Mathematical models for the dynamics of production and utilization of forage for sustainable livestock industry in Semi-arid Region of Uganda

Abstract

A considerable amount of work in ecological theory is based on the simple Lotka- Volterra, with assumptions (identical individuals, equal birth and death rates, consume resources equally and live in homogeneous environments), to approximate complex population models. The over-simplifying assumptions in such models, produce results that are not biologically sensible. We applied this fundamental paradigm to a grazing system describing the interaction between livestock and forage resources and extended it to study the dynamics of physiologically structured model to approximate a three-stage population model. In this model, we have two stages of juveniles and one stage for adults. Based on the grazing practice of the pastoral communities, we obtained a system of six ordinary differential equations with a somewhat close-to-reality description of the traditional grazing system in the semi-arid region with 27 parameters. However, due to a lack of data and parameter values, we reformulated the system using delay differential equations with weaning as the delay in the livestock life history. This reformulation reduced the number of variables to four and the parameters to twenty. We also incorporated trampling in the modelling process to assess its impact on the intrinsic growth, which influences the net primary production of the grassland. The results of numerical simulations, on a trial field of 10 hectares, revealed that there was over-utilization caused by the rapidly growing livestock population. This was controlled by pulsed harvesting that alternates between juveniles and adults, to reduce the livestock population to 45 to 55 animals. More so, we recommend the livestock composition should be more juveniles than adults, as hard trampling is a function of hoof size and occurs in the face of limited forage and water. With these recommendations, our results indicate minimal over-grazing and trampling effects, thereby minimising their impacts on the grasslands and environmental degradation caused by humans.