Ecosystem Viable Yields

Abstract

The World Summit on Sustainable Development (Johannesburg, 2002) encouraged the application of the ecosystem approach by 2010. However, at the same summit, the signatory States undertook to restore and exploit their stocks at maximum sustainable yield (MSY), a concept and practice without ecosystemic dimension, since MSY is computed species by species, on the basis of a monospecific model. Acknowledging this gap, we propose a definition of “ecosystem viable yields” (EVY) as yields compatible (a) with biological safety levels (over which biomasses can be maintained for all times) and (b) with an ecosystem dynamics. The difference from MSY is that this notion is not based on equilibrium but on viability theory, which offers advantages for robustness. For a generic class of multispecies models with harvesting, we provide explicit expressions for the EVY. We apply our approach to the anchovy–hake couple in the Peruvian upwelling ecosystem.

Appendices

Appendix 1: Discrete Time Viability

Let us consider a nonlinear control system described in discrete time by the dynamic equation

$$ \left\{ \begin{array}{l} x(t+1)={f}\big(x(t),u(t)\big) \mbox{ for all } t \in \mathbb{N} ,\\ x(0)=x_0 \mbox{ given,} \end{array} \right . \label{eq:generaldyn} $$

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where the state variable x(t) belongs to the finite dimensional state space ${\mathbb X}=\mathbb{R}^{n_{{\mathbb X}}}$, the control variable u(t) is an element of the control set ${\mathbb U}=\mathbb{R}^{n_{{\mathbb U}}}$ while the dynamics f maps ${\mathbb X} \times {\mathbb U}$ into ${\mathbb X}$.

A controller or a decision maker describes “acceptable configurations of the system” through a set ${\mathbb D} \subset {\mathbb X} \times {\mathbb U}$ termed the acceptable set

$$\label{eq:constraint} \big(x(t),u(t)\big) \in {\mathbb D}\; \mbox{ for all }\; t \in \mathbb{N} \; , $$

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where ${\mathbb D}$ includes both system states and controls constraints.

The state constraints set $\mathbb{V}^0$ associated with ${\mathbb D}$ is obtained by projecting the acceptable set ${\mathbb D}$ onto the state space ${\mathbb X}$:

$$ \mathbb{V}^0 {:=} {\rm Proj}_{{\mathbb X}}({\mathbb D} ) = \{x \in {\mathbb X} \mid \exists u \in {\mathbb U} \, , \, (x,u) \in {\mathbb D} \} \; . \label{eq:state_constraints_set} $$

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Viability is defined as the ability to choose, at each time step t ∈ ℕ, a control $u(t) \in {\mathbb U}$ such that the system configuration remains acceptable. More precisely, viability occurs when the following set of initial states is not empty:

$$ \mathbb{V}({f},{\mathbb D} ) {:=} \left\{x_0\in {\mathbb X} \left| \begin{array}{l}\exists\; (u(0),u(1), \ldots ) \mbox{ and } \\ (x(0),x(1), \ldots )\\ \mbox{ satisfying\ Eqs.\ } 16 \mbox{ and } 17 \end{array} \right. \right\} \; . \label{eq:viability_kernel} $$

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The set $\mathbb{V}({f},{\mathbb D} )$ is called the viability kernel [2] associated with the dynamics f and the acceptable set ${\mathbb D}$. By definition, we have $\mathbb{V}({f},{\mathbb D}) \subset \mathbb{V}^0 ={\rm Proj}_{{\mathbb X}}({\mathbb D} )$, but in general, the inclusion is strict. For a decision maker or control designer, knowing the viability kernel is of practical interest since it describes the initial states for which controls can be found that maintain the system in an acceptable configuration forever. However, computing this kernel is not an easy task in general.

We now focus on some tools to achieve viability. A subset $\mathbb{V}$ is said to be weakly invariant for the dynamics f in the acceptable set ${\mathbb D}$ or a viability domain of f in ${\mathbb D}$, if

$$ \forall x \in \mathbb{V} \, , \quad \exists u \in {\mathbb U} \, , \quad (x,u) \in {\mathbb D} \mbox{ and } {f}(x,u) \in \mathbb{V} \; . \label{eq:viability_domain} $$

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That is, if one starts from $\mathbb{V}$, an acceptable control may transfer the state in $\mathbb{V}$. Moreover, according to viability theory [2], the viability kernel $\mathbb{V}({f},{\mathbb D} )$ turns out to be the union of all viability domains or also the largest viability domain:

$$\begin{array}{rll} \mathbb{V}({f},{\mathbb D}) &=& \bigcup \bigl\{ \mathbb{V},\;\mathbb{V} \subset \mathbb{V}^0,\;\mathbb{V}\mbox{ viability domain}\\ && {\kern18pt} \mbox{for ${f}$ in ${{\mathbb D}}$} \bigr\} \; . \label{eq:viability_kernel_union} \end{array} $$

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Viable controls are those controls $u \in {\mathbb U}$ such that $(x,u) \in {\mathbb D}$ and ${f}(x,u) \in \mathbb{V}({f},{\mathbb D})$.

A major interest of such a property lies in the fact that any viability domain for the dynamics f in the acceptable set ${\mathbb D}$ provides a lower approximation of the viability kernel. An upper approximation $\mathbb{V}_k$ of the viability kernel is given by the so-called viability kernel until time k associated with f in ${\mathbb D}$:

$$ \mathbb{V}_{k} {:=} \left\{x_0\in {\mathbb X} \left| \begin{array}{l}\exists\; (u(0),u(1), \ldots, u(k) ) \mbox{ and }\\ (x(0),x(1), \ldots, x(k) )\\ \mbox{ satisfying Eq.} \text{\ref{eq:generaldyn}} \mbox{ for } t=0,\ldots,k-1 \\ \mbox{ and Eq.\ } \text{\ref{eq:constraint}} \mbox{ for } t=0,\ldots,k \end{array} \right. \right\} \, . \label{eq:Vtau} $$

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We have

$$ \mathbb{V}({f},{\mathbb D} ) \subset \mathbb{V}_{k+1} \subset \mathbb{V}_{k} \subset \mathbb{V}_0 = \mathbb{V}^0 \, \mbox{ for all } k \in \mathbb{N} \; . \label{eq:inclusionsVtau} $$

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It may be seen by induction that the decreasing sequence of viability kernels until time k satisfies

$$\begin{array}{rll} \mathbb{V}_0& =& \mathbb{V}^0 \mbox{ and } \mathbb{V}_{k+1} \\&=& \big\{ x \in \mathbb{V}_k \; \left| \; \exists u \in {\mathbb U} \, , \, (x,u) \in {\mathbb D} \mbox{ and } {f}(x,u) \in \mathbb{V}_k \right. \big\} \; .\\ \label{eq:induction} \end{array} $$

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By Eq. 23, such an algorithm provides approximation from above of the viability kernel as follows:

$$ \mathbb{V}({f},{\mathbb D} ) \subset \displaystyle \operatornamewithlimits{\bigcap}\limits_{k \in \mathbb{N}} \mathbb{V}_k = \lim\limits_{k \to +\infty} \! \downarrow \! \mathbb{V}_k \; . \label{eq:algo1} $$

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Conditions ensuring that equality holds may be found in [32]. Notice that, when the decreasing sequence $(\mathbb{V}_{k})_{k \in \mathbb{N}}$ of viability kernels up to time k is stationary; its limit is the viability kernel. Indeed, if $\mathbb{V}_{k} = \mathbb{V}_{k+1}$ for some k, then $\mathbb{V}_{k}$ is a viability domain by Eq. 24. Now, by Eq. 19, $\mathbb{V}({f},{\mathbb D} )$ is the largest of viability domains. As a consequence, $\mathbb{V}_{k} = \mathbb{V}({f},{\mathbb D} )$ since $\mathbb{V}({f},{\mathbb D} ) \subset \mathbb{V}_{k}$ by Eq. 23. We shall use this property in the following Appendix 2.

Appendix 2: Viable Control of Generic Nonlinear Ecosystem Models with Harvesting

For a generic ecosystem model 1, we provide an explicit description of the viability kernel. Then, we shall specify the results for predator–prey systems, in particular, for discrete time Lotka–Volterra models.

The acceptable set ${\mathbb D}$ in Eq. 17 is defined by minimal biomass levels ${B_{y}}{^{\flat}}\geq 0$, ${B_{z}}{^{\flat}}\geq 0$ and minimal catch levels ${C_{y}}{^{\flat}}\geq 0$, ${C_{z}}{^{\flat}}\geq 0$:

$$\begin{array}{rll} {\mathbb D}&=&\big\{\,({y},{z},{v},{w})\in \mathbb{R}^4 \mid {y}\geq {B_{y}}{^{\flat}},\\ &&{\kern7pt} {z}\geq {B_{z}}{^{\flat}}, \; {v} {y}\geq{C_{y}}{^{\flat}}, \; {w} {z}\geq{C_{z}}{^{\flat}}\,\big\} \; . \label{eq:thresholds} \end{array} $$

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1.1 2.1   Expression of the Viability Kernel

The following Proposition 5 gives an explicit description of the viability kernel, under some conditions on the minimal levels.

Proposition 5

Assume that the function R _y :ℝ³ →ℝ is continuously decreasing in the control v and satisfies lim _{v → + ∞} R _y (y,z,v) ≤ 0, and that R _z :ℝ³ →ℝ is continuously decreasing in the control variable w and satisfies lim _{w → + ∞} R _z (y,z,w) ≤ 0. If the minimal levels in Eq. 26 are such that the following growth factors are greater than one

$$ {R_{{y}}}\left({B_{y}}{^{\flat}},{B_{z}}{^{\flat}},\frac{{C_{y}}{^{\flat}}}{{B_{y}}{^{\flat}}}\right) \geq 1, {R_{{z}}}\left({B_{y}}{^{\flat}},{B_{z}}{^{\flat}},\frac{{C_{z}}{^{\flat}}}{{B_{z}}{^{\flat}}}\right) \geq 1 \; , \label{eq:favorable_conditions2} $$

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the viability kernel associated with the dynamics f in Eq.  1 and the acceptable set ${\mathbb D}$ in Eq. 26 is given by

$$\begin{array}{rll} \mathbb{V}({f},{\mathbb D})& =& \left\{ \vphantom{\frac{{C_{y}}{^{\flat}}}{{y}}} ({y},{z}) \mid {y}\geq {B_{y}}{^{\flat}}, \; {z}\geq {B_{z}}{^{\flat}}, \right.\\ &&{\kern3.6pt} \left. \; {y}{R_{{y}}}\left({y},{z},\frac{{C_{y}}{^{\flat}}}{{y}}\right)\geq {B_{y}}{^{\flat}},\right.\\ && {\kern5.6pt}\left. {z}{R_{{z}}}\left({y},{z},\frac{{C_{z}}{^{\flat}}}{{z}}\right)\geq {B_{z}}{^{\flat}} \right\}. \label{eq:predator_prey_viability_kernel} \end{array} $$

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Proof

According to induction 24, we have:

$$\begin{array}{rll} \mathbb{V}_0 &=& \big\{\,({y},{z}) \left| {y}\geq {B_{y}}{^{\flat}}, {z}\geq {B_{z}}{^{\flat}} \right. \,\big\},\\[1mm] \mathbb{V}_1 {\kern-2pt}&=&{\kern-3pt} \left\{{\kern-1pt} ({y},{z}) {\kern-1pt}\left|{\kern-1pt} \begin{array}{lll} {y} {\kern-1pt} \geq{\kern-1pt} {B_{y}}{^{\flat}}, {z} {\kern-1pt}\geq{\kern-1pt} {B_{z}}{^{\flat}} \mbox{ and,}\\[2mm] {\kern10pt}\mbox{for some }\, ({v},{w})\geq 0,\\[1mm] {v} {y} {\kern-1pt}\geq{\kern-1pt} {C_{y}}{^{\flat}}, {w} {z} {\kern-1pt}\geq{\kern-1pt} {C_{z}}{^{\flat}}, {y}{R_{{y}}}({y},{z},{v}) {\kern-1pt}\geq{\kern-1pt} {B_{y}}{^{\flat}},\\[1mm] {z}{R_{{z}}}({y},{z},{w})\geq {B_{z}}{^{\flat}} \end{array} \right. \right\} \\[1mm] &=& \left\{({y},{z}) \left| {y}\geq {B_{y}}{^{\flat}}, {z}\geq {B_{z}}{^{\flat}}, {y}{R_{{y}}}{\kern-1pt}\left({\kern-1pt}{y},{z},\dfrac{{C_{y}}{^{\flat}}}{{y}}{\kern-1pt}\right){\kern-1pt}\geq {B_{y}}{^{\flat}}, \right.\right.\\[3mm] &&{\kern32.8pt} \left.\left. {z}{R_{{z}}}\left({y},{z},\dfrac{{C_{z}}{^{\flat}}}{{z}}\right)\geq {B_{z}}{^{\flat}} \right. \right\}\\ && \mbox{ because } {v} \mapsto {R_{{y}}}({y},{z},{v}) \mbox{ and } {w} \mapsto {R_{{z}}}({y},{z},{w}) \\ && {\kern3pt}\mbox{are decreasing, and thus we may select } \\ && {\kern3pt} {v}=\frac{{C_{y}}{^{\flat}}}{{y}}, \; {w} =\frac{{C_{z}}{^{\flat}}}{{z}}. \\ && {\kern3pt} \mbox{Denoting } {y}'= {y}{R_{{y}}}({y},{z},{v}), \,\, {z}'={z}{R_{{z}}}({y},{z},{w}),\\ &&{\kern3pt} \mbox{we obtain,}\\[4mm] \mathbb{V}_2 &=& \left\{({y},{z}) \left| \begin{array}{ll} {y}\geq {B_{y}}{^{\flat}}, {z}\geq {B_{z}}{^{\flat}}\\[3mm] {\kern10pt} \mbox{ and, for some }\, ({v},{w})\geq 0,\\[3mm] {v} {y}\geq {C_{y}}{^{\flat}},\,\, {w} {z}\geq {C_{z}}{^{\flat}}\\[3mm] {y}'\geq {B_{y}}{^{\flat}},\,\, {y}'{R_{{y}}} \left({y}',{z}',\dfrac{{C_{y}}{^{\flat}}}{{y}'}\right)\geq {B_{y}}{^{\flat}},\\[4mm] {z}'\geq {B_{z}}{^{\flat}},\,\, {z}'{R_{{z}}}\left({y}',{z}',\dfrac{{C_{z}}{^{\flat}}}{{z}'}\right)\geq {B_{z}}{^{\flat}} \end{array} \right. \right\} \; . \end{array}$$

We shall now make use of the property, recalled in Appendix 1, that when the decreasing sequence $(\mathbb{V}_{k})_{k \in \mathbb{N}}$ of viability kernels up to time k is stationary, its limit is the viability kernel $\mathbb{V}({f},{\mathbb D} )$. Hence, it suffices to show that $\mathbb{V}_1\subset \mathbb{V}_2$ to obtain that $\mathbb{V}({f},{\mathbb D}) = \mathbb{V}_1$. Let $({y},{z})\in \mathbb{V}_1$, so that

$$\begin{array}{lll} && {y}\geq {B_{y}}{^{\flat}}, \quad {z}\geq {B_{z}}{^{\flat}} \mbox{ and }~ {y}{R_{{y}}}\left({y},{z},\frac{{C_{y}}{^{\flat}}}{{y}}\right)\geq {B_{y}}{^{\flat}}, \\ && {z}{R_{{z}}}\left({y},{z},\frac{{C_{z}}{^{\flat}}}{{z}}\right)\geq {B_{z}}{^{\flat}} \; . \end{array}$$

Since R _y :ℝ³ →ℝ is continuously decreasing in the control variable, with lim _{v → + ∞} R _y (y,z,v) ≤ 0 and since ${y}{R_{{y}}}({y},{z},\frac{{C_{y}}{^{\flat}}}{{y}}) \geq {B_{y}}{^{\flat}}$, there exists a $\hat{{v}} \geq \frac{{C_{y}}{^{\flat}}}{{y}}$ (depending on y and z) such that ${y}'={y}{R_{{y}}}({y},{z},\hat{{v}})= {B_{y}}{^{\flat}}$. The same holds for R _z :ℝ³ →ℝ and ${z}'={z}{R_{{z}}}({y},{z},\hat{{w}})= {B_{z}}{^{\flat}}$. By Eq. 27, we deduce that

$$\begin{array}{lll} &&{y}'{R_{{y}}}({y}',{z}',\frac{{C_{y}}{^{\flat}}}{{y}'}) = {B_{y}}{^{\flat}} {R_{{y}}}\left({B_{y}}{^{\flat}},{B_{z}}{^{\flat}},\frac{{C_{y}}{^{\flat}}}{{B_{y}}{^{\flat}}}\right) \geq {B_{y}}{^{\flat}} \;\mbox{ and }\\ && {z}'{R_{{z}}}\left({y}',{z}',\frac{{C_{z}}{^{\flat}}}{{z}'}\right) = {B_{z}}{^{\flat}} {R_{{z}}}\left({B_{y}}{^{\flat}},{B_{z}}{^{\flat}},\frac{{C_{z}}{^{\flat}}}{{B_{z}}{^{\flat}}}\right) \geq {B_{z}}{^{\flat}} \; . \end{array}$$

The inclusion $\mathbb{V}_1\subset \mathbb{V}_2$ follows.□

Corollary 6

Suppose that the assumptions of Proposition 2 are satisfied. Denoting

$$ \left\{ \begin{array}{rcl} \hat{{v}}({y},{z}) &=& \max \left\{ {v} \geq \dfrac{{C_{y}}{^{\flat}}}{{y}} \mid {y}{R_{{y}}}({y},{z},{v})= {y}{^{\flat}} \right\} \; , \\[4mm] \hat{{w}}({y},{z}) &=& \max \left\{ {w} \geq \dfrac{{C_{z}}{^{\flat}}}{{z}} \mid {z}{R_{{z}}}({y},{z},{w})={z}{^{\flat}} \right\} \; , \end{array} \right. $$

the set of viable controls is given by

$$ {\mathbb U}_{\mathbb{V}({f},{\mathbb D})}({y},{z}) = \left\{({v},{w}) \left| \begin{array}{l} \hat{{v}}({y},{z}) \geq {v}\geq \dfrac{{C_{y}}{^{\flat}}}{{y}}, \\[3mm] \hat{{w}}({y},{z}) \geq {w}\geq \dfrac{{C_{z}}{^{\flat}}}{{z}}, \\[3mm] {y}'{R_{{y}}}\left({y}',{z}',\dfrac{{C_{y}}{^{\flat}}}{{y}'}\right)\geq {y}{^{\flat}},\\[3mm] {z}'{R_{{z}}}\left({y}',{z}',\dfrac{{C_{z}}{^{\flat}}}{{z}'}\right)\geq {z}{^{\flat}} \end{array} \right. \right\} \; , $$

where y′ = yR _y (y,z,v), z′ = zR _z (y,z,w) .

1.2 2.2   Proof of Proposition 4

Proof

By Eq. 9 and the property that both R _y and R _z are decreasing in the control variable, the quantities 8 exist.

Also since both R _y and R _z are decreasing in the control variable, we obtain that

$$\begin{array}{rll} &&{\kern-6pt} {R_{{y}}}\left({B_{y}}{^{\flat}},{B_{z}}{^{\flat}}, \frac{{C_{y}}{^{\flat,\star}}({y}_0,{z}_0)}{{B_{y}}{^{\flat}}}\right) \\ &&{\kern4pt} \geq {R_{{y}}}\left({B_{y}}{^{\flat}},{B_{z}}{^{\flat}}, \frac{{C_{y}}{^{\flat,\star}}}{{B_{y}}{^{\flat}}}\right) = 1 \mbox{ and }\\ &&{\kern-6pt} {R_{{z}}}\left({B_{y}}{^{\flat}},{B_{z}}{^{\flat}}, \frac{{C_{z}}{^{\flat,\star}}({y}_0,{z}_0)}{{B_{z}}{^{\flat}}}\right) \\ && {\kern4pt} \geq {R_{{z}}}\left({B_{y}}{^{\flat}},{B_{z}}{^{\flat}}, \frac{{C_{z}}{^{\flat,\star}}}{{B_{z}}{^{\flat}}}\right) = 1 \; . \end{array}$$

To end, the above inequalities and the assumption that ${y}_0 \geq {B_{y}}{^{\flat}}$ and ${z}_0 \geq {B_{z}}{^{\flat}}$ allow us to conclude, thanks to Proposition 5, that (y ₀,z ₀) belongs to the viability kernel $\mathbb{V}({f},{\mathbb D})$ given in Eq. 28.

In other words, starting from the initial point (y(t ₀),z(t ₀)) = (y ₀,z ₀) , there exists an appropriate harvesting path which can provide, for all times, at least the catches Eq. 8.□