Fisheries provide a good example of the exploitation of a renewable resource
and of the problem of stock exhaustion under overexploitation [Clark, 1990].
The standard, but static, approach is based on the concept
of a maximum
sustainable yield [Gordon, 1954,Schaefer, 1954].
According to these classical views, [Clark, 1990], economic
overfishing is simply the fact that maximum sustainable yield,
or rather maximum economic yield,
is not achieved because agents keep on increasing their fishing
effort as long as any profit can be made. Any factor decreasing costs or
increasing revenues, such as
technological improvements, could then drive the ecological-economical
equilibrium close to resource exhaustion.
But the maximum sustainable yield assumption is essentially
an equilibrium assumption which
does not take into account the fact that the population level
of the resource, namely the fish stock, depends upon previous
catches, and thus requires a dynamic, rather than static, approach [Allen and McGlade, 1986,Larkin, 1977].
A number of approaches toward understanding the dynamics of fisheries
have been proposed in recent years [Mangel, 1992] which depart from
the maximum sustainable yield framework, and most of these
recent modeling efforts fall into one of two
categories: the control approach [Clark, 1990,Charles, 1989];
or the case history
approach for describing a specific place and time sequence [Bousquet 
, 1993,Clark, 1990,Opsomer and Conrad, 1994,Durand 
, 1989].
(Of course, some authors do use parameters estimated from
real case studies in the control approach, e.g.[Charles, 1983b]).
However, the idea of control presupposes a central authority
able to enforce the rules that optimize the catches,
a situation seldom encountered in reality, except
in places where access is restricted to a small number
of boats.
The case history approach often relies upon including
such a large number of details about the particular case being
studied that both interpretation and generalization of the
results to other situations sometimes become difficult.
Furthermore, one of the purposes of a case study is
to validate a model by adjusting its parameters in order to obtain
a best fit to available data, while the approach taken by
this paper is more general: our main purpose is to get some insight
in the mechanisms responsible for resource depletion
and the efficiency of factors that could decrease it.
In our view, resource depletion is most often due to
inertial factors in the dynamics of exploitation, such as labor
and capital. Since these aspects are seldom taken into account
in previous modeling studies, one famous exception being
[Meadows 
, 1992,Meadows 
, 1972] for instance,
the emphasis in this paper is on simplification
to improve understanding.
Dealing with problems characterized by a lot of variability and
uncertainties, we are looking for semi-qualitative properties such as
the qualitative behavior of the observed dynamical regimes with respect
to parameters and their scaling laws.
We consider the case where the resource is harvested under conditions of open access, without a central control. The main difference between our approach and the standard one as described in [Clark, 1990] for instance is that we do not suppose here that fishing effort adjusts instantly and rationally to changing circumstances. Our assumption is that the industry can only readjusts with some inertia caused by the capital and labour already engaged in the economic activity. The word inertia is used here in a straightforward extension from physics: in physical systems, forces do not directly control velocity but its time derivative, acceleration, because of the inertia of finite masses. In the dynamics of resource exploitation, inertia of capital and labour already engaged in the fishing activity prevent instant change of effort, but agents can change the time derivative of the effort by re-investing or alternately not renewing their equipment. We will discuss further how capital inertia results into dangerous oscillations of the resource level and of production before the system stabilizes to equilibrium. Most bio-economical models taking into account dynamics of capital, labour or prices are within the framework of a perfectly fluid market and economic rationality [Clark, 1990]. As one can imagine, a number of real fisheries are working far from these assumptions [Charles, 1988]. Labor and capital market are not perfectly fluid:
The issue of the irreversibility of investment in resource management has been discussed by a few authors, including [Smith, 1968] and [Charles, 1983a]. The simplified model that we use in this paper to describe fisheries basic dynamics resembles their basic models. But those previous authors where interested in equilibrium properties of the model [Smith, 1968] or supposed optimal control of exploitation by the investment level [Charles, 1983a]
The present paper is organized in sections describing models of increasing complexity. To simplify the analysis, we have chosen capital as the only economic variable whose dynamics is coupled to that of the resource. In order to derive general results, and facilitate their interpretation, we work with a pair of differential equations which describe the coupled time dynamics of the resource, which is the fish population or its total mass in the case of fisheries, and the capital. The behavior of this simple system of differential equations is easily analysed by linearizing near the equilibria and by doing computer simulations using a standard but robust o.d.e. integrator [DeBoer, 1983], basically variable time step Runge Kutta or Rosenbrock integrator. We start in section 2 from a very simple reallocation rule such that profits from the fishery, namely catch minus capital depreciation, are shared according to a constant fraction between consumption and capital re-investment. This automatic reallocation rule corresponds in fact to an extremely myopic behavior where agents only take into account present profits when deciding how to reallocate them and to unperfect rationality since they don't care about capital and labour market. In real life all these aspects influence the behavior of the agents. But rather than refining traditional economic modeling to better fit insufficient data from some given fishery, we want to understand the possible consequences of capital and labour inertia. In some sense, rather than supposing that fisheries are perfectly coupled to the rest of the economy, we start from the opposite simplifying assumption, namely that they are completely decoupled. In section 2, the oversimplification of a first model proposed by Roughgarden and Brown [Roughgarden and Brown, 1993] allows us to gain some insight in the dynamical behavior of the model and to derive analytical expressions of the relevant quantities. This simple model evolves towards equilibrium, but only after overshooting the equilibrium significantly. Following the overshoot period, the resource is dramatically depleted. Our main focus is the study of the depletion that follows the overshoot. The interest of this first section is mainly heuristic: since the differential system only contains one reduced parameter, the influence of this parameter on depletion is easily demonstrated.
By introducing several refinements in the subsequent sections, 3 to 5, we are able to use the methods and the concepts defined in section 2 to investigate how the dynamics described by each model depend on the particular processes that are included in it as well as how they change with the parameter values. We will consider one-by-one the influence of a natural carrying capacity for the resource (section 3), the effect of harvest prices on the market (section 4), and the effect of processes that keep consumption by the labor force always above some minimum level (section 5). We investigate to what degree these economic and biological mechanisms attenuate the initial resource depletion observed in the simplest model. The role that some of the market control measures commonly employed by planning authorities, such as governments, might play in stabilizing the market will be also discussed. In each case we concentrate on the implication of this series of models for fisheries with open access. However, one aim of the present study is to obtain results of a general nature that apply to other renewable natural resources such as forests or clean water.