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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]

In this paper, the smoothness properties of semiflows on C 1 -solution submanifold of a parabolic partial differential equations with state-dependent delay are investigated. The problem is formulated as an abstract ordinary retarded functional differential equation of the form d u ( t ) / d t = A u ( t ) + F ( u t ) with a continuously differentiable map G from an open subset U of the space C 1 ( [ − h , 0 ] , L 2 ( Ω ) ) , where A is the infinitesimal generator of a compact C 0 -semigroup. The present study is continuation of a previous work [14] that highlights the classical solutions and C 1 -smoothness of solution manifold. Here, we further prove the continuous differentiability of the semiflow. We finally verify all hypotheses by a biological example which describes a stage structured diffusive model where the delay, which is the time taken from birth to maturity, is assumed as a function of a immature species population. [ABSTRACT FROM AUTHOR]