The Tragedy of Open Ecosystems

Abstract

This paper investigates the role played by cooperation for the sustainable harvesting of an ecosystem. To achieve this, a bio-economic model based on a multi-species dynamics with interspecific relationships and multi-agent catches is considered. A comparison between the non-cooperative and cooperative optimal strategies is carried out. Revisiting the Tragedy of Open Access and over-exploitation issues, it is first proved analytically how harvesting pressure is larger in the non-cooperative case for every species. Then it is examined to what extent gains from cooperation can also be derived for the state of the ecosystem. It turns out that cooperation clearly promotes the conservation of every species when the number of agents is high. When the number of agents remains limited, results are more complicated, especially if a species-by-species viewpoint is adopted. However, we identify two metrics involving the state of every species and accounting for their ecological interactions which exhibit gains from cooperation at the ecosystem scale in the general case. Numerical examples illustrate the mathematical findings.

Appendices

Appendix 1: Proofs

1.1 Proof of Proposition 1

The resolution of the model follows the method proposed by [25] (see Section 3.1.3 ‘Some technical notes on feedback strategies in fishery problems,’ pp. 82–84).

First set the vector $y(t)=\log (x(t)')=(\log (x_{1}),\ldots ,\log (x_{m}))^{\prime }$. Taking the logarithm of ecosystem dynamics (2) controlled by the harvesting rate $F=(F_{1},\ldots ,F_{m})^{\prime }$ gives the linear dynamics written in matrix form

$$\begin{aligned} y(t+1)=r+M' \log (1-F(t))+M' y(t). \end{aligned}$$

(27)

where we use the notation $M=( I+S)'$ as defined in Eq. (8). Using the change of variable from x(t) to $y(t)$, Bellman equation corresponding to the non-cooperative optimization problem (5) can be written as follows

$$\begin{aligned} V_{i}\left( y\right) =\max _{F_{i}}\left\{ a^{\prime }(y+\log (F_i))+\rho V_{i}\left( \log \left( G\left( \left( 1-F_i-F_{-i}\right) x\right) \right) \right) \right\} . \end{aligned}$$

where $F_{-i}$ stands for the aggregate catch rate of players different than i. Using the dynamics (27), it reads

$$\begin{aligned} V_{i}(y)=\max _{F_{i}}\left( a^{\prime }\log (F_i)+a^{\prime }y+\rho V_{i}\left( r+M' \log \left( 1-F_i-F_{-i}\right) +M' y\right) \right) . \end{aligned}$$

Following [25] or [15], we now prove that the value function (assumed to be unique⁹) takes a log-linear form; namely, it is a linear combination of logarithms y in the sense that

$$\begin{aligned} V_{i}(y)=\upsilon +w^{ \prime }y\end{aligned}$$

where $\upsilon$ and $w$ are vectors of size $(m\times 1)$. We determine the coefficients $\upsilon$ and $w$ by applying the Bellman principle. The Bellman equation for every agents i becomes

$$\begin{aligned} V_{i}(y)=\max _{F_{i}}\left( \begin{array}{c} a^{\prime }y+a^{\prime }\log (F_i)+\rho \upsilon \\ +\rho w^{\prime }\biggl ( r+M' \log \left( 1-F_i-F_{-i}\right) +M' y\biggr ) \end{array} \right) . \end{aligned}$$

First-order optimality conditions give for every species j

$$\begin{aligned} \frac{a_{j}}{F_{ij}}=\frac{\rho \left( Mw\right) _{j}}{1-F_{ij}-F_{(-i)j}} \end{aligned}$$

We deduce that users are identical in the sense that $F_{ij}=F_{j}$ for every i. Thus, $F_{(-i)j}=(n-1)F_{ij}$ and we obtain

$$\begin{aligned} F_{ij}^\mathrm{nc}=\frac{a_{j}}{ na_{j}+\rho \left( Mw\right) _{j}}. \end{aligned}$$

The aggregate non-cooperative harvesting rate is

$$\begin{aligned} F_{j}^\mathrm{nc}=\frac{na_{j}}{na_{j}+\rho \left( Mw\right) _{j}}. \end{aligned}$$

as required. The scarcity constraint $F_{j}^\mathrm{nc}\le 1$ is satisfied because of assumption $\rho \left( Mw\right) _{j}>0$.

The vector $w$ is obtained by identification with the form of the value function $V(y)=\upsilon +w^{\prime }y$. We obtain¹⁰

$$\begin{aligned} a^{\prime }+\rho w^{\prime }M' =w^{\prime }, \end{aligned}$$

or equivalently $w=\left( I-\rho M\right) ^{-1}a$ as required.

1.2 Proof of Proposition 5

In the cooperative case, we know from Proposition 2 that

$$\begin{aligned} F_{j}^\mathrm{c}=\frac{a_{j}}{a_{j}+\rho \left( Mw\right) _{j}}. \end{aligned}$$

(28)

Consequently from assumption $Mw> 0$, we derive that

$$\begin{aligned}F_{j}^\mathrm{c}<1,\; \forall j\end{aligned}$$

Assume now for a moment that $\displaystyle \lim _{n\rightarrow +\infty } x_{j}^\mathrm{c} (1)=0$. From Gompertz dynamics (1), this implies that

$$\begin{aligned} x_j^\mathrm{c}(0)=0 \text{ or } F_{j}^\mathrm{c}(0)=1 \text{ or } \exp \left( 1+ r_j + \sum _{k=1}^{m}s_{jk}\log ( x_{k}(0)(1-F_k^\mathrm{c}(0))\right) =0 \end{aligned}$$

This is contradictory since the initial state $x_j^\mathrm{c}(0)$ is supposed to be strictly positive in all of its components and the exponential is also strictly positive.

We proceed iteratively to obtain the assertion for every time $t=2, \ldots$.

1.3 Proof of Proposition 7

By taking the logarithm of the exploited dynamics (2) at the steady state $x_j(t+1)=x_j(t)={x_{*}}_j$, we obtain

$$\begin{aligned} \log ({x_{*}}_{j})=\log ({x_{*}}_{j}-h_{j})+r_{j}+\sum _{k}s_{jk}\log ({x_{*}}_{k}-h_{k}) \end{aligned}$$

Since $h_{j}=F_{j}x_{j}$ it yields

$$\begin{aligned} \log ( {x_{*}}_{j})= & {} \log (\left( 1-F_{j}\right) {x_{*}}_{j})+r_{j}+\sum _{k}s_{jk}\log (\left( 1-F_{k}\right) {x_{*}}_{k}) \\ 0= & {} \log \left( 1-F_{j}\right) +r_{j}+\sum _{k}s_{jk}\log \left( 1-F_{k}\right) +\sum _{k}s_{jk}\log {x_{*}}_{k} \end{aligned}$$

In matrix form, it gives

$$\begin{aligned} 0= & {} (I+S)\log \left( 1-F\right) +r+S\log (x_{*}) \\ -S\log (x_{*})= & {} (I+S)\log \left( 1-F\right) +r \end{aligned}$$

where the notation $\log \left( x\right)$ means the vector of logarithms by species, namely $(\log \left( x\right) )_j=\log \left( x_j\right)$. Assuming that S is invertible, this reads:

$$\begin{aligned} \log (x_{*})=-S^{-1}L \end{aligned}$$

with $L=r+M' \log \left( 1-F\right)$. The comparison between species states in the cooperative $x_{*}^\mathrm{c}$ and non-cooperative $x_{*}^\mathrm{nc}$ cases yields

$$\begin{aligned} \log (x_{*}^\mathrm{c})-\log (x_{*}^\mathrm{nc})=-S^{-1}\left( L^\mathrm{c}-L^\mathrm{nc}\right) =-(I+S^{-1})\log \left( \dfrac{1-F^\mathrm{c}}{1-F^\mathrm{nc}}\right) . \end{aligned}$$

We deduce that

$$\begin{aligned} \begin{array}{rcl} \dfrac{\textsc {er}(x_{*}^\mathrm{c})}{\textsc {er}(x_{*}^\mathrm{nc})}&{}=&{}\dfrac{\exp \left( -(1, \ldots , 1).(I+S^{-1})^{-1}\log (x_{*}^\mathrm{c})\right) }{\exp \left( -(1, \ldots , 1).(I+S^{-1})^{-1}\log (x_{*}^\mathrm{nc})\right) }\\ \\ &{}=&{} \exp \left( -(1, \ldots , 1).(I+S^{-1})^{-1}\left( \log (x_{*}^\mathrm{c})-\log (x_{*}^\mathrm{nc})\right) \right) \\ &{}=&{}\exp \left( (1, \ldots , 1).\log \left( \dfrac{1-F^\mathrm{c}}{1-F^\mathrm{nc}}\right) \right) . \end{array} \end{aligned}$$

Since for every species $\dfrac{1-F^\mathrm{c}}{1-F^\mathrm{nc}}\ge 1$, we conclude with

$$\begin{aligned} \frac{\textsc {er}(x_{*}^\mathrm{c})}{\textsc {er}(x_{*}^\mathrm{nc})}\ge 1 \end{aligned}$$

1.4 Proof of Proposition 8

Consider the optimal cooperative $x^\mathrm{c}(t)$ and non-cooperative $x^\mathrm{nc}(t)$ trajectories starting from the same initial state $x_0$. Let us prove that

$$\begin{aligned} \textsc {Ecos}(x^\mathrm{nc}(t))\le \textsc {Ecos}(x^\mathrm{c}(t)),\;\forall t=0,1,\ldots \end{aligned}$$

Taking the logarithm formulation of Eq. (2), we can derive by iteration that

$$\begin{aligned} y(t)= M'^ty_0+\sum _{s=0}^{t-1}M'^s r+ \sum _{s=1}^{t}M'^s \log (1-F) \end{aligned}$$

We deduce that

$$\begin{aligned}\begin{array}{ccc} \textsc {Ecos}(x^\mathrm{nc}(t))-\textsc {Ecos}(x^\mathrm{c}(t))= & {} \displaystyle w'\sum _{s=1}^{t}M'^s \left( \log (1-F^\mathrm{nc})-\log (1-F^\mathrm{c})\right) \end{array}\end{aligned}$$

since the cooperative and non-cooperative initial states $y_0^\mathrm{nc}=y_0^\mathrm{c}$ coincide. Using matrix properties, the difference reads as follows:

$$\begin{aligned}\begin{array}{ccc} \textsc {Ecos}(x^\mathrm{nc}(t))-\textsc {Ecos}(x^\mathrm{c}(t)) &{}=&{} \displaystyle \sum _{s=1}^{t}w'M'^s \biggl (\log (1-F^\mathrm{nc})-\log (1-F^\mathrm{c})\biggr )\\ &{}=&{} \displaystyle \sum _{s=1}^{t}(M^sw)' \biggl (\log (1-F^\mathrm{nc})-\log (1-F^\mathrm{c})\biggr ) \end{array}\end{aligned}$$

The assumption of Proposition 8 guarantees that vector $M^sw$ is positive for every species j and every time s. Moreover, from Proposition 3 related to the gain from cooperation for catch rates, the difference $\log (1-F^\mathrm{nc})-\log (1-F^\mathrm{c})$ is always non-positive for every species j.

Appendix 2: Scilab Code for the Simulations

Below is the Scilab code used for the simulations. Scilab is an open-source software for numerical computation available at http://www.scilab.org/en/download/latest