Analysis of two types of cyclic biological system models with time delays
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In this thesis, we perform the stability analysis of two types of cyclic biological processes involving time delays. We analyze the genetic regulatory network having nonlinearities with negative Schwarzian derivatives. Using preliminary results on Schwarzian derivatives, we present necessary conditions implying the global stability and existence of periodic solutions regarding the genetic regulatory network. We also analyze homogenous genetic regulatory network and prove some stability conditions which only depend on the parameters of the nonlinearity function. In the thesis, we also perform a local stability analysis of a dynamical model of erythropoiesis which is another type of cyclic system involving time delay. We prove that the system has a unique fixed point which is locally stable if the time delay is less than a certain critical value, which is analytically computed from the parameters of the model. By the help of simulations, existence of periodic solutions are shown for delays greater than this critical value.
Poincar´e Bendixson Theorem
Gene Regulatory Networks
Monotone Dynamical System