\;t \geq T_{1}\;\;\text{for some}\;\;T_{1}\geq 0.\]  
Moreover, from Lemma 3 and [43], we have \(P\leq K\) for \(t\geq T_{1}\), implying that \(\frac{caP}{b+P}\leq\frac{caK}{b+K}\) for all \(t\geq T_{1}\). Then, we can observe that  
\[\frac{dL}{dt}\leq L\bigg{[}c\frac{a}{b+(K+\epsilon_{1})}(K+\epsilon_{1})-d- \upmu\bigg{]} \tag{18}\]  
Hence, for \(\mathcal{R}_{0}\leq 1\), \(  
Theorem 3 states that when the birth rate of livestock is lower than the mortality rate, the number of livestock dying surpasses the number of newborns. Consequently, in the long term, this circumstance leads to the growth of the plant population up to the maximum capacity of the environment, while the livestock population eventually faces extinction.