## 3 Numerical Simulations and Discussion
### Parameter Values
#### 3.1.3 Effect of Harvest Rate (\(\upmu\))  
In this section, we examine the effect of varying livestock harvest (takeoff) rates in a model where all other parameters are kept constant. Our goal is to understand how the onset of rainfall affects livestock, enabling us to encourage farmers to harvest their livestock strategically to avoid losses due to food shortages, water shortages, and diseases, which are heavily influenced by the timing of rainfall. We analyze how delays or early onset of rainfall impact plant biomass, the primary food source for livestock. This analysis aims to help farmers minimize losses from drought or overgrazing, which are prevalent  
Figure 9: Figure showing effect of rainfall timing in (**a**) Kgagodi, (**b**) Letlhakeng, (**c**) Mabutsane, and (**d**) Sehitlwa. The rainfall time is when the rain comes at the expected time; the case is when rain is delayed by 30 days, and lastly, when the rain comes 30 days early. issues caused by unpredictable rainfall patterns. Such variability in rainfall often results in farmers losing a significant portion of their livestock. In our model, we assume a threshold of 2000 units of livestock across four regions. When livestock numbers reach or exceed this threshold, farmers are advised to start changing their harvesting strategy. We use a base harvest rate of \(\mu_{0}=3.2/365\), suggesting that in a \(4\times 4\) km grid with 500 animals, farmers should harvest at least 32 cattle equivalent livestock over ten years. We maintain consistent initial numbers of livestock and biomass across all four regions while testing various harvest rates: \(\mu=3.2/365\), \(\mu_{1}=0.2/365\), \(\mu_{2}=0.4/365\), and \(\mu_{3}=0.8/365\). Analyzing the results presented in Figure 10, we observe the effect of an increasing harvest rate when rainfall occurs at the anticipated time, typically at the onset of the spring season in Botswana. Notably, the regions of Sehitlwa, Letthakeng, and Mabutsane show similar trends in biomass retention. However, Kgagodi's pattern differs significantly due to lower rainfall levels recorded in 2020, which adversely impact vegetation growth and, thus, livestock sustainability. In Kgagodi, without a harvest rate adjustment, plant biomass is severely affected, especially between days 200 and 250. However, increasing the harvest rate in this region mitigates plant depletion, indicating that by reducing livestock numbers, the remaining animals can be supported by the region's available resources. In Kgagodi, where the timing of rainfall has no significant effect, the maximum livestock capacity is around 3000 units. This capacity can only be maintained for a limited duration; if livestock numbers are not carefully managed, there will be a rapid decline. In contrast, other regions are capable of sustaining more than 3000 units for extended periods. Increasing the harvest rate aligns Kgagodi's biomass patterns more closely with the other regions, indicating that reducing livestock numbers brings them to a sustainable level for the area. This adjustment reduces plant depletion and overgrazing, allowing vegetation to recover and stabilize. During the late onset of the rainy season, as depicted in Figure 11, livestock in Kgagodi barely reach the 3000-mark, and without harvesting, numbers may drop below 1000. In other  
Figure 10: Figure showing effect of increasing harvest rate when there is no variations in rainfall pattern. regions, livestock levels remain more stable due to sufficient food and water, even with late rains. Increasing the harvest rate in Kgagodi reduces livestock loss, and at a maximum rate of \(\mu=4/365\), the region can sustain livestock without significant losses. For the early onset of the rainy period, as shown in, Figure 12, livestock numbers rise to 3000 or more across all regions. In Kgagodi, this increase in livestock population causes severe depletion of plant biomass. Without intervention, livestock could face significant mortality due to food and water shortages, especially during the mid-year.