A model for the population dynamics of plants under extensive livestock grazing

Extensive livestock production on rangelands involves continuous biomass extraction, as various plant species serve as food for different animal populations. Unlike agricultural systems, rangeland biomass extraction reduces plant size without their complete removal, leading to more complicated management strategies. Mathematical models could predict where plant biomass is available, to relocate animals accordingly, but the current state of the art offers plant population models with a single variable, confusing growth rate, fitness, and carrying capacity. This study addresses this limitation through a model that divides plant populations into two state variables: i) total biomass ( ) and ii) the number of individuals/vegetation cover ( ). Biomass follows a standard logistic population dynamic constrained by the carrying capacity of the ecosystem, while represents population spread and resource acquisition. The model integrates Schaeffer and Noy-Meir logistic population models with biomass extraction and includes a seeding term ( ) to account for human interventions. Results showed that system stability and equilibrium depend on the efficiency parameter ( ), which links and . Higher values reduced the system's maximum yield under biomass extraction, highlighting the trade-off between vegetation cover and biomass productivity. This model provides a promising alternative for describing rangeland dynamics under extensive livestock production.