We consider diffusion in a disk, representing a cell with a circular interior compartment. Using bipolar coordinates, we perform exact calculations, not restricted by the size or location of the intracellular compartment. We find Green functions, hitting densities and mean times to move from the compartment to the cellular surface, and vice versa. For molecules with diffusivity D, mean times are proportional to R 2 /D, where R is the radius of the cell. We find explicit expressions for the dependence on a 2 (the fraction of the cell occupied by the intracellular compartment) and on the displacement of the compartment from the center of the cell. We consider distributions of initial conditions that are (i) uniform on the nuclear surface, (ii) uniform on the cellular surface, or (iii) given by the hitting density of particles diffusing from the nuclear to the cellular surface.

Global mean temperatures have increased by 0.72 • C since the 1950s, and climate warming is resulting in geographical shifts in the range limits of many species. Climate velocity is estimated to be 0.42 km/year, and if a species fails to adapt to the new climate, it must track the location of its climatically constrained niche in order to survive. Dispersal has an important role to play in enabling a population to shift is geographical range limits, but many species are partially sedentary, with only a fraction of the population dispersing each year. We ask, can partially sedentary populations keep pace with climate or will such populations be more vulnerable to extinction? Through the development of a moving-habitat integrodifference equation model, we show that, provided climate velocity is not too large, partially sedentary populations can outperform fully dispersing populations in one of two ways: (i) by persisting at climate speeds where a fully dispersing population cannot, and (ii) exhibiting higher population densities. Moreover, we find that positive density-dependent dispersal can further improve the likelihood a population can persist. Our results highlight the positive role that non-dispersers may play in mitigating the effects of overdispersal and facilitating population persistence in a warming world.

When the mutation rate is high and/or the population size is large, recurrent mutation can lead to multiple, independently generated copies of the same beneficial allele spreading through the population. However, classical analyses of fixation probability and time assume that the mutation rate is low and therefore, that fixation and extinction of a beneficial allele occur faster than the appearance of additional copies. We developed a diffusion equation approximation for the fixation probability and time that accounts for recurrent mutation, incomplete fixation, and fixation from standing genetic variation. Our results show that when the number of new beneficial alleles per generation in the population is greater than one, fixation is guaranteed, and fixation time is significantly lower than expected by the standard approximation. Moreover, we show that fixation time is significantly shorter if the initial allele frequency is greater than 0, or if fixation is defined for an allele frequency lower than 1.

We give exact Green’s functions in two space dimensions. We work in a scaled domain that is a circle of unit radius with a smaller circular “inclusion”, of radius a, removed, without restriction on the size or position of the inclusion. We consider the two cases where one of the two boundaries is absorbing and the other is reflecting. Given a particle with diffusivity D, in a circle with radius R, the mean time to reach the absorbing boundary is a function of the initial condition, given by the integral of Green’s function over the domain. We scale to a circle of unit radius, then transform to bipolar coordinates. We show the equivalence of two different series expansions, and obtain closed expressions that are not series expansions.