In the present paper a nonlinear size-structured population dynamical model with size and density dependent vital rate functions is considered. The linearization about stationary solutions is analyzed by semigroup and spectral methods. In particular, the spectrally determined growth property of the linearized semigroup is derived from its long-term regularity. These analytical results make it possible to derive linear stability and instability results under biologically meaningful conditions on the vital rates. The principal stability criteria are given in terms of a modified net reproduction rate.

Abstract. Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.

We apologize for the error in the references. The correct citation is:Yochelis A., Bar-On T., Gov N.S. (2016) Reaction-diffusion-advection approach to spatially localized treadmilling aggregates of molecular motors. Physica D: Nonlinear Phenomena 318-319:84-90.The original article can be found online at https://doi

We employ semigroup and spectral methods to analyze the linear stability of positive stationary solutions of a generalized size-structured Daphnia model. Using the regularity properties of the governing semigroup, we are able to formulate a general stability condition which permits an intuitively clear interpretation in a special case of model ingredients. Moreover, we derive a comprehensive instability criterion that reduces to an elegant instability condition for the classical Daphnia population model in terms of the inherent net reproduction rate of Daphnia individuals.

Abstract. We introduce a nonlinear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment that carries mass, hence the model is equipped with generalized Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for example Wolbachia in a mosquito population. Therefore the (infinite dimensional) nonlinearity arises in the recruitment term. First we establish global existence of solutions and the Principle of Linearised Stability for our model. Then, in our main result, we formulate simple conditions, which guarantee the existence of non-trivial steady states of the model. Our method utilizes an operator theoretic framework combined with a fixed point approach. Finally, in the last section we establish a sufficient condition for the local asymptotic stability of the positive steady state.