Size-structured populations: immigration, (bi)stability and the net growth rate

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 All\'{e}e-effect.Comment: to appear in Journal of Applied Mathematics and Computin

Advancing biological understanding and therapeutics discovery with small-molecule probes

Small-molecule probes can illuminate biological processes and aid in the assessment of emerging therapeutic targets by perturbing biological systems in a manner distinct from other experimental approaches. Despite the tremendous promise of chemical tools for investigating biology and disease, small-molecule probes were unavailable for most targets and pathways as recently as a decade ago. In 2005, the NIH launched the decade-long Molecular Libraries Program with the intent of innovating in and broadening access to small-molecule science. This Perspective describes how novel small-molecule probes identified through the program are enabling the exploration of biological pathways and therapeutic hypotheses not otherwise testable. These experiences illustrate how small-molecule probes can help bridge the chasm between biological research and the development of medicines but also highlight the need to innovate the science of therapeutic discovery