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1 Fitness landscapes in biology

Fitness landscapes are often conceived of as ranges of mountains. There exist local peaks (points from which all paths are downhill, i.e. to lower fitness) and valleys (regions from which most paths lead uphill). A fitness landscape with many local peaks surrounded by deep valleys is called rugged. If all genotypes have the same replication rate, on the other hand, a fitness landscape is said to be flat.

An evolving population typically climbs uphill in the fitness landscape, until a local optimum is reached (Fig. 1). There it remains, unless a rare mutation opens a path to a new, higher fitness peak. Note, however, that at high mutation rates this picture is somewhat simplistic. A population may not be able to climb a very sharp peak if the mutation rate is too high, or it may drift away from a peak it had already found. The process of drifting away from a peak is often referred to as Muller's ratchet.

2 Fitness landscapes in evolutionary optimization

Apart from the field of evolutionary biology, the concept of a fitness landscape has also gained importance in evolutionary optimization methods such as genetic algorithms or evolutionary strategies. In evolutionary optimization, one tries to solve real-world problems (e.g., engineering or logistics problems) by imitating the dynamics of biological evolution. For example, a delivery truck with a number of destination addresses can take a large variety of different routes, but only very few will result in a short driving time. In order to use evolutionary optimization, one has to define for every possible solutionIn mathematics, the solution point is a point that satisfies an equation. For example, the point (2,1) is a solution point of the equation 4''x + y 9 because if x 2 and y 1 then 4''x + y 4· 2 + 1 8 + 1 9. Some equations have no solution points — for examp s to the problem of interest (i.e., every possible route in the case of the delivery truck) how 'good' it is. This is done by introducing a scalarAbstract algebra Algebra Linear algebra The concept of a scalar is used in mathematics and physics. The concept used in physics is a more concrete version of the same idea that goes by that name in mathematics. In mathematics, the meaning of scalar depend-valued functionIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtu f(s) (scalar valued means that f(s) is a simple number, such as 0.3, while s can be a more complicated object, for example a list of destination addresses in the case of the delivery truck), which is called the fitness function or fitness landscape. A high f(s) implies that s is a good solution. In the case of the delivery truck, f(s) could be the number of deliveries per hour on route s. The best, or at least a very good, solution is then found in the following way. Initially, a population of random solutions is created. Then, the solutions are mutated and selected for those with higher fitness, until a satisfying solution has been found.

Evolutionary optimization techniques are particularly useful in situations in which it is easy to determine the quality of a single solution, but hard to go through all possible solutions one by one (it is easy to determine the driving time for a particular route of the delivery truck, but it is almost impossible to check all possible routes once the number of destinations grows to more than a handful).

The concept of a scalar valued fitness function f(s) also corresponds to the concept of a potential or energy function in physicsPhysics (from the Greek, physikos , "natural", and physis , "Nature") is the science of Nature in the broadest sense. Physicists study the behavior and properties of matter in a wide variety of contexts, ranging from the sub-microscopic particles from whi. The two concepts only differ in so far that physicists traditionally think in terms of minimizing the potential function, while biologists prefer the notion that fitness is being maximized. Therefore, multiplying a potential function by -1 turns it into a fitness function, and vice versa.