## 3 Numerical Simulations and Discussion
### Parameter Values
#### 3.1.1 Effects of Change in Rainfall Intensity in Botswana  
This study examines the effects of rainfall patterns in Botswana, with a focus on four distinct types of vegetation cover, shown in (Figure 1). Our analysis involves varying the intensity of rainfall data collected from 2008 to 2023. It should be noted that an increase in the intensity of rainfall generally benefits both plant and livestock populations in a plant-livestock ecosystem. For plants, the additional rainwater availability enhances growth in biomass production and reproductive success, leading to a higher plant population density. This abundance of plant resources, in turn, provides livestock with more food, improving their survival rates, health, and reproductive success. An increase in rainfall intensity also makes water available for livestock to drink. Consequently, the livestock population is likely to increase in number due to the ample food supply and water available. In such a case, farmers are able to have more harvest from their animals. In contrast, a reduction in rainfall intensity results in detrimental effects on the ecosystem. Decreased rainfall diminishes soil moisture and groundwater, leading to stunted plant growth and forage scarcity. These conditions can significantly impair livestock populations by limiting both their food and drinking water availability, ultimately compromising their health and reproductive potential. Through this investigation, we aim to contribute to a comprehensive understanding of how fluctuations in rainfall patterns impact the intricate balance of the plant-livestock ecosystem in Botswana. Figure 5 reveals a clear relationship between rainfall intensity, livestock population, and plant biomass. When rainfall intensity is decreased on average by 10%, the total number of livestock in the region remains mostly below 2000. During this scenario, there are only six instances where plant biomass drops below 2000 units. These occurrences coincide with periods where livestock units hover around 2000, indicating a balance between the availability of forage and livestock population under reduced rainfall. Figure 5 also shows that when rainfall intensity is increased by 10% on average, we observe a significant rise in livestock units, which exceeds 2000 for the majority of the time. However, around the 3000-day mark, the livestock population temporarily dips below 2000 units. This suggests that the increase in rainfall supports higher livestock populations, although it may not sustain them uniformly over time. In Kgagodi, Figure 6 demonstrates the dynamics of the livestock-plant system, revealing a distinct limit cycle that captures the interdependence between plant biomass and livestock populations. This pattern is consistent throughout Figure 6, with the exception of a disruption observed around day 5000. When rainfall intensity is increased by 10%, we observe a noticeable growth in livestock populations in Kgagodi (Figure 6b). The increase in rainfall enhances plant biomass availability, allowing livestock to sustain higher population levels. This result underscores the critical role of rainfall intensity in determining the carrying capacity of the ecosystem. In Sehitlwa Figure 7, the livestock population exhibits a general downward trend starting just after 1000 days, a pattern observed regardless of whether rainfall intensity is increased or decreased. This indicates that other factors, such as competition for resources or natural population dynamics, may contribute to this decline over time. As depicted in Figure 7, an increase in rainfall intensity by 10% correlates with a significant rise in the average livestock population, approaching a mean of approximately 4000 units of animals. This growth highlights the beneficial impact of enhanced rainfall on  
Figure 5: Plant–livestock population dynamics in Letlhakeng when rainfall intensity is (**a**) decreased by 10%, (**b**) increased by 10%, and (**c**) at baseline.