## 4 Conclusions  
This study is primarily motivated by the observed effects of climate variability, with a particular focus on the impact of erratic rainfall patterns on livestock production in Botswana. By estimating relevant model parameters from literature and calculating others using online data, the research demonstrates how fluctuating rainfall directly affects livestock populations. Botswana's diverse vegetation is classified into four major areas, as shown in Figure 1, with data points Kgagodi, Sehitlwa, Mabutsane, and Letlhakeng, which provides a detailed context for understanding these effects. The study presents a mathematical model based on a non-autonomous ordinary differential equation that includes a periodic equation for plant growth rates. The model uses rainfall data from Climate Engine ([[http://ClimateEngine.org](http://ClimateEngine.org)]([http://ClimateEngine.org](http://ClimateEngine.org)), accessed on 15 February 2025) across four distinct study sites to estimate plant growth rates. The model is analyzed for positivity and boundedness. An average reproduction ratio, \(\mathcal{R}_{0}\) value, based on the average livestock ratio, for the coexistence of plant and livestock populations was established, and a trapping region was identified. The general pattern of the livestock basic reproduction number \(\mathcal{R}_{0}\) is observed to decrease through time. This decrease could be due to some environmental and climatic variations in the region, serving as an important signal to policymakers. The model's sensitivity to changes in key parameters such as harvest rate, rainfall intensity (increase or decrease), and the timing of rainfall (early or late onset) was explored. Sensitivity analysis revealed that early rainfall positively impacts livestock production across all regions, as does increased rainfall intensity. Furthermore, adjusting the harvest rate in response to changes in the rainfall pattern significantly enhances livestock management and mitigates overgrazing, demonstrating its importance in maintaining sustainable plant-livestock dynamics. Numerical simulations underscore that the adverse effects of climatic variations on livestock production can be mitigated by optimizing livestock harvest rates. Increasing the harvest rate reduces livestock numbers, allowing plant biomass to recover and sustain  
Figure 12: Figure showing effect of increasing harvest rate when rainfall comes early than expected in different regions of Botswana. the ecosystem whenever necessary. For example, the analysis of livestock dynamics across different locations highlights the impact of harvesting and rainfall variability on herd numbers. In Kgagodi, without harvesting, livestock numbers decline by approximately 37% due to feed shortages, while moderate harvesting (\(\mu=2/365\)) reduces this loss to 17%. With significant harvesting (\(\mu=8/365\)), livestock numbers remain stable at their peak levels. In Sehitluwa, biomass declines without harvesting, but livestock numbers show minimal reduction. When rainfall is delayed, Kgagodi experiences a sharper decline (over 55%), with numbers briefly dropping below 1000 before recovering. Letlhakeng, in contrast, sees a 6% increase in livestock when rainfall is delayed, leading to a sharp decrease in biomass, which can be stabilized by harvesting. Mabutsane and Letlhakeng generally maintain their peak numbers with minimal variation. The study shows that in these regions, the growth rate of plants, which serves as the primary food source for livestock, is significantly impacted by inconsistent rainfall. This, in turn, affects the availability of drinking water, resulting in compounding the challenges faced by livestock. The results align well with observed field data, confirming the physical soundness and applicability of the model. These findings are particularly beneficial for traditional farmers in Botswana, whose livelihoods largely depend on livestock production and who often lack access to advanced climate data or technologies for managing weather variability.