Farkas JZ (2011) Size-structured populations: immigration, (bi)stability and the net growth rate. Journal of Applied Mathematics and Computing, 35 (40940), pp. 617-633. http://www.springerlink.com/content/1598-5865/; https://doi.org/10.1007/s12190-010-0382-y
We consider a class of physiologically structured population models, a first order nonlinear partial differential equation equipped with a nonlocal boundary condition, with a constant external inflow of individuals. We prove that the linearised system is governed by a quasicontraction semigroup. We also establish that linear stability of equilibrium solutions is governed by a generalized net reproduction function. In a special case of the model ingredients we discuss the nonlinear dynamics of the system when the spectral bound of the linearised operator equals zero i.e. when linearisation does not decide stability. This allows us to demonstrate, through a concrete example, how immigration might be beneficial to the population. In particular, we show that from a nonlinearly unstable positive equilibrium a linearly stable and unstable pair of equilibria bifurcates. In fact, the linearised system exhibits bistability, for a certain range of values of the external inflow, induced potentially by Allee-effect.
Structured population dynamics; Population inflow; Bistability; Quasicontraction semigroups; Positivity; Spectral methods; Net growth rate; Population dynamics
Journal of Applied Mathematics and Computing: Volume 35, Issue 40940