VOLUME 75, NUMBER 10 PHYSICAL REVIEW LETTERS 4 SEPTEMBER 1995 Fitness Optimization and Decay of Extinction Rate Through Biological Evolution Paolo Sibani, ' Michel R. Schmidt, ' and Preben Alstrgm 'Physics Department, Odense University, DK-5230 Odense M, Denmark The Niels Bohr Institute, DK 2100-Copenhagen g, Denmark (Received 22 February 1995) We present a simple theoretical model of evolution featuring a decreasing extinction rate due to an increasing average fitness of the species. The dynamics is based on a random walk on a rugged fitness landscape, with evolutionary jumps for each species triggered by the achievement of fitness records during the walk. We analyze two different rules for extinction. In the first, an evolutionary jump leads to an extinction with a given probability, while in the second a specific competition mechanism is considered. In both cases temporal power laws are found to describe the evolution. Extensive simulation results from our second model are in reasonable agreement with paleontological data, showing that the background extinction rate has decreased since Cambrian time. PACS numbers: 87. 10. +e, 05.40.+j Some years ago, Raup and Sepkoski [1] suggested that the apparent decrease in the extinction rate of species since Cambrian time could be viewed as a natural consequence of fitness optimization leading to prolonged survival of species through evolutionary time. In this Letter we investigate this idea on the basis of a very simple nonstationary model of evolution, which incorporates the concept of punctuated equilibrium as, for example, discussed by Eldredge and Gould [2,3]. Our approach is in sharp contrast to the recently suggested view of evolution as fluctuation driven, self-organized critical dynamics [4 — 6]. Following a discussion by Weisbuch [7], we con- sider a species as a "cloud" of points (individuals) in an abstract genetic configuration space — the fitness land- scape — which is equipped with a fitness function f. Ran- dom mutations and Darwinian competition continuously change the fitness at the level of individuals. However, only those mutations leading to individuals with fitness higher than previously seen in the whole population trig- ger changes at the level of a species: They lead to evolu- tionary events shifting the average of the cloud to a new point of the landscape with a higher fitness value. Once a change has been triggered, it proceeds very quickly on the time scales appropriate for evolution [7], and we thus model it by a jump in time. By the above mechanism, each species maintains a record of the best fitness value achieved so far. Past history thus strongly constrains the future, both in terms of the time needed for the next evo- lutionary step, which becomes on the average increasingly longer, and in terms of the actual fitness values which can be reached. The magnitude of the fitness and/or the cor- responding genetic configuration change induced by the jump is quite immaterial to the model. The discontinu- ity should, however, be large enough to stand out from fluctuations in the distribution of fitness characterizing the steady state of a species. We now consider two model versions. Introducing for convenience the term "agent" to represent the average evolutionary state of a species, we assume in the first version that, within an ensemble of agents, a fraction n of the jumps leads to extinctions. Although this model is obviously oversimplified, it allows an analytical discussion of several evolUtionary measures. Moreover, it provides a limit to our second model, which is studied numerically, and which explicitly includes competition as a coevolution and extinction mechanism. Assuming that the fitness landscape is rugged [8], the jump statistics are the same as that of magnitude records in an infinite sequence of independent, identically distributed noise spikes. These statistics have recently been studied by Sibani and Littlewood [9], who show that the probability P„(t) of n records (jumps) happening within time t is asymptotically a Poisson distribution in logt, P„(t) = t '(log t)"/n! This means that long time correlations induced by the record dynamics disappear when time is measured on a logarithmic scale. The average number of jumps n prior to time t grows logarithmically at large times, n(t) = g nP„(t) = log t, (1) n=O independently of the distribution from which the fitness values are drawn [9,10]. Here, we select a distribution of fitness values which is initially uniform between 0 and 1. The fitness f„after n jumps then becomes uniformly distributed in the interval ( f„ i, 1). It follows that in the course of time the fitness approaches unity, the average of — log(1 — f) diverging as — log(1 — f) = logt (2) Consider the extinction rate, i.e. , the number of extinc- tions per time unit relative to the number of species. In our first approach it is simply [9] r(t) = adn/dt = nt (3) 0031-9007/95/75(10)/2055(4)$06. 00 1995 The American Physical Society 2055