Fig. 4.

a Effects of mean dispersal distance ($\beta$) and intrinsic growth rate (R) on persistence and critical wave speeds. In the black region all populations go extinct. In the dark grey region, a fully dispersing population cannot persist, but a partially sedentary population can, and $c_{p=1}^{\ast }<c_{p_{\ne }1}^{\ast }$. In the light grey region all populations can persist, but $c_{p=1}^{\ast }<c_{p_{\ne }1}^{\ast }$, while in the white region all populations persist, but $c_{p=1}^{\ast }>c_{p_{\ne }1}^{\ast }$. In b and c total population density of the travelling pulse solution is plotted as a function of climate speed. In (b), $R=7.5$ and $\beta =2.5$, corresponding to the scenario where $c_{p=1}^{\ast }>c_{p_{\ne }1}^{\ast }$. In c, $R=6$ and $\beta =2.5$, corresponding to the scenario where $c_{p=1}^{\ast }<c_{p_{\ne }1}^{\ast }$. The dash-dot line indicates the population size of the partially sedentary population ($p=0.6$), the solid line denotes the size of the fully dispersing population ($p=1$), and the large dots correspond to the size of population exhibiting positive density-dependent dispersal, with $a=1$