Fig. 2

Collapse of breeding migration (Model 1). Panels a–c show stationary solutions for \(\bar {N}\) from Eqs. (1) & (4) as a function of mortality, for individual accuracies of a _o =0.0, 0.4 and 0.8 respectively. Results including collective navigation (a(U) as in Eq. (3)) are blue while those with independent navigation (a(U)=a _o ) are grey. The shaded blue regions depict an unstable limit cycle, inside of which the dynamics are locally attracted to the stable equilibrium within, while globally (and in response to extreme perturbations) the system collapses to the stable equilibrium where the population is extinct. Vertical lines correspond to the boundaries (bifurcation points) in panel d. Note in panel a the grey curve is not visible because it is equal to zero for all h. Panel d traces the branching point (solid line – Eq. (17)), Hopf point (dotted line) and limit point (dashed line) through h- a _o space. Faint horizontal lines correspond to the cross-sections depicted in panels a–c. The qualitatively distinct states of the system in the different parameter regions are: i) Group navigation is bistable [high N _h |N _h extinct], solo navigation is not possible; ii) Both group and solo navigation are possible, group navigating population densities are higher; iii & iv) Neither group nor solo migration can persist