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Considering the mature condition of any individual to have eaten a specific amount of food during the entire period that it can spend at its immature stage, we propose a size-structured model by a first-order quasi-linear partial differential equation. The model can be firstly reduced to a single state-dependent delay differential equation and then to a constant delay differential equation. The state-dependent delay represents intra-specific competition among individuals for limited food resources. A complete analysis of the global dynamics on the positivity and boundedness of solutions, global stability for each equilibrium and Hopf bifurcation is carried out. Our results imply that the delay leads to instability that is shown by a simple example of a certain structured population model. [ABSTRACT FROM AUTHOR]

In this paper, the stability properties of a parabolic partial differential equation with state-dependent delay are investigated by the heuristic approach. The previous works [1,2] obtained a continuously differentiable semiflow with continuously differentiable solution operators defined by the classical solutions, and resolved the problem of linearization for this equation. Here, we clarify the relation between the spectral properties of the linearization of the semiflow at a stationary solution and the strong continuous semigroup defined by the solutions of the linearization of this equation, and consider the local stable and unstable invariant manifolds of the semiflow at a stationary solution. By a biological application, we finally verify all hypotheses for an age structured diffusive model with state-dependent delay and consider its stability behavior. [ABSTRACT FROM AUTHOR]

Background: As a highly efficient and specific gene regulation technology, RNAi has broad application fields and good prospects. The effect of RNAi enhances as the dosage of siRNA increases, while an exorbitant siRNA dosage will inhibit the RNAi effect. So it is crucial to formulate a dose-effect model to describe the degradation effects of the target mRNA at different siRNA dosages. Results: In this work, a simple RNA interference model with hill kinetic function (Giulia Cuccato et al. (2011)) is extended. Firstly, by introducing both the degradation time delay τ1 of mRNA caused by siRNA and the transportation time delay τ2 of mRNA from the nucleus to the cytoplasm during protein translation, one acquires a novel delay differential equations (DDEs) model with physiology lags. Secondly, qualitative analyses are executed to identify regions of stability of the positive equilibrium and to determine the corresponding parameter scales. Next, the approximate period of the limit cycle at Hopf bifurcation points is computed. Furthermore we analyze the parameter sensitivity of the limit cycle. Finally, we propose an optimal strategy to select siRNA dosage which arouses significant silencing efficiency. Conclusions: Our researches indicate that when the dosage of siRNA is large, oscillating periods are identical for disparate number of siRNA target sites even if it greatly impacts the critical siRNA dosage which is the switch of oscillating behavior. Furthermore, parametric sensitivity analyses of limit cycle disclose that both of degradation lag and maximum degradation rate of mRNA due to RNAi are principal elements on determining periodic oscillation. Our explorations will provide evidence for gene regulation and RNAi. [ABSTRACT FROM AUTHOR]

Abstract: An impulsive delayed SI model with variable coefficients and a nonlinear incidence is formulated and analyzed. By introducing three thresholds, we obtain sufficient conditions for eradication and permanence of the disease, respectively. It is shown that the conditions depend on time delay for both the global attractivity of the positive infection-free periodic solution and permanence of the model. Furthermore, our results indicate that the disease will disappear if the ratio of the maximum to minimum of the pulse vaccination rate is lager than some value. The main feature of this paper is that we introduce multi-delays and variable coefficients into the SI model, and exhibit a new method which is applied to investigate this model. Numerical results show that the system we considered has complex dynamics including periodic and quasi-periodic oscillations. [Copyright &y& Elsevier]