## 2 Materials and Methods
### Model Formulation and Description
A certain minimal amount of moisture is needed to activate seed germination and reproduction. That is, they become abundantly green and grow during the rainy season, and they die out or do not grow during droughts. The rate or amount in which rain is received at a time could be too much or too low for the environment. Too much water injures plants, compacts soils, and leads to soil erosion [34]. Root loss occurs when excess water reduces oxygen in the soil. Extreme summer rain can leach nitrogen out of the soil. Excessive rainfall is not necessarily useful or desirable at all times. Some of it is unavoidably wasted, while some may cause destruction [34]. If the average rainfall is much lower than the 'ideal' amount needed, it could lead to the dying of plants and grasses [35]. Most grasses in Botswana are seasonal; that is, they grow in summer during the rainy season and become dry and die out during the winter season when rainfall reduces or is not at all. Therefore, in this paper, we consider rainfall to be a factor that affects plant growth rate. The plant growth in this study refers to the increase in plant biomass that is edible by the livestock, and we represent this information by the model below for \(r(t)\):  
\[\frac{dr}{dt}=\delta\left(\cos\left(\omega t+\frac{\pi}{6}\right)\right)^{n}+ \left(A\left(\frac{R(t)-\overline{R}}{\sigma}\right)-\beta\right)r, \tag{3}\]  
where \(A\) is a given constant, and \(\beta\) is the value that takes the effects of soil type and soil texture on the growth rate. \(R(t)\) represents daily rainfall data, \(\overline{R}\) is the mean of rainfall data, \(\sigma\) represents the standard deviation from \(R(t)\) while \(\delta,\beta\in[0,1]\). We also assume that \(R(t)\) is \(\omega\)-periodic, with \(\omega=365\) days (or 1 year). * **Sinusoidal response term:** The term \(\delta(\cos{(\omega t+\frac{\pi}{6})})^{n}\) represents the rate of plant growth at the beginning of the spring season. Plants are structurally smart and can sense their environments changing, and those changes signal them to start to grow again. For example, when plants experience a number of warm days, they are prompted to start growing again. Buds begin to appear on trees. Glucose that was produced during photosynthesis and stored in the roots will provide plants with energy and food for budding. The delay of rain will make plants finish stored glucose and begin to die [36]. * **Soil structure, \(\beta\)r:** This term represents the physical characteristics of soil in reducing the growth of plants. According to [PERSON] [37], roots grow most rapidly in very friable soil, but their uptake of water and nutrients may be limited by inadequate contact with the solid and liquid phases of the soil. With the above modifications, the model system that we study, therefore, takes the following form:  
\[ \begin{split}\frac{dP}{dt}&=r(t)P\bigg{(}1-\frac{P} {K}\bigg{)}-\frac{a}{b+P}LP\\ \frac{dL}{dt}&=L\bigg{[}c\frac{a}{b+P}P-d-\mu \bigg{]}\\ \frac{dr}{dt}&=\delta(\cos{(\omega t+\frac{\pi}{6} )})^{n}+Ar\bigg{(}\frac{R(t)-\overline{R}}{\sigma}\bigg{)}-\beta r,\end{split}  \tag{4}\]  
where the plant growth rate function (3) is included in the system to obtain a non-autonomous ordinary differential equations.