A simple model of unlimited growth can be designed as follows. Let r > 0 be the rate of growth, IR+ the set of nonnegative real numbers, and f the function f: 1R+ ! IR+
For any given x , one can iterate f so as to get, for example, fff( X))) or ffffX)))). Writing « for the number of iterations, one can define a function :
This can be written as
The exponential function provides a fundamental model of unlimited growth. It is worth noticing that the Euler number e and the exponential function itself were introduced into mathematical physics about three centuries ago on the basis of analyses of a key phenomenon of economic growth: the mechanism of compound interest. Later on, Malthus (1798) noticed that in many circumstances the exponential function was a good model for the dynamics of biological populations, including humans. Malthus also insisted that population growth could not go on without limits and therefore expected serious famines that would ultimately stabilize human population worldwide.
In 1838, the Belgian mathematician Verhulst modified Euler's exponential function so as to represent limits to growth. Call that limit K and let the variable rate of growth r obey the formula r ' — r ( 1--
This implies that the rate of growth approaches 0 as the growing variable approaches its limit, while it approaches r as the growing variable approaches zero. Instead of  we get x ! 1 + r ( 1 ) ? x
By introducing an auxiliary variable z,  can be simplified as follows:
As long as r is smaller than 2, the variable growth rate r gradually diminishes as x increases. The result is an S-shaped curve that converges from below to the level x = K. In 1974, R. M. May discovered that for r above a threshold slightly larger than 3.57 the process generates a chaotic dynamics: it jumps to levels above K and from there falls back to levels below K, and it does so without repeating itself in a regular fashion. For values between 2 and about 3.57, the process generates various forms of oscillations. These findings make the argument about limits to growth more intricate, but the key idea can be developed with the simpler S-shaped curve.
Now suppose that some agent (e.g., the fisheries industry) is faced with a population whose dynamics can be described by  with 1 < r<2. The agent wants to maximize its utility from catching fish over an indefinite period. Make the usual assumption that the utility function is additive across time, concave and monotonically
increasing in each period, discounting future utility by a discount factor 0 < ft <1:
If the agent is sufficiently greedy, that is, if ft is rather small, he will try to catch as much fish as he can and as fast as he can. He will then choose his strategy h accordingly. Suppose that it takes time to build up fishing capabilities. Then the strategy space H will be such that increasing catch becomes feasible in the course of time. This, however, will lead the agent to fish until the population is exhausted according to a pattern known as overshoot and collapse. The upper panel in Figure 1 gives an illustration.
If the agent cares more about the distant future, that is, if ft is close enough to 1, a different strategy will be appropriate: build up fishery capacity up to the point where a constant catch is possible so as to maintain the population indefinitely. The lower panel in Figure 1 gives an illustration.
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