The thesis consists of two parts. In the first part, I investigate the developmental mechanisms that regulate the nutritional plasticity of organ sizes in Drosophila melanganster, fruit fly. Here I focus on the insulin-like signalling pathway through which the developmental nutrition is signalled to growing organs. Two mathematical models, an ODE model and a PDE model, are established based on the IIS pathway. In the ODE model, the circulating gene expression of each components in IIS pathway is considered as model variables. By analyzing the steady states of the ODE model under different parameter settings, the hypothesis that the difference of the nutritional plasticity among all organs of Drosophila is due to the variation of the total gene expressions of components in IIS pathway is verified. Furthermore, the forkhead transcription factor FOXO, a negative growth regulator that is activated when nutrition and insulin signaling are low is a key factor to maintain organ-specific differences in nutritional-plasticity and insulin-sensitivity. In the PDE model, I focus more on the molecule structure within each individual cell. The transportation of proteins between nucleus and cell membrane is modelled in the system. In simulations of the PDEs system, the hypothesis that the concentration of FOXO decrease as the concentration of insulin increase is verified.In the second part of the thesis, I study the bifurcation properties of the nonlocal diffusion equation:\[ L_{\epsilon} u + \lambda (u - u^3) = 0. \]where $L_{\epsilon} u$ is an integral defined as \[ L_{\epsilon} u = \int_{0}^{\pi} \epsilon^{-3} J( \frac{y-x}{\epsilon} ) ( u(y) - u(x) ) dy. \]and $J(x)$ is a non-negative radially symmetric function with $J(0) > 0$. It is shown that as the scaling parameter $\epsilon$ is small enough the equation has the pitchfork bifurcations at the spectrum of the operator $L_{\epsilon} u$. A concrete example is considered. The bifurcations result is verified in the concrete example by solving the equation with Newton's Method.
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