Modeling natural forage dependent livestock production in arid and semi-arid regions: analysis of seasonal soil moisture variability and environmental factors

Livestock agriculture in sub-Sahara region is the most important industry that represents a considerable portion of the overall GDP of the region. Livestock activities in the region are conducted using traditional methods, and animal production is primarily reliant on the availability of natural forage. This paper introduces a modified mathematical model for plant-herbivore interactions to explore the influence of environmental factors on livestock production. The model incorporates seasonal variation in plant growth rate, and makes use of the soil moisture data obtained from satellite observations for regions with different soil structure and climate. The non-autonomous model is analysed for positivity, as well as for existence and uniqueness of solutions. Additionally, essential threshold values for co-existence are derived. The trapping region for the solution, that ensures coexistence of population is obtained. The analysis shows that if the mean livestock reproduction ratio (R_o) is greater than unity and if the initial population density satisfies a given condition, coexistence equilibrium can be achieved. The sensitivity of the seasonality parameters representing the growth rate of the resilient proportion of plant types at a given grid-cell (m₀), and the seasonal forcing amplitude (δ) were also studied. To validate the results of the mathematical analysis, numerical simulations are executed, and the temporal profiles of plant biomass and livestock density are also presented for benchmark parameters. The numerical solution of the problem is used to study the plant-livestock interaction dynamics and the corresponding livestock production level under different scenarios. In this regard, the effect of adaptation in dry regions, variation in soil moisture amount, effect of harvest rate at various representative grid-cells are numerically investigated. The study shows that planting and adapting more resilient plant species in relatively dry regions can increase the livestock production in the region. Moreover, an increase in the soil moisture is counter productive in the wet regions, while it favours the regions with medium and dry soil moisture. It is also shown that the yield in livestock production can be managed by carefully selecting the harvest (off-take) rate corresponding to the climatic variations.