On the validation, stability and control of certain biological systems

This thesis is concerned with the long-term behavior of certain mathematical models describing predator-prey interactions, virus propagation in vivo and integrated pest management strategies. Of interest are the stability and impulsive controllability of such models, since the corresponding mathematical findings are then easily interpretable in terms of biological concepts with major relevance such as disease endemicity, species extinction or permanence and pest eradication. Threshold conditions for the global stability of the equilibria are obtained by means of Lyapunov’s direct method combined with LaSalle invariance principle. These results, stated in terms of a biologically significant key parameter called the basic reproduction number, are then reconfirmed by using monotonicity methods. The models of concern are formulated in a general way, no specialization being made, for instance, on the incidence rate of the infection and on the removal rate of the virus (for the virus model) or on the functional response of the predator (for the predator-prey model). This makes the findings are applicable to a large class of real-life interactions. Also, threshold conditions with immediate biological significance which guarantee the global success of integrated pest management strategies are derived using Floquet theory. The corresponding impulsive controllability results are then obtained by using comparison arguments. Also, a bifurcation analysis is performed via an operator theoretic approach and some situations leading to a chaotic behavior of the solutions are investigated by means of numerical simulations.