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Modelling of Bad Biomass Invasion of a Food Chain Ecosystem with Optimal Control

In this paper we provide a model to describe the dynamics of the species of the ecosystem after it has been raided by a bad competing specie. The competing specie invades the native plants for nutrition, carbon dioxide and space. This affects the population of the native species of the ecosystem. The effect of the bad biomass on the ecosystem is examined by considering it's equilibrium points as well as its stability. An optimal control system is developed by using Pontryagin's maximum principle to construct a Hamiltonian function which minimizes the spread of the bad biomass. Numerical simulations were conducted to analyze the results. It was established that the intrinsic growth rate r of the good biomass was responsible for the sustenance and continuous survival of the ecosystem. If growth rate of the good biomass is increased, the other species showed positive growth. In addition to the growth rate of the good biomass, death rates of both fish and birds also affected the state of the ecosystem. Simulation results also revealed that the invasive bad species biomass introduction affected the growth of the good biomass. This was as a result of the competition for nutrition, carbon dioxide, space between the good and the bad biomases. After implementing the control units, simulation results showed an improvement in the growth of the good biomass, fish and birds populations whereas it showed a decline in the bad biomass growth.