Mathematical Modelling of Harvesting of a Stage and Size - Structured Fish Population: the Case of African Catfish and Nile Tilapia

Abstract

Over fishing has a great impact on the fish population because it results into decrease of fish stock and extinction of fish species. Consumer-resource systems with harvesting, in which African catfish (Glarias~gariepinus) and Nile tilapia (Oreochromis~niloticus) consumed food resources were studied. The cannibalistic behavior of African catfish was captured by using a four stage-structured system. The dynamics of the food resource and African catfish was described using a system of ordinary differential equations referred to as stage - structured fish population model. Quantitative analysis coupled with numerical simulations were used with the aim of determining the optimal harvest size for harvesting African catfish. Eight different harvesting scenarios which account for yield of the fish stock were investigated. Results from the simulations showed that, harvesting large juveniles of size (10-29.9 cm) and small adults of size (30-170 cm) under equal harvesting rates, gives the highest maximum sustainable yield. A stage - structured model for African catfish and Nile tilapia together with two food resources was also formulated. This model was used to compare financial profit and biomass outtake in a two species system versus single species systems. The model dynamics included cannibalism, predator-prey, feeding, reproduction, maturation development, mortality, and harvesting. Well posedness of the model was proved. The model was analyzed qualitatively and results showed that, the fish-free equilibrium point was locally asymptotically stable when the recovery potentials of African catfish and Nile tilapia were respectively less than unity. The recovery potential is a measure of how quickly a fish stock recovers from overfishing or any other major disturbance. The simulation results revealed asymptotically stable solutions, with co-existence of African catfish, Nile tilapia, and the two food resources. Further simulation results using patternsearch showed that, harvesting in a system with both species yields more biomass and financial profits than harvesting in a system with only one species. As a result of conducting this study, the following recommendations were proposed: In a cannibalistic fish population system, small juveniles of size (1-1.9 cm) and large adults that have reached a maximum size of 170 cm should not be harvested for sustainability. Poly-culture of both African catfish and Nile tilapia should be implemented for better revenue.