A mathematical model is proposed and analysed to study the dynamics of two-prey one predator system of fishery model with Holling type II function response. The effect of harvesting was incorporated to both populations and thoroughly analysed. We study the ecological dynamics of the Nile perch, cichlid, and tilapia fishes as prey-predator system of lake Victoria fishery in Tanzania. In both cases, by nondimensionalization of the system, the equilibrium points are computed and conditions for local and global stability of the system are obtained. Condition for local stability was obtained by eigenvalue approach and Routh-Hurwitz Criterion. Moreover, the global stability of the coexistence equilibrium point is proved by defining appropriate Lyapunov function. Bioeconomic equilibrium is analysed and numerical simulations are also carried out to verify the analytical results. The numerical results indicate that the three species would coexist if cichlid and tilapia fishes will not be overharvested as these populations contribute to the growth rates of Nile perch population. The fishery control management should be exercised to avoid overharvesting of cichlid and tilapia fishes.

We describe transmission dynamics of yellow fever (YF) within two host populations, and build up a deterministic SVEIRS model with vaccination to the entire new born. The model aims at clarifying contributions of mathematical ideas in studying the impact of YF disease dynamics. We examine existence of equilibrium solutions and give out conditions that are sufficient for existence of realistic equilibria. Threshold of the form R0 = √(Rhv + Rvm) is obtained, where Rhv and Rvm are reproduction numbers for human-vector and vector-primate compartments respectively. Stability analysis of the equilibria is presented; trace-determinant approach is used in determining local stability of DFE and Metzler matrix for global stability. Lyapunov function is used in establishing conditions for global stability of EE. Our results suggests that eradication of the infection to human population is possible only if Rhv < 1 and Rvm <1. Due to new births and immunity loss to YF after ten years, susceptible class will always be refilled and hence continuous vaccination is essential.

A deterministic mathematical model was formulated using non-linear ordinary differential equations to gain an insight of dynamics of yellow fever (YF) between primates, human beings and Aedes mosquito for the purpose of controlling the disease. Basic reproduction number, R0, was computed and its sensitivity analysis with respect to epidemiological parameters was performed to study the effect of model parameters to R0. Results showed that R0 is most sensitive to daily biting rate of mosquitoes, recruitment rate of vectors, probability of transmission of infection, recruitment of unvaccinated immigrants and the incubation period for both vector and humans. Thus, for the minimization of YF transmission, these parameters should closely be monitored. Stability analysis of disease-free equilibrium (DFE) and endemic equilibrium (EE) points were performed to study perseverance and conditionnecessaryfordiseaseinterruptionandcontrol. ResultsshowedthattheDFEislocally asymptotically stable if the rate of new infection from infected monkey to vector is less than unity, and is globally asymptotically stable if the rate of new infection from infected vector to human is less than unity. Lyapunov stability theory and LaSalles Invariant Principle were used to investigate stability of EE. Results show that EE is globally asymptotically stable whenever R0 > 1. To assess the impact of control measures on YF dynamics, we derived and analysed the necessary conditions for optimal control using optimal control theory. Results show that multipleoptimalcontrolstrategyisthemosteffectivetobringastabledisease-freeequilibrium compared to single and two controls. However, spray of insecticides alone was not effective without personal protection, and optimal use of personal protection alone is beneficial to minimize transmission of the infection to the community. Furthermore, cost-effectiveness analysis of the optimal control measures was considered. We used incremental cost-effectiveness ratio to investigate and compare the costs required against the health benefits achieved between two or more alternative intervention strategies that compete for the same resource. Results showed that combination of all strategies is the most cost-effective compared to others.

Interaction between prey and predator species is a complex and non-linear process. Understanding various phenomena in the dynamics of prey-predator systems is vital to both mathematical ecology and conservation biology. Mathematical models on prey-predator systems have been the hot sport providing important information regarding the interactions of prey and predator species in various ecosystems. In this paper, a review of the available mathematical models on prey-predator systems was done. Our aim was to assess their structure, behaviour, available control strategies, population involved and their ability in predicting the future behaviour of the ecosystems. We observed diversities in the reviewed mathematical models, some model incorporated factors such as drought, harvesting and prey refuge as the factors that affect ecosystems, some ignored the contribution of environmental variations while others considered the variable carrying capacity. Most of the models reviewed have not considered the contribution of diseases and seasonal weather variation in the dynamics of prey predator systems. Some of the reviewed models do not match the real situation in most modelled ecosystems. Thus, to avoid unreliable results, this review reveals the need to incorporate seasonal weather variations and diseases in the dynamics of prey predator systems of Serengeti ecosystem.