Fig. 4.

Continual evolution with a single evolving trait consisting of two phenotypes. (A) A species with only two possible phenotypes, $A$ and $B$, which both show fast positive feedbacks on their own phenotype. The arrows signify the different processes/feedbacks, with thickness representing the strength of the interaction, and the number of dashes representing the timescale. Continuous arrows are the fast positive feedback. Phenotype $A$ shows a strong negative feedback on an external factor $\phi$ (thick dashed line), while phenotype $B$ shows a weaker (or nonexistent), negative feedback (thin dashed line); this feedback could result from the phenotype having a positive effect on the environment and the environment having a negative effect on the phenotype (as in the figure), or vice versa. Mutations are represented with the thinnest and most spaced dashed lines, because they are the weakest and slowest interactions; they are only possible between the two phenotypes, although rates are low. (B) Approximate phase planes. Block arrows and colored areas show the sign of the derivatives. Blue and red lines show the isoclines where the derivatives are 0. The isoclines for $d{u}_A/dt=0$ and $d{u}_B/dt=0$ cross as shown, because the fast positive feedback ensures that the growth of the subpopulation $u_B$ decreases faster with increasing $u_A$ than the growth of the subpopulation $u_A$, and vice versa. Black dashed lines show the direction of change over time. Different phase planes with black solid lines above show how the phase planes will change over time.