## 2 Materials and Methods
### Model Formulation and Description  
To formulate a mathematical model for the interaction between plants and livestock in the given spatial area, we used the classical predator-prey structure as a base model with logistic growth dynamics for plant biomass. By employing the process-based modeling approach, the plant growth rate is assumed to depend on the seasonal effects of the environment as well as the actual precipitation. In this section, we proceed to the construction of a mathematical model for the plant-livestock interaction dynamics in order to estimate the potential livestock production in Botswana supported by natural forage in climate variation. The basic mathematical system we consider is the classical Lotka-Volterra type model in the version presented by [PERSON] et al. [9; 16; 30]. We assume that the following hypothesis holds for our model. 1. Livestock depend on the availability of food to reproduce. Their nutrients are mostly found on natural forage if they are not given supplementary feeds [31]. 2. There is a relationship between seasonal variation of rainfall and plant growth. There is a significant season-to-season variability in rainfall amount and distribution [32]. Within the semi-arid forests, there may be little seasonal rhythm in the growth and development of vegetation as a whole. Summer, spring, and autumn are seasons experiencing rainfall; particularly, the summer season is known for extreme rainfall and is well distributed in most parts of the country [12]. A season varies from three to four months. More vegetative growth starts around spring, and flourishing occurs mid-spring up to autumn. In winter, there is no vegetation growth because of no rainfall and extreme cold temperature variation. Assume that the state variables for the model system are given by the plant biomass represented by \(P\) and the livestock density represented by \(L\). Then, using the above assumptions, the dynamics of the system at any time \(t>0\) can be written as:  
Figure 1: Sites location on a map of Botswana with different environmental vegetation taken from [26]. \[\frac{dP}{dt} =r(t)P\left(1-\frac{P}{K}\right)-LF(P) \tag{1}\] \[\frac{dL}{dt} =L[cF(P)-d-\mu]\] (2) \[P(0)\geq 0,L(0)\geq 0,\]  
where the values \(K,c,\mu\), and \(d\) are positive constants, with \(K\) representing the environmental carrying capacity for plant biomass, \(c\) is the conversion rate of the consumed plant into livestock, \(\mu\) is the off-take rate, and \(d\) is the mean natural death rate for livestock. We take these parameters as constant because the effect of rainfall in them is very insignificant to the level that it can be ignored. \(r(t)\) is the rainfall-dependent plant growth rate, and \(F(P)\) is the functional response of consumption of plants by livestock through grazing and browsing. Now we give descriptions of important parameters used in the model and their assumptions. * **A constant per-capita death rate of livestock (\(d\)):**, It is assumed in this study that the mean death rate of livestock is constant as used in [33]. However, some authors [9; 16] consider a non-constant or plant-dependent death rate with the assumption that livestock die off exponentially in the absence of plants. If livestock eat more, they will be less likely to die from starvation or from the consequences of weakness brought by hunger. * **Harvest rate**, \(\mu\): This is considered a constant, and it represents the human interference in the system by removing livestock from the system at a constant rate. It is the removal of animals through consumption and/or selling. * **Functional response function, \(F(P)\):** This is the intake rate of the livestock as a function of plant biomass. It is also associated with the reproduction rate of livestock as a function of plant biomass. In this work, we consider the Holling type II function, \(F(P)=\frac{aP}{b+P}\), as described in [9; 16]. * **Plant growth rate function, \(r(t)\):** The amount and distribution of rainfall have a strong impact on the development of plants.