Applied Mathematics Publications

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    Modelling and analysis of Limnothrissa Miodon population in Lake Kariba with harvesting predation and environmental factors
    (2022) Mutasa, F.K.
    Limnothrissa miodon, called kapenta locally, is a natural resource which provides Zimbabwean and Zambian communities with protein and is a source of livelihoods to fisherman, wholesalers and retailers. The kapenta catches from the Lake Kariba fishery have been declining and it has been attributed to harvesting, climate change, predation and reduced nutrient inflow into the lake. Studies have been done using regression, surplus production models and analytical models. No studies have been done involving a nutrient, plankton, kapenta and tigerfish food chain as a dynamical system. The aim of this research is to formulate a deterministic, non-linear model of ordinary differential equations and analyse the impact of nutrients, harvesting, predation and lake surface water temperature on the population density of kapenta. Mathematical preliminaries such as positivity and existence of solutions are done. Local and global stability analysis of equilibrium points are done using the Routh-Hurwitz criterion and Lyapunov method respectively. Numerical simulations are done using Runge-Kutta method of order 4 in MATLAB and Wolfram Mathematica. Results show that nutrients are key to the productivity of the water body and kapenta will continue to thrive as long as the nutrient inflow rate is greater than some threshold value. Results also show that the coexistence equilibrium is stable provided certain conditions are satisfied and globally unstable when feasible. A maximum sustainable yearly catch of 34668.59 tonnes is obtained and is within the range obtained by other authors. Optimal control theory applied to a kapenta model with harvesting shows that not more than 505 fishing units to be licensed to operate on Lake Kariba, with 278 on the Zimbabwean side and 227 on the Zambian side and currently there is overcapacity in the lake. Bifurcation analysis of the kapenta model with tigerfish predation shows existence of an Poincar ́e-Androv-Hopf bifurcation with a possibility of chaos for the zooplankton growth rate parameter. Bifurcation type, point, existence, angular frequency, period, stability and direction are determined for some model control parameters. Lyapunov exponents are used for determining stability of periodic orbits and to check for possibility of chaos. Simulation results show that predator-prey dynamics of tigerfish and kapenta show oscillatory behaviour which is ecologically stable and agrees with actual data and therefore reflects reality. Lake surface water temperature was added as an environmental factor to the kapenta model with harvesting and predation. Numerical results show that the population density of kapenta declines after a lake surface water temperature of 30◦C. Warming of the lake has a negative effect on the more palatable Chlorophyceae and this results in a decrease in the density of kapenta in Lake Kariba.