Acta Biotheoretica

, Volume 62, Issue 3, pp 325–338

Fishermen’s Profits Maximization: Case of Generalized Nash Equilibrium of a Non-symmetrical Game

Regular Article

DOI: 10.1007/s10441-014-9223-y

Cite this article as:
El Foutayeni, Y. & Khaladi, M. Acta Biotheor (2014) 62: 325. doi:10.1007/s10441-014-9223-y
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Abstract

In the present paper, we consider a bio-economic equilibrium model which describes the dynamics of a fish population fished by several fishermen seeking to maximize their profits. Each fisherman tries to find the fishing effort which maximizes his profit at biological equilibrium without any consultation with others, but all of them have to respect two constraints: (1) the sustainable management of the resources ; and (2) the preservation of the biodiversity. With all these considerations, our problem leads to a generalized Nash equilibrium problem. The objective is to show that even when a fisherman \(i\) provides a fishing effort equal to twice the fishing effort of a fisherman \(j\), then the profit of fisherman \(i\) is not necessarily double that of fisherman \(j\).

Keywords

Population dynamics Bio-economic equilibrium model Generalized nash equilibrium Sustainable management of the resources Preservation of the biodiversity Fisherman’s profit maximization 

1 Introduction

The management of renewable biological resources implies both the objectives of the conservation of a population and the maximization of the benefits coming from its exploitation. To achieve these objectives, there are several studies of bio-economic models using various mathematical tools. In Purohit et al. (2007) deal with the problem of optimal harvesting of a fishery consisting of two competing fish species, each of them obeys the Gompertz law of growth. A dynamic reaction model is developed from a capital-theoretic view point taking taxation as a control measure to regulate exploitation of the fishery. The existence, as well as the stability of the possible steady states is examined. Bionomic equilibrium of the system is determined and the optimal harvest policy is studied with the help of Pontryagin’s Maximum Principle. In Merinoa et al. (2007) use the GAMEFISTO simulation model to present a tool to improve the small scale fisheries bio-economic simulation techniques. The main novelty of the model is the implementation of game theoretic techniques for forecasting the fishing effort trends and consequently, the fish population levels and the economic outcome, including landings, profit and net profits. The model assigns individual fishing strategies to individual vessels according to their technical characteristics. The fishermen within a fishing fleet exploiting a single stock are assumed to be the decision agents, who share not only a fish population (modeled through stock externality) but also a market, through an offer-demand function (market externality). In previous work Bulte (2003), extended the traditional G-S model of open access by defining a non-concave harvesting function. The main of the paper was to demonstrate the possible existence of multiple equilibria and perverse comparative statics and show that small changes in the underlying economic parameters may trigger large jumps in species abundance. More recently, in the paper El Foutayeni et al. (2012), we defined a bio-economic equilibrium model for several fishermen who catch three species; these species compete with each other for space and/or food. The natural growth of each species is modeled using a logistic law. We find the fishing effort that maximizes the profit of each fisherman. The existence of the steady states and its stability are studied using eigenvalue analysis. In El Foutayeni and Khaladi (2012) we presented a bio-economic equilibrium model for several fish populations taking into consideration the fact that the prices of fish populations vary according to the quantity harvested. We introduced a mathematical model; studied the existence and stability of the equilibrium point and calculating the fishing effort that maximizes the profit of the fisherman exploiting all fish populations.

The goal of the present paper is to consider a bio-economic equilibrium model which describes the dynamics of a fish population fished by several fishermen seeking to maximize their profits. Specifically, we consider a bio-economic equilibrium model of a fish population exploited by several fishermen represented by their fishing efforts \((E_{i})_{i=1,\ldots ,n}\), where \(n\) is the number of fishermen. The objective is to find the fishing effort \(E_{i}^{*}\) maximizing each fisherman’s profit \(\pi _{i}^{*}(E^{*})\), at biological equilibrium, without any consultation between the fishermen, but all of them have to respect the sustainable management of the resources. Each fisherman strives to maximize its profit choosing a fishing effort strategy. With these considerations, our problem leads to a generalized Nash equilibrium problem. The fishing effort \(E^{*}=\left( E_{1}^{*},\ldots ,E_{n}^{*}\right) \) depends on: (a) the catchability coefficients \(q_{i}\); (b) the costs of fishing \(c_{i}\); and (c) the price of fish population \(p\). It allows us to discuss the trends of individual fishing effort in terms of competitiveness. In our previous works El Foutayeni et al. (2012), El Foutayeni and Khaladi (2012), it has been shown that if the fishermen provides the same fishing effort \(E_{i}=E_{j}\), then all fishermen earn the same profits \(\pi _{i}=\pi _{j}\). In this work we assume that the fishermen provides distinct fishing efforts \(E_{i}\ne E_{j}\). The main objective of this paper is to show that even when a fisherman \(i\) provides a fishing effort equal to twice the fishing effort of a fisherman \(j \), while not necessarily the first (fisherman \(i\)) earns double the second (fisherman \(j\)).

The paper is organized as follows. In the next section, we present a bio-economic equilibrium model describing the dynamics of a fish population exploited by several fishermen seeking to maximize their profits. We give the steady states of biological model and present bio-economic equilibrium model of fishery. We show that finding a solution of bio-economic equilibrium model is equivalent to solving a generalized nash equilibrium problem (GNEP). In Sect. 3 we show that the problem GNEP has a unique solution and we compute this solution. In Sect. 4 we give numerical simulations of the mathematical model and discussion of the results. Finally, we give a conclusion in Sect. 5.

2 The Bio-Economic Equilibrium Model

The description of the bio-economic equilibrium model is divided into three parts: The first one is the mathematical model and hypotheses; the second one is the steady states of the system; and the third one is the bio-economic equilibrium model of fishery.

2.1 The Mathematical Model and Hypotheses

The evolution of the biomass \(B\) at time \(t\) of a given fish population is modelled by the following equation (Clark 1990).
$$\begin{aligned} \dot{B}(t)=rB\left( 1-\frac{B}{K}\right) -F_{m}B, \end{aligned}$$
(1)
where \(r\) is the intrinsic growth rate, \(K\) is the area’s environmental carrying capacity or saturation level for a given fish population, and \(F_{m} \) is the fishing mortality. Note that the fishing mortality applied to a population is composed by a catchability coefficient \(q\) and a fishing effort \(E\) term
$$\begin{aligned} F_{m}=qE. \end{aligned}$$
(2)
The first has been defined as the mortality generated by a unit of fishing effort and its dynamics have already been explained. Note that if the fishing effort is measured, for example, in trawl hours, then catch could be measured in kg or tonnes. Fishing effort and catch should both be related to the same unit of time, which could be a day or a week. Under these assumptions, biomass changes through time can be expressed as
$$\begin{aligned} \dot{B}(t)=rB\left( 1-\frac{B}{K}\right) -qEB. \end{aligned}$$
(3)
Furthermore, the total fishing mortality suffered by an exploited population \(F_{m}^{total}\) is the sum of the mortalities generated by each fisherman \( F_{m}^{i}\) and each of them is the product of the individual catchability \( q_{i}\) and the individual activity or fishing effort \(E_{i}\)
$$\begin{aligned} F_{m}^{total}=\sum \limits _{i=1}^{n}F_{m}^{i}=\sum \limits _{i=1}^{n}q_{i}E_{i}. \end{aligned}$$
(4)
We have to note that the fishing effort is one of the easiest concepts to understand and most difficult to define and quantify in fishery science; but in general, we can say that fishing effort is defined as the product of a fishing activity and a fishing power. The fishing effort exerted by a fleet is the sum of these products over all fishing units in the fleet. The fishing activity is in units of time. The fishing power is the ability of a fishing unit to catch fish and it is a complex function depending on vessel, gear and crew. However, since measures of fishing power may not be available, activity (such as hours or days fished) has often been used as a substitute for effort. We can also hypothesize that fishing effort behaves as a function of profits (Fig. 1), either recent profits or profit expectation. This dependency has been focused firstly with the extension of the Gordon-Schaefer’s model proposed by Smith (1969).
Fig. 1

Static (equilibrium) and dynamic trajectories of revenues and costs resulting from the application of different fishing effort levels

It is interesting to note that according to the literature, the fishing effort depends on several variables, namely for example: number of hours spent fishing; search time; number of hours since the last fishing; number of days spent fishing; number of operations; number of sorties flown; ship; technology; fishing gear; crew; etc. However, in this paper, the fishing effort is treated as a unidimensional variable which includes a combination of all these factors.

2.2 The Steady States of the System

When the fish population is at biological equilibrium, i.e., the variation of the biomass of fish population is zero: \(\dot{B}(t)=0\), and thus losses by natural and fishing mortalities are compensated by the fish population increase due to individual growth and recruitment. The equation can be defined as
$$\begin{aligned} rB\left( 1-\frac{B}{K}\right) -qEB=0. \end{aligned}$$
(5)
The solutions of this equation are \(B_{1}^{*}=0\) and \(B_{2}^{*}=K \left( 1-\frac{1}{r}qE\right) \).
Note that \(B_{1}^{*}\) and \(B_{2}^{*}\) are not only solutions of the algebraic equation (5) but also represent constant solutions of the differential equation (3) since all we have done in Eq. (5) is to algebraically manipulate the right-hand side of Eq. (3) to equal forms. That is, it has been shown that the differential equation (3) can be rewritten in the form
$$\begin{aligned} \dot{B}(t)=rB\left( 1-\frac{1}{r}qE\right) \left[ 1-\frac{B}{K\left( 1-\frac{1}{r}qE\right) }\right] . \end{aligned}$$
(6)
Furthermore, it is easy to show that the first constant solution \(B_{1}^{*}=0\) is unstable and that the second constant solution \(B_{2}^{*}=K\left( 1-\frac{1}{r}qE\right) \) is asymptotically stable. Under exploitation, the fish population behavior through time is described as shown in Fig. 2.
Fig. 2

Population logistic growth model (for r = 0.85 per year, K = 1,000 kg, q = 0.06 kg per unit of effort and E = 1.5) depending on the initial condition \(B_{0}:\)a for \(B_{0}<B_{2}^{*}\); b for \(B_{0}=B_{2}^{*}\) and c for \(B_{0}>B_{2}^{*}\)

2.3 The Bio-Economic Equilibrium Model of Fishery

The description of the bio-economic equilibrium model of fishery is divided into four parts: The total revenue, the total cost, the profit and the bio-economic equilibrium model optimization.

2.3.1 Total Revenue

Let \(E_{i}\) be the fishing effort strategy of fisherman \(i\) and let \(q_{i}\) be the catchability coefficient of fisherman \(i\). The total revenue of a fishery \(R\), equals quantity harvested \(H\) multiplied by the price of the fish population \(p\). The price of the fish population from a particular stock is hardly affected by quantity fished if the fish is sold in a competitive market with many sellers and buyers and in competition with similar types of fish from other stocks. In the following analysis we shall assume that the price of fish population, \(p\), is constant across time and quantity. Under these hypotheses, the total revenue to fisherman \(i\), at biological equilibrium, can be represented as
$$\begin{aligned} R_{i}&=pH_{i} \nonumber \\&=pq_{i}E_{i}B \nonumber \\&=pq_{i}E_{i}\left(-\frac{1}{r}\sum \limits _{j=1}^{n}q_{j}E_{j}+1\right)K \nonumber \\&=pKq_{i}E_{i}\left(-\frac{1}{r}q_{i}E_{i}+1 -\frac{1}{r}\sum \limits _{j=1,j\ne i}^{n}q_{j}E_{j}\right) \nonumber \\&=-\frac{pK}{r}q_{i}^{2}E_{i}^{2} +pKq_{i}\left(1-\frac{1}{r}\sum \limits _{j=1,j\ne i}^{n}q_{j}E_{j}\right)E_{i}, \end{aligned}$$
(7)

2.3.2 Total Cost

The total cost function is also built into the model. As the resource fish population density goes down, the fishing effort \((E_{1},E_{2},\ldots ,E_{n})\) needed to capture it goes up. In the simplest model we will suppose that the total cost of fishing effort is proportional to this fishing effort: the more fishing effort we use, the higher cost. This cost would be the product of the harvesting costs per fishing effort \(c\) multiplied by the fishing effort \(E\). Thus the \(C\) curve would have a positive constant slope. In reality this curve may be more complex: the cost per unit of effort may go down if the capture capacity goes up (if there are economies of scale). In this work we will keep to the simplest hypothesis that the total costs to fisherman \(i\) is proportional to fishing effort \(E_{i}\), expressed mathematically as
$$\begin{aligned} C_{i}=c_{i}E_{i}, \end{aligned}$$
(8)
where \(C_{i}\) is the total costs to fisherman \(i\) and \(c_{i}\) is the harvesting costs per fishing effort employed by fisherman \(i\).

2.3.3 Profit

The profit of fisherman \(i\) is then given by the equation
$$\begin{aligned} \pi _{i}=R_{i}-C_{i}. \end{aligned}$$
(9)
It follows from (7) and (8) that
$$\begin{aligned} \pi _{i}(E)=-\frac{pK}{r}q_{i}^{2}E_{i}^{2}+pKq_{i}\left( \frac{pKq_{i}-c_{i}}{pKq_{i}}-\frac{1}{r}\sum \limits _{j=1,j\ne i}^{n}q_{j}E_{j}\right) E_{i}. \end{aligned}$$
(10)

2.3.4 Bio-Economic Equilibrium Model Optimization

The objective of the work is to find the fishing effort that maximizes the profit of each fisherman, but we must respect two constraints:
  1. (a)

    the first one is the preservation of biodiversity of fish population, expressed mathematically as \(\frac{B}{K}=1-\frac{1}{r}\sum \nolimits _{j=1}^{n}q_{j}E_{j}>0\);

     
  2. (b)
    and the second one is that the optimal fishing effort \(E^{*}=(E_{1}^{*},\ldots ,E_{n}^{*})\) must satisfy the following conditions
    $$\begin{aligned} \pi _{i}(E_{1}^{*},\ldots ,E_{i}^{*},\ldots ,E_{n}^{*})\geqslant \pi _{i}(E_{1}^{*},\ldots ,E_{i},\ldots ,E_{n}^{*})\quad {\text{ for\,all }}\quad i=1,\ldots ,n; \end{aligned}$$
    with the conditions that \((E_{i})_{i=1,\ldots ,n}\geqslant 0\).
     
Note that an important but self-evident component of this work is that fishermen are jointly constrained by the population dynamics of the fish stock.

With all these considerations, our bio-economic equilibrium model can be translated into the following mathematical problem

Each fisherman \(i\in \{1,\ldots ,n\}\) must solve problem \((P_{i})\)
$$\begin{aligned} (P_{i})\left\{ \begin{array}{ll} \max \pi _{i}(E)=-&{}\frac{pK}{r}q_{i}^{2}E_{i}^{2}+pKq_{i}\left( \frac{pKq_{i} -c_{i}}{pKq_{i}}-\frac{1}{r}\sum \limits _{j=1,j\ne i}^{n}q_{j}E_{j}\right) E_{i} \\ \text {{subject\;to}} &{}\\ &{}\frac{1}{r}q_{i}E_{i}<-\frac{1}{r}\sum \limits _{j=1,j\ne i}^{n}q_{j}E_{j}+1 \\ &{}E_{i}\geqslant 0 \\ &{}(E_{j})_{j=1,\ldots ,n;\;j\ne i}{\ \ }\text {{are given}}{.}\end{array}\right. \end{aligned}$$

3 Computing the Generalized Nash Equilibrium

We recall that the GNEP is an extension of the nash equilibrium problem (NEP), in which each fisherman’s strategy (fishing effort) set is dependent on the rival fishermen strategies. Mathematically, \((E_{1}^{*},\ldots ,E_{n}^{*})\) is called generalized Nash equilibrium point, if and only if, \(E_{i}^{*}\) is a solution of the problem \((P_{i})\) for \({(}E_{j}^{*})_{j=1,\ldots ,n;\;j\ne i}{\ }\)are given.

The optimality conditions of Karush-Kuhn-Tucker for \((P_{i})\) are that if \(E_{i}^{*}\) is a solution of the problem \((P_{i})\), then there exist constants \(u_{i}\in \mathrm {IR}_{+}, v\in \mathrm {IR}_{+}\) and \(\lambda _{^{i}}\in \mathrm {IR}_{+}\) such that the following relations hold
$$\begin{aligned} \text {for\, all}\,i\,&=1,\ldots ,n \nonumber \\&\left\{ \begin{array}{l} 2\frac{pKq_{i}^{2}}{r}E_{i}^{*}+pKq_{i}\left( \frac{c_{i}-pKq_{i}}{pKq_{i}} +\frac{1}{r}\sum \limits _{j=1,j\ne i}^{n}q_{j}E_{j}^{*}\right) -u_{i}+\frac{q_{i}}{r}\lambda _{^{i}}=0 \\ \frac{1}{r}\sum \limits _{j=1}^{n}q_{j}E_{j}^{*}+v=1 \\ <u_{i},E_{i}^{*}>=<\lambda _{i},v>=0 \\ u_{i},E_{i}^{*},\lambda _{i},v\geqslant 0\end{array}\right. \end{aligned}$$
(11)
To maintain the biodiversity of the fish population, it is natural to assume that the biomass remain positive, that is \(B>0\); therefore \(v=\frac{B}{K}>0\). As the inner product \(<\lambda _{^{i}},v>=0\), so \(\lambda _{^{i}}=0\) for all \(i=1,\ldots ,n\).
The problem (11) reduces to the following expressions
$$\begin{aligned} \text {for \, all}\,i\,&=1,\ldots ,n \nonumber \\&\left\{ \begin{array}{l} 2q_{i}E_{i}^{*}+\left( r\frac{c_{i}-pKq_{i}}{pKq_{i}}+\sum \limits _{j=1,j\ne i}^{n}q_{j}E_{j}^{*}\right) -\frac{r}{pKq_{i}}u_{i}=0 \\ \frac{1}{r}\sum \nolimits _{j=1}^{n}q_{j}E_{j}^{*}+v=1 \\ <u_{i},E_{i}^{*}>=0 \\ u_{i},E_{i}^{*},v\geqslant 0\end{array}\right. \end{aligned}$$
(12)
If we set \(\bar{E}_{i}=q_{i}E_{i}^{*}, \bar{u}_{i}=\frac{r}{pKq_{i}}u_{i}\) and \(\bar{v}=rv\), then (12) can be rewritten in the following form
$$\begin{aligned} \left\{ \begin{array}{l} \bar{u}=A\bar{E}+b \\ \bar{v}=r-\sum \nolimits _{i=1}^{n}\bar{E}_{i} \\ <\bar{u},\bar{E}>=0 \\ \bar{u},\bar{E},\bar{v}\geqslant 0\end{array}\right. \end{aligned}$$
(13)
where
  • \(A=(a_{ij})_{1\leqslant i,j\leqslant n}\) where \(a_{ii}=2\) and \(a_{ij}=1\) for all \(i\ne j\);

  • \(b=(b_{i})_{1\leqslant i\leqslant n}\) where \(b_{i}=r(\frac{c_{i}}{pKq_{i}}-1) \).

Now, let us set that \(z=(\bar{E},0)^{T}\) and \(w=(\bar{u},\bar{v})^{T}\), then (13) can be rewritten in the following form (subscript \(T\) denotes the vector transpose)
$$\begin{aligned} \left\{ \begin{array}{l} z\geqslant 0 \\ w=Mz+q\geqslant 0 \\ z^{T}w=0\end{array}\right. \end{aligned}$$
(LCP)
where \(M=\left[ \begin{array}{cc} A &{} \mathbf {0} \\ -\mathbf {1} &{} 1\end{array} \right] \) and \(q=\left[ \begin{array}{c} b \\ r\end{array} \right] \).

Here, \(\mathbf {-1}\) denotes the vector \((-1,\ldots ,-1)\in \mathrm {IR}^{n}\) and \(\mathbf {0}\) denotes the vector \((0,\ldots ,0)^{T}\in \mathrm {IR}^{n}\).

We note that the \((LCP)\) problem is called a Linear Complementarity Problem \(Lcp(M,q)\) (for more details see Cottle and Dantzig (2006), Cottle et al. (1992), El Foutayeni and Khaladi (2010), El Foutayeni and Khaladi (2012), and Lemke (1965)).

For solving the linear complementarity problem \(Lcp(M,q)\) we can demonstrate that the matrix \(M\) is P-matrix (Recall that a matrix \(M\) is called P-matrix if all of its principal minors are positive) and we will use the following result: a linear complementarity problem \(Lcp(M,q)\) has a unique solution for every \(q\) if and only if \(M\) is a P-matrix (For demonstration we can see Murty (1971)).

To prove that \(M\) is P-matrix, we note by \((A_{i})_{i=1,\ldots ,n}\) the submatrix of \(A\) and by \((M_{i})_{i=1,\ldots ,n+1}\) the submatrix of \(M\), then we obtain \(det(M_{i})=det(A_{i})=i+1>0\) for all \(i=1,\ldots ,n\) and \(det(M)=det(A)=n+1\). So the matrix \(M\) is P-matrix and therefore the \(Lcp(M,q) \) admits one and only one solution. This solution is given by (see El Foutayeni and Khaladi (2013))
$$\begin{aligned} \text {for all }i&= 1,\ldots ,n \nonumber \\&\left\{ \begin{array}{l} z_{i}=\frac{r}{(n+1)pK\prod _{k=1}^{n}q_{k}}\left[ \prod _{j\ne i}^{n}q_{j}(pKq_{i}-nc_{i})+\sum \nolimits _{j\ne i}^{n}c_{j}\prod _{k\ne j}^{n}q_{k}\right] \\ w_{i}=0\end{array}\right. \\&\text {and } \nonumber \end{aligned}$$
(14)
$$\begin{aligned}&\left\{ \begin{array}{l} z_{n+1}=0 \\ w_{n+1}=r-\frac{r}{(n+1)pK\prod _{k=1}^{n}q_{k}}\sum \nolimits _{i=1}^{n}\prod _{j\ne i}^{n}q_{j}(pKq_{i}-c_{i})\end{array}\right. \end{aligned}$$
(15)
furthermore, the generalized Nash equilibrium point is given by
for all \(i=1,\ldots ,n\)
$$\begin{aligned} E_{i}^{*}=\frac{r}{(n+1)pK\prod _{k=1}^{n}q_{k}}\frac{\left( \prod _{j\ne i}^{n}q_{j}(pKq_{i}-nc_{i})+\sum \nolimits _{j\ne i}^{n}c_{j}\prod _{k\ne j}^{n}q_{k}\right) }{q_{i}} \end{aligned}$$
(16)
and the profit of fisherman \(i\) is then given by
for all \(i=1,\ldots ,n\)
$$\begin{aligned} \pi _{i}(E^{*})=\frac{r}{pK(n+1)^{2}\prod _{k=1}^{n}q_{k}^{2}}\left( \prod _{j\ne i}^{n}q_{j}(pKq_{i}-nc_{i})+\sum \limits _{j\ne i}^{n}c_{j}\prod _{k\ne j}^{n}q_{k}\right) ^{2} \end{aligned}$$
(17)
It is important to remark that if \(c_{i}=pKq_{i}\), then \(\prod _{j\ne i}^{n}q_{j}(pKq_{i}-nc_{i})+\sum \nolimits _{j\ne i}^{n}c_{j}\prod _{k\ne j}^{n}q_{k}=0\) and therefore \(E^{*}=\pi ^{*}=0\).

4 Numerical Simulations of the Mathematical Model and Discussion of the Results

We consider four different fishermen (players), represented by their fishing effort \(E_{i}, i\in \{1,\ldots ,4\}\). The fishermen are assumed to fish the same fish population, they differ, however, in terms of their (a) \(q_{i}\) the catchability coefficient of each fisherman; (b) \(c_{i}\) the harvesting costs per fishing effort employed by each fisherman; (c) and \(E_{i}^{*}\) the fishing effort strategy.

We take it for granted that each of these fishermen seeks to maximize his profit from the fishery. The strategy space open to each fisherman is his harvesting quantity. More precisely, he can choose any positive level of harvesting bounded only by two constraints: the first one is the sustainable management of the resources; and the second one is the preservation of the biodiversity. In other words, each fisherman can choose any positive level of harvesting bounded only by availability at equilibrium biologic of fish. Each fisherman can choose his strategy only once. This strategy will be calculated by: (1) the intrinsic growth rate \(r\); (2) the area’s environmental carrying capacity \(K\); (3) the catchability coefficients \(q_{i}\); (4) the costs of fishing \(c_{i}\); (v) and the price of fish population \(p\). It allows us to discuss the trends of individual fishing effort in terms of competitiveness. Under these hypotheses and from (16) the corresponding fishing effort \(E^{*}\) is the following
$$\begin{aligned} \left\{ \begin{array}{c} E_{1}^{*} =\frac{r}{5pKq_{1}^{2}q_{2}q_{3}q_{4}}[q_{2}q_{3}q_{4}(pKq_{1} -4c_{1})+c_{2}q_{1}q_{3}q_{4}+c_{3}q_{1}q_{2}q_{4}+c_{4}q_{1}q_{2}q_{3}] \\ E_{2}^{*}=\frac{r}{5pKq_{1}q_{2}^{2}q_{3}q_{4}}[c_{1}q_{2}q_{3}q_{4} +q_{1}q_{3}q_{4}(pKq_{2}-4c_{2})+c_{3}q_{1}q_{2}q_{4}+c_{4}q_{1}q_{2}q_{3}]\\ E_{3}^{*}=\frac{r}{5pKq_{1}q_{2}q_{3}^{2}q_{4}}[c_{1}q_{2}q_{3}q_{4} +c_{2}q_{1}q_{3}q_{4}+q_{1}q_{2}q_{4}(pKq_{3}-4c_{3})+c_{4}q_{1}q_{2}q_{3}]\\ E_{4}^{*}=\frac{r}{5pKq_{1}q_{2}q_{3}q_{4}^{2}}[c_{1}q_{2}q_{3}q_{4} +c_{2}q_{1}q_{3}q_{4}+c_{3}q_{1}q_{2}q_{4}+q_{1}q_{2}q_{3}(pKq_{4}-4c_{4})] \end{array} \right. \end{aligned}$$
and from (17) the corresponding profit \(\pi ^{*}\) is the following
$$\begin{aligned} \left\{ \begin{array}{c} \pi _{1}^{*} =\frac{r}{25pKq_{1}^{2}q_{2}^{2}q_{3}^{2}q_{4}^{2}}[q_{2}q_{3}q_{4}(pKq_{1} -4c_{1})+c_{2}q_{1}q_{3}q_{4}+c_{3}q_{1}q_{2}q_{4}+c_{4}q_{1}q_{2}q_{3}]^{2} \\ \pi _{2}^{*} =\frac{r}{25pKq_{1}^{2}q_{2}^{2}q_{3}^{2}q_{4}^{2}}[c_{1}q_{2}q_{3}q_{4} +q_{1}q_{3}q_{4}(pKq_{2}-4c_{2})+c_{3}q_{1}q_{2}q_{4}+c_{4}q_{1}q_{2}q_{3}]^{2} \\ \pi _{3}^{*} =\frac{r}{25pKq_{1}^{2}q_{2}^{2}q_{3}^{2}q_{4}^{2}}[c_{1}q_{2}q_{3}q_{4} +c_{2}q_{1}q_{3}q_{4}+q_{1}q_{2}q_{4}(pKq_{3}-4c_{3})+c_{4}q_{1}q_{2}q_{3}]^{2} \\ \pi _{4}^{*} =\frac{r}{25pKq_{1}^{2}q_{2}^{2}q_{3}^{2}q_{4}^{2}}[c_{1}q_{2}q_{3}q_{4} +c_{2}q_{1}q_{3}q_{4}+c_{3}q_{1}q_{2}q_{4}+q_{1}q_{2}q_{3}(pKq_{4}-4c_{4})]^{2} \end{array} \right. \end{aligned}$$
The values of the model biological and economic parameters are given in Tables 1 and 2.
Table 1

Model biological parameters

Biological parameters

Value

Unit

\(r\)

0.850000

Per year

\(q_{1}\)

0.004000

Kg per unit of effort

\(q_{2}\)

0.002828

Kg per unit of effort

\(q_{3}\)

0.003464

Kg per unit of effort

\(q_{4}\)

0.002000

Kg per unit of effort

\(K\)

1,000.000

Kg

\(B_{0}\)

100.0000

Kg

Table 2

Model economic parameters

Economic parameters

Value

Unit

\(p\)

1.000000

Euros per Kg

\(c_{1}\)

0.500000

Euros per unit of effort

\(c_{2}\)

0.186760

Euros per unit of effort

\(c_{3}\)

0.071996

Euros per unit of effort

\(c_{4}\)

0.250000

Euros per unit of effort

These biological and economic values require that (a) the first fisherman must provide a fishing effort (which maximizes the profit) which is equal to 30.252051 (unit of effort), in this case, its harvest is equal to 32.353068 kg and its profit is equal to 17.227042 Euros; (b) the second fisherman must provide a fishing effort which is equal to 60.504659 (unit of effort), in this case, its harvest is equal to 45.754568 kg and its profit is equal to 34.454718 Euros; (c) the third fisherman must provide a fishing effort which is equal to 60.504105 (unit of effort), in this case, its harvest is equal to 56.037159 kg and its profit is equal to 51.681129 Euros; (d) and the fourth fisherman must provide a fishing effort which is equal to 60.504103 (unit of effort), in this case, its harvest is equal to 32.353068 kg and its profit is equal to 17.227042 Euros as shown in Table 3.
Table 3

The fishing effort, harvest, revenue, cost and profit

Fisherman No. 1

\(E_{1}^{*}\)

30.252051

Unit of effort

\(H_{1}^{*}\)

32.353068

Kg

\(R_{1}^{*}\)

32.353068

Euros

\(C_{1}^{*}\)

15.126026

Euros

\(\pi _{1}^{*}\)

17.227042

Euros

Fisherman No. 2

\(E_{2}^{*}\)

60.504659

Unit of effort

\(H_{2}^{*}\)

45.754568

Kg

\(R_{2}^{*}\)

45.754568

Euros

\(C_{2}^{*}\)

11.299850

Euros

\(\pi _{2}^{*}\)

34.454718

Euros

Fisherman No. 3

\(E_{3}^{*}\)

60.504105

Unit of effort

\(H_{3}^{*}\)

56.037159

Kg

\(R_{3}^{*}\)

56.037159

Euros

\(C_{3}^{*}\)

04.356029

Euros

\(\pi _{3}^{*}\)

51.681129

Euros

Fisherman No. 4

\(E_{4}^{*}\)

60.504103

Unit of effort

\(H_{4}^{*}\)

32.353068

Kg

\(R_{4}^{*}\)

32.353068

Euros

\(C_{4}^{*}\)

15.126026

Euros

\(\pi _{4}^{*}\)

17.227042

Euros

It is important to remark that the fishing efforts \(E_{1}^{*}=30.25, E_{2}^{*}=60.50, E_{3}^{*}=60.50\) and \(E_{4}^{*}=60.50\), given in Table 3, maximize the profits \(\pi _{i}(E)=-\frac{pK}{r}q_{i}^{2}E_{i}^{2}+pKq_{i}(\frac{pKq_{i} -c_{i}}{pKq_{i}}-\frac{1}{r}\sum \nolimits _{j=1,j\ne i}^{n}q_{j}E_{j})E_{i}\) as shown in Fig. 3. We note that fishing efforts \(E_{1}^{*}, E_{2}^{*}, E_{3}^{*}\) and \(E_{4}^{*}\) maximize the profits \(\pi _{1}^{*}, \pi _{2}^{*}, \pi _{3}^{*}\) and \(\pi _{4}^{*}\) and does not maximize the harvests \(H_{1}^{*}, H_{2}^{*}, H_{3}^{*}\) and \(H_{4}^{*} \).
Fig. 3

Profit and revenue curves of the fishermen

The result to highlight is even when the fisherman \(i\) provides the fishing effort equal to twice the fishing effort of the fisherman \(j\), then it will not necessarily to conclude that the first (fisherman \(i\)) earns double the second (fisherman \(j\)). To shed light on this result (see Fig. 4), we consider as a benchmark the first fisherman and we consider three scenarios: (a) the first one is that the second fisherman provides the fishing effort \(E_{2}=2E_{1}\) and it earns double profit \(\pi _{2}=2\pi _{1}\); (b) the second one is that the third fisherman provides the fishing effort \(E_{3}=2E_{1}\) but it earns the triple profit \(\pi _{3}=3\pi _{1}\); (c) and the third one is that the fourth fisherman provides the fishing effort \(E_{4}=2E_{1}\) but it earns the same profit \(\pi _{4}=\pi _{1}\) as shown in Fig. 4.
Fig. 4

Comparison of the harvests, the revenues and the fishing

Figure 4 shows also that if the fishermen provide a fishing efforts \(E_{2}=E_{3}=E_{4}=2E_{1}=60.504659\) (unit of effort), then (1) the first fisherman spends \(C_{1}=15.126026\) Euros, its income equal to \(R_{1}=32.353068\) Euros, its harvest is equal to \(H_{1}=32.353068\,\hbox {kg}\) (because we have taken \(p=1\) Euros per kg) and its profit is equal to \(\pi _{1}=17.227042\) Euros; (2) the second fisherman spends \(C_{2}= 0.747000C_{1}\,\bar{(}=11.299850\) Euros), its income equal to \(R_{2}=1.414200R_{1}\) (=45.754568 Euros), its harvest is equal to \(H_{2}=1.414200H_{1}\) (= 45.754568 kg) and its profit is equal to \(\pi _{2}=2\pi _{1}\) (= 34,454718 Euros); (3) the third fisherman spends \(C_{3}=0.288000C_{1}\,\bar{(}=04.356029\) Euros), its income equal to \(R_{3}=1.732100R_{1}\) (= 56.037159 Euros), its harvest is equal to \(H_{3}=1.732100H_{1}\) (= 56.037159 kg) and its profit is equal to \(\pi _{3}=3\pi _{1}\) (= 51.681129 Euros); (4) and the fourth fisherman spends \(C_{4}=C_{1}\,(=15.126026\) Euros), its income equal to \(R_{4}=R_{1}\) (= 32.353068 Euros), its harvest is equal to \(H_{4}=H_{1}\) (= 32.353068 kg) and its profit is equal to \(\pi _{4}=\pi _{1}\) (= 17.227042 Euros). So we can conclude that even when the fisherman \(i\) (\(i=2,3,4\)) provides the fishing effort equal to twice the fishing effort of the fisherman \(1\), then it will not necessarily to conclude that the fisherman \(i\) earns double the fisherman \(1\).

5 Conclusion

In this paper, we have considered a bio-economic equilibrium model which describes the dynamics of a fish population fished by several fishermen seeking to maximize their profits. At biological equilibrium, we have calculated the fishing effort which maximizes the profit for each fisherman. To do so, we took into consideration the fact that all the fishermen have respected the sustainable management of the resources and the preservation of the biodiversity. This gave us a generalized Nash equilibrium problem. In section Numerical simulations, we have shown that even when the fisherman \(i\) provides the fishing effort equal to twice the fishing effort of the fisherman \(j\), then it will not necessarily to conclude that the first (fisherman \(i\)) earns double the second (fisherman \(j\)). Specifically we have shown that (scenario 1) if fisherman \(2\) provides fishing effort \(E_{2}=2E_{1}\), then he earns double profit \(\pi _{2}=2\pi _{1} \); (scenario 2) if fisherman \(3\) provides fishing effort \(E_{3}=2E_{1}\), then he earns the triple profit \(\pi _{3}=3\pi _{1}\); (scenario 3) and if fisherman \(i\) provides fishing effort \(E_{4}=2E_{1}\), then he earns the same profit \(\pi _{4}=\pi _{1}\).

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Analysis, Modeling and Simulation Laboratory LAMSHassan II UniversityCasablancaMorocco
  2. 2.Mathematical Populations Dynamics Laboratory LMDPCadi Ayyad UniversityMarrakechMorocco
  3. 3.UMI UMMISCO, IRD - UPMCParisFrance

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