In the simplest case of a one-dimensional landscape a local minimum can disappear in a saddle-node bifurcation: The deformation of the landscape is such that the minimum corresponding to a local optimum for the fitness parameter collides with a nearby local maximum and both disappear (Figure 1). The system experiences again a gradient force and moves on to the next lower minimum again with an exponential rate. This bifurcation is called local, since all the changes happened locally in the neighborhood of the local optimum.
Figure 1: Example of a generalized fitness landscape Vk in a one-dimensional variable q. The (saddle node) bifurcation parameter is denoted by k. For sub-critical values of the bifurcation parameter (k < 0) the landscape has a minimum (i.e. best fitness) at q @0.235 and a maximum (worst fitness) at q @ 0.19. At the bifurcation point (k=0) both fixed points collide and annihilate each other at q*. Notice that the minimum becomes very shallow giving rise to critical fluctuations and slowing down. For supercritical values (k > 0) the slope is positive everywhere in this region and the system will "slide down" the landscape coming to rest at a new minimum (not shown in this figure).
Other local bifurcations (e.g. pitchfork bifurcations) create new minima at the same time, as the old minimum becomes a maximum (Figure 2). In other words: what used to be a best practice is being replaced by an innovative modification that represents a significant improvement. Since in this case the system always remains at the location of the local minimum its change is determined by the change of the shape of the fitness landscape. Typically there is no fixed exponent associated with that transition and therefore no fixed time scale is defined. Instead, the innovation induced improvement curve generically will follow a power-law. In a physics context the associated exponents are known as "critical exponents" and contain important information about the system.
Figure 2: Same as in figure 1 but for a different (pitchfork) bifurcation. For sub-critical values of the bifurcation parameter (k < 0) the landscape has a minimum at q=0. For super-critical parameter values the former minimum turns into a maximum while two new minima emerge at q+, q-. Their distance from the old minimum increases with a power-law as a function of the bifurcation parameter k.
Figure 3 shows the graphs of both function types to illustrate their different behavior at initial and asymptotic times. In order to discriminate between the two function types one often uses log-linear and log-log representations of the data. A linear graph in the first case is characteristic for exponential behavior (incremental innovations) whereas a linear graph in the second type of representation is characteristic for power laws and might be an indication for a more fundamental innovation.
Figure 3: Illustration of exponential (solid line) vs. power law (dashed line) asymptotic behavior of generalized fitness functions. The examples have been chosen to start at zero and asymptotically tend to one.