Bifurcation diagrams showing the interplay between negative carry-over effects and stability.

<p>In (<i>a</i>)–(<i>b</i>), the long-term behavior of the solutions is plotted as the strength of the carry-over effects |<i>a</i>| is increased; red dashed lines correspond to the unstable equilibrium and some unstable 2-cycles. (<i>a</i>) Density-independent mortality using <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0155579#pone.0155579.e003" target="_blank">Eq (3)</a> with <i>K</i> = 1, <i>r</i> = 3.7, <i>d</i>(<i>N</i>_<i>b</i>) ≡ <i>d</i> = 0.1; carry-over effects are stabilizing but the route from chaos to stability is not the usual one since stability switches can appear once the periodic regime reaches a period-two solution. (<i>b</i>) We use <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0155579#pone.0155579.e003" target="_blank">Eq (3)</a> with density-dependent mortality <i>d</i>(<i>N</i>_<i>b</i>) = 1 − exp(−<i>αN</i>_<i>b</i>), <i>K</i> = 1, <i>r</i> = 3.65, and <i>α</i> = 0.8. Increasing the strength of carry-over effects can destabilize the equilibrium and give rise to the coexistence of two stable attractors for |<i>a</i>|∈ [1.16, 1.54]: a positive equilibrium and a 2-cycle.