Fig 1.
Sewell Wright’s original fitness landscape.
Sewell Wright first proposed the use of a fitness landscape for visualizing the interactions of particular genetic combinations in 1932 [72]. His visualization (left) presented the fitness landscape as a two-dimensional model with continuous expression of the two genes along the respective axes. Under specific environmental conditions, for given organisms, particular combinations of genetic expression are adaptive, whereas others are maladaptive, and the adaptive landscape helps visualize this. Combinations of high adaptive value are expressed as high points, or peaks, whereas relatively maladaptive combinations are low spots, or valleys. The peaks and valleys are represented by + and -. Although the original visual model uses a two-dimensional landscape, Wright notes that the thousands of genes with their millions of potential combinations meant that organisms operated in an n-dimensional landscape. Wright points out that the peaks are in no way fixed and operate in an unseen temporal framework. Should the environment change over time, the peak might move (or disappear), such that a different combination of allele frequencies would constitutes a new peak, or optimum value.
Fig 2.
Simulated one-dimensional model of the fitness landscape.
The origin (O) represents a peak that has disappeared. The population must then move to a new peak in the future (time is represented by distance along the X-axis). Peak A is fixed in fitness (expressed by height / vertical axis) and position relative to the origin (O). Peak B varies in position and strength, starting at the same location and strength as Peak A, then iteratively becoming more distant from the origin (O). At each greater distance, the height of the peak is increased until the strength of selection is sufficient to draw the population from A to B. In this figure, the likelihood of survival and reproduction for an agent is expressed as height on the y-axis.
Fig 3.
Results of the simulated fitness landscape model (n = 500 agents; 5,000 generations).
A typical example of a cycle through the model. In this instance, the distant peak (B) is twice as far from the origin (O) as the near peak (A). The colored lines represent the populations at any given time, and each color is a different iteration (varying strength of B relative to A). The horizontal axis is time, and the vertical axis is the location of the population in adaptive space. For each iteration, the height of Peak B is increased, increasing its ‘attractiveness,’ starting equal to A and ending at B, having 20x the attractiveness of A. Most populations remain at the closer peak (A), and only when the ‘height’ of the more distant peak (B) is 1.5x or greater do the populations migrate through Peak A and into Peak B.
Fig 4.
Height of the peak necessary to attract the population.
The height of the far peak (Peak B) necessary to attract the population as that peak becomes more distant over multiple repeated iterations, is shown. Each iteration is represented by a different color. The more distant the far peak, the greater the height of that peak is necessary. Here, the distant peak starts at the same location as the near peak (X-axis value of 1) and increases. The X-value is the distance factor, so ‘2’ is twice as far as the near peak. The height of the peak necessary to attract the population is on the Y-axis. As the far peak recedes, the strength necessary increases asymptotically. For this analysis, whenever the population arrived within one standard deviation of the distant peak, the population was deemed to have ‘arrived’.
Fig 5.
Model of potential outcomes over time.
Here, the model of potential outcomes over time (the phase space) for any single parameter, using a bounded logistic model is derived from empirical analyses of predictive uncertainty in climatology, is visualized. In this figure, using time flows from left to right. Future outcomes (right) are more unpredictable the further they are from the present but ultimately are bounded. The different color curves represent varying degrees of predictive imprecision, which reflects varying values for z in the logistic function. The green curve is least affected by uncertainty, and the blue the most.
Fig 6.
a and b–Models Employing Uncertainty. Here, visualizations of the two models employing uncertainty are represented using the fitness landscape. In the first model (upper), the height of the temporally distant peak is probabilistically attenuated according to the logistic function. In the second, the distant peak is variably absent, also following the probability of the logistic function.
Fig 7.
a and b–Effect of Uncertainty on Selection for Future Events. Here, the effect of uncertainty on selection for events in the future using models in which the height of the distant peak is affected (a), and the presence/absence of the peak is affected (b), are shown. The left axis is as in Fig 4 (the strength of the distant peak necessary to reach the destination peak). The lower axis (as in Fig 4) is the distance of that peak. That number indicates how far into the future a peak could be reached for a given influence of uncertainty (upper number, corresponding to z in the formula in the text). A distribution that fails to reach horizontally across the graph indicates that, even with maximum selective pressure (20x), the population could not reach a peak beyond the number indicated on the lower horizontal axis. So, for example, in Fig 7b, the uncertainty (z) factor of .5 prevented the population from finding a peak beyond 2.4 times the original distance from the origin (O) to Peak 1, irrespective of the selection strength for that peak.
Table 1.
The effects of uncertainty on the adaptive peak model after multiple iterations of each model, for populations of 500 (5000 generations, 100 cycles of selection strength increase in peak 2 height, by factors of .2, for 50 iterations of increasing distance for peak 2, repeated 50x).
In this table, the column headers represent the degree of uncertainty, from 0 (no uncertainty) to .99 (maximum uncertainty). The number in the cells are the average distances achievable under each level of uncertainty, after increasing selection strength for Peak B up to 20x the strength of Peak 1. With no uncertainty, the populations could reach peaks 4x as far as Peak 1, but this reduces to only 1.4 as far under the maximum uncertainty, using Model B.