Robust Viable Analysis of a Harvested Ecosystem Model

Abstract

The World Summit on Sustainable Development (Johannesburg, 2002) encouraged the adoption of an ecosystem approach. In this perspective, we propose a theoretical management framework that deals jointly with three issues: (i) ecosystem dynamics, (ii) conflicting issues of production and preservation, and (iii) robustness with respect to dynamics uncertainties. We consider a discrete-time two-species dynamic model, where states are biomasses and where two controls act as harvesting efforts of each species. Uncertainties take the form of disturbances affecting each species growth factors and are assumed to take their values in a known given set. We define the robust viability kernel as the set of initial species biomasses such that at least one harvesting strategy guarantees minimal production and preservation levels for all times, whatever the uncertainties. We apply our approach to the anchovy-hake couple in the Peruvian upwelling ecosystem. We find that accounting for uncertainty sensibly shrinks the deterministic viability kernel (without uncertainties). We comment on the management implications of comparing robust viability kernels (with uncertainties) and the deterministic one (without uncertainties).

Appendices

Appendix A: The Deterministic Viability Kernel

The deterministic viability kernel, $\mathbb {V}iab(t_{0})$, associated with the following dynamics (19) and constraints (20) and (21), for t = t ₀,…,T, is the set of viable states defined as follows. A couple (y ₀, z ₀) of initial biomasses is said to be a viable state if there exist a trajectory of harvesting efforts (controls) (υ _y (t),υ _z (t))∈[0, 1], t = t ₀,…,T−1, such that the state path $\{\left (y(t),z(t)\right )\}_{t=t_{0},\ldots ,T},$ and control path $\{\left (\upsilon _{y}(t),\upsilon _{z}(t)\right )\}_{t=t_{0},\ldots ,T-1}$, solution of³

$$ \left\{ \begin{array}{rcl} y(t+1) &=& y(t) \mathcal{R}_{y}\left(y(t),z(t)\right)\left(1-\upsilon_{y}(t)\right) \; , \\ z(t+1) &=& z(t) \mathcal{R}_{z}\left(y(t),z(t)\right)\left(1-\upsilon_{z}(t)\right) \; , \end{array} \right. $$

(19)

starting from (y(t ₀),z(t ₀))=(y ₀, z ₀) satisfy the following goals:

• Preservation (minimal biomass levels): for all t = t ₀,…,T

  $$ y(t) \geq y^{\flat} \; , \quad z(t) \geq z^{\flat} \; , and $$

  (20)

• Production requirements (minimal catch levels): for all t = t ₀,…,T−1

  $$\begin{array}{@{}rcl@{}} \upsilon_{y}(t) y(t) \mathcal{R}_{y}\left(y(t),z(t)\right)&\geq& Y^{\flat} \; ,\\ \quad \upsilon_{z}(t) z(t) \mathcal{R}_{z}\left(y(t),z(t)\right)&\geq& Z^{\flat} \; . \end{array} $$

  (21)

We now turn to the proof of Proposition 1 in Section 2.3.

Proof

Consider y ^♭ ≥ 0, z ^♭ ≥ 0, Y ^♭ ≥ 0, Z ^♭ ≥ 0. We set

$$\mathbb{V}_{0} = \left\{ (y,z) \in \mathbb{R}_{+}^{2} \left| y\geq y^{\flat}, z\geq z^{\flat} \right. \right\} $$

and we define a sequence $(\mathbb {V}_{k})_{k\in \mathbb {N}}$ inductively by

$$\begin{array}{@{}rcl@{}} \mathbb{V}_{k+1}\!&=&\!\left\{\!\right.(y,z)\!\in\mathbb{V}_{k}\! \left|\! \right.\!\exists (\upsilon_{y}, \upsilon_{z})\!\in\![0,\!1] \text{ such\hspace*{-.5pt} that}\\[2pt] y\upsilon_{y} \mathcal{R}_{y}\hspace*{-.5pt}(y,\!z) &\geq& y^{\flat}, z\upsilon_{z}\mathcal{R}_{z}(y,z)\geq z^{\flat}, \text{and}\\[2pt] y^{\prime}&=&\!y\mathcal{R}_{y}(y,z)(1\!-\!\upsilon_{y}), z^{\prime}\!=\!z\mathcal{R}_{z}(y,z)\hspace*{-.5pt}(1\!-\!\upsilon_{z}\hspace*{-.5pt}),\\[2pt] &&\left. \text{are such that} (y^{\prime},z^{\prime})\in\mathbb{V}_{k} \right\}.\end{array} $$

For k = 0, we obtain

$$\begin{array}{@{}rcl@{}} \mathbb{V}_{1} &=& \left\{(y,z) \left| \begin{array}{lll} y\geq y^{\flat}, z\geq z^{\flat} \text{ and, for some }\, (\upsilon_{y},\upsilon_{z})\in[0, 1],\\ \upsilon_{y}y\mathcal{R}_{y}(y,z)\geq Y^{\flat}, \upsilon_{z}z\mathcal{R}_{z}(y,z)\geq Z^{\flat},\\ y\mathcal{R}_{y}(y,z)(1-\upsilon_{y})\geq y^{\flat}, z\mathcal{R}_{z}(y,z)(1-\upsilon_{z})\geq z^{\flat} \end{array} \right. \right\} \\ &=& \left\{(y,z) \left| \begin{array}{lll} y\geq y^{\flat}, z\geq z^{\flat} \text{ for which there exist} ~(\upsilon_{y},\upsilon_{y}) \text{such that}\\ \frac{Y^{\flat}}{y\mathcal{R}_{y}(y,z)}\leq \upsilon_{y}\leq \frac{y\mathcal{R}_{y}(y,z)-y^{\flat}}{y\mathcal{R}_{y}(y,z)}\quad \text{and}\quad 0\leq \upsilon_{y}\leq 1,\\ \frac{Z^{\flat}}{z\mathcal{R}_{z}(y,z)}\leq \upsilon_{z}\leq \frac{z\mathcal{R}_{z}(y,z)-z^{\flat}}{z\mathcal{R}_{z}(y,z)}\quad \text{and}\quad 0\leq \upsilon_{z}\leq 1 \end{array} \right. \right\} \\ &=& \left\{(y,z) \left| \begin{array}{lll} y\geq y^{\flat}, z\geq z^{\flat}, \\ \sup\left\{0,\frac{Y^{\flat}}{y\mathcal{R}_{y}(y,z)}\right\}\leq\inf\left\{1,1-\frac{y^{\flat}}{y\mathcal{R}_{y}(y,z)}\right\} \\ \sup\left\{0, \frac{Z^{\flat}}{z\mathcal{R}_{z}(y,z)}\right\}\leq\inf\left\{1,1- \frac{z^{\flat}}{z\mathcal{R}_{z}(y,z)}\right\} \end{array} \right. \right\} \\ &=& \left\{(y,z) \left| y\geq y^{\flat}, z\geq z^{\flat}, \frac{Y^{\flat}}{y\mathcal{R}_{y}(y,z)}\leq \frac{y\mathcal{R}_{y}(y,z)-y^{\flat}}{y\mathcal{R}_{y}(y,z)}, \frac{z^{\flat}}{z\mathcal{R}_{z}(y,z)}\leq \frac{z\mathcal{R}_{z}(y,z)-z^{\flat}}{z\mathcal{R}_{z}(y,z)} \right. \right\}\\ &=& \left\{(y,z) \left| y\geq y^{\flat}, z\geq z^{\flat}, y^{\flat}\leq y\mathcal{R}_{y}(y,z)-y^{\flat}, z^{\flat}\leq z\mathcal{R}_{z}(y,z)-z^{\flat} \right. \right\} \; . \end{array} $$

Then, for k = 1, we obtain

$$\begin{array}{@{}rcl@{}} \mathbb{V}_{2} &=& \left\{(y,z) \left| \begin{array}{l} y\geq y^{\flat}, z\geq z^{\flat}\text{ and, for some }\, (\upsilon_{y},\upsilon_{z})\in[0, 1],\\ \upsilon_{y} y\mathcal{R}_{y}(y,z)\geq Y^{\flat},\,\, \upsilon_{z}z\mathcal{R}_{z}(y,z)\geq Z^{\flat} \\ \text{and such that } (y^{\prime},z{^{\prime}})\in\mathbb{V}_{1} \\ \text{where } y{^{\prime}}= y\mathcal{R}_{y}(y,z)(1-\upsilon_{y}), \,\, z{^{\prime}}=z\mathcal{R}_{z}(y,z)(1-\upsilon_{z}) \end{array} \right. \right\} \\ &=& \left\{(y,z) \left| \begin{array}{l} y\geq y^{\flat}, z\geq z^{\flat}\text{ and, for some }\, (\upsilon_{y},\upsilon_{z})\in[0, 1],\\ \upsilon_{y} y\mathcal{R}_{y}(y,z)\geq Y^{\flat},\,\, \upsilon_{z}z\mathcal{R}_{z}(y,z)\geq Z^{\flat},\,\, y{^{\prime}}\geq y^{\flat},\,\, z{^{\prime}}\geq z^{\flat},\\ Y^{\flat}\leq y{^{\prime}}\mathcal{R}_{y}(y{^{\prime}},z{^{\prime}})- y^{\flat},\,\, Z^{\flat}\leq z{^{\prime}}\mathcal{R}_{z}(y{^{\prime}},z{^{\prime}})- z^{\flat} \\ \text{where } y{^{\prime}}= y\mathcal{R}_{y}(y,z)(1-\upsilon_{y}), \,\, z{^{\prime}}=z\mathcal{R}_{z}(y,z)(1-\upsilon_{z}) \end{array} \right. \right\} \; . \end{array} $$

We now make use of the property (see [9]) that, when the decreasing sequence $(\mathbb {V}_{k})_{k \in \mathbb {N}}$ is stationary, its limit is the viability kernel $\mathbb {V}iab(t_{0})$. Hence, it suffices to show that $\mathbb {V}_{1}\subset \mathbb {V}_{2}$ to obtain that $\mathbb {V}iab(t_{0}) = \mathbb {V}_{1}$.

Let $(y,z)\in \mathbb {V}_{1}$. We have that

$$\begin{array}{@{}rcl@{}} y&\geq& y^{\flat}, \quad z\geq z^{\flat}~\text{and }y\mathcal{R}_{y}(y,z)\,-\,y^{\flat}\!\geq\! Y^{\flat},\! \quad z\mathcal{R}_{z}(y,z)\,-\,z^{\flat}\!\geq\! Z^{\flat}. \end{array} $$

Let us set $\hat {\upsilon }_{y}= \frac {y\mathcal {R}_{y}(y,z)-y^{\flat }}{y\mathcal {R}_{y}(y,z)}$, which has the property that $y{^{\prime }}=y\mathcal {R}_{y}(y,z)(1-\hat {\upsilon }_{y})= y^{\flat }$. We prove that $\hat {\upsilon }_{y}\in [0, 1]$. Indeed, on the one hand, we have that $\hat {\upsilon }_{y}\leq 1$ since $1-\hat {\upsilon }_{y}=y^{\flat } / y\mathcal {R}_{y}(y,z)$, where y ^♭ ≥ 0. On the other hand, since by assumption $y\mathcal {R}_{y}(y,z)-y^{\flat }\geq Y^{\flat }\geq 0$, we deduce that $\hat {\upsilon }_{y}\geq 0$. The same holds true for $\hat {\upsilon }_{z}$ and $z{^{\prime }}=z\mathcal {R}_{z}(y,z)(1-\hat {\upsilon }_{z})= z^{\flat }$. By Eq. 9, we deduce that

$$\begin{array}{@{}rcl@{}} y{^{\prime}}\mathcal{R}_{y}\left(y{^{\prime}},z{^{\prime}}\right)-y^{\flat} &=& y^{\flat}\mathcal{R}_{y}\left(y^{\flat},z^{\flat}\right)-y^{\flat}\geq Y^{\flat} \text{ and }\\ z{^{\prime}}\mathcal{R}_{z}\left(y{^{\prime}},z{^{\prime}}\right)-z^{\flat} &=& z^{\flat} \mathcal{R}_{z}\left(y^{\flat},z^{\flat}\right)-z^{\flat} \geq Z^{\flat} \; . \end{array} $$

The inclusion $\mathbb {V}_{1}\subset \mathbb {V}_{2}$ follows; hence, $\mathbb {V}iab(t_{0}) = \mathbb {V}_{1}$, and (10) holds true. □

The viable controls attached to a given viable state (y, z)∈Vi a b(t ₀) are the admissible controls (υ _y , υ _z ) such that the image by the dynamics (19) is in $\mathbb {V}iab(t_{0})$.

Corollary 1

Suppose that the assumptions of Proposition 1 are satisfied. The set of viable controls associated with the state (y,z) is

$$\left\{(\upsilon_{y},\upsilon_{z}) \in [0, 1]^{2} \left| \begin{array}{l} \frac{y\mathcal{R}_{y}(y,z)-y^{\flat}}{y\mathcal{R}_{y}(y,z)} \geq \upsilon_{y}\geq \frac{y^{\flat}}{y\mathcal{R}_{y}(y,z)}, \quad \frac{z\mathcal{R}_{z}(y,z)-z^{\flat}}{z\mathcal{R}_{z}(y,z)} \geq \upsilon_{z}\geq \frac{z^{\flat}}{z\mathcal{R}_{z}(y,z)}, \\ y{^{\prime}}\mathcal{R}_{y}(y{^{\prime}},z{^{\prime}})-y^{\flat}\geq y^{\flat},\,\, z{^{\prime}}\mathcal{R}_{z}(y{^{\prime}},z{^{\prime}})-z^{\flat}\geq z^{\flat} \end{array} \right. \right\} \; , $$

where $y{^{\prime }}= y\mathcal {R}_{y}(y,z)(1-\upsilon _{y}),\,\, z{^{\prime }}=z\mathcal {R}_{z}(y,z)(1-\upsilon _{z})$.

Appendix B: Numerical Computation of Robust Viability Kernels

We first sketch how to establish a dynamic programming equation associated with dynamics (1), preservation (7), and production (8) minimal thresholds. Then, we depict a numerical discretization scheme to solve this equation numerically.

2.1 B.1 Dynamic Programming Equation

The dynamic programming equation associated with dynamics (1), preservation (7), and production (8) minimal thresholds is given by⁴

$$\begin{array}{@{}rcl@{}} V_{T}(y,z)&=& 1_{\mathbb{A}}(y,z),\\ V_{t}(y,z)&=& 1_{\mathbb{A}}(y,z) \max_{(\upsilon_{y},\upsilon_{z}) \in [0, 1]^{2}} \min_{(\varepsilon_{y},\varepsilon_{z}) \in \mathbb{S}(t)}\\ \left[ 1_{\mathbb{B}(y,z,\varepsilon_{y},\varepsilon_{z})}(\upsilon_{y},\upsilon_{z}) V_{t+1}\!\left(G(y,z,\upsilon_{y},\upsilon_{z},\varepsilon_{y},\varepsilon_{z}) \right)\right]\!, \end{array} $$

(22)

for all t = t ₀,…,T−1, where the continuous function G denotes the dynamics (1)

$$ G(y,z,\upsilon_{y},\upsilon_{z},\varepsilon_{y},\varepsilon_{z}) = \left\{ \begin{array}{l} y \mathcal{R}_{y}\left(y ,z ,\varepsilon_{y} \right)\left(1-\upsilon_{y} \right) \; , \\ z \mathcal{R}_{z}\left(y ,z ,\varepsilon_{z} \right)\left(1-\upsilon_{z} \right) \; , \end{array} \right. $$

(23)

where $\mathbb {A}$ stands for the subset of biomass satisfying conservation objectives (7)

$$ \mathbb{A}=\left\{ (y,z) \mid y \geq y^{\flat} \; , z \geq z^{\flat} \right\} = y^{\flat} , +\infty \times z^{\flat} , +\infty \; , $$

(24)

and where $\mathbb {B}(y,z,\varepsilon _{y},\varepsilon _{z})$ stands for the subset of catches satisfying minimal production requirements (8)

$$\begin{array}{@{}rcl@{}} \mathbb{B}(y,z,\varepsilon_{y},\varepsilon_{z}) &=& \{ (\upsilon_{y},\upsilon_{z}) \in [0, 1]^{2} \mid \upsilon_{y} y \mathcal{R}_{y}(y,z ,\varepsilon_{y} )\\&\geq& Y^{\flat},\, \upsilon_{z}z \mathcal{R}_{z}(y,z ,\varepsilon_{z} )\geq Z^{\flat}\} \; . \end{array} $$

(25)

The notation $1_{\mathbb {A}}(y,z)$ is the indicator function of the set $\mathbb {A}$: it takes the value 1 when $(y,z)\in \mathbb {A}$ and 0 else. The same holds for $1_{\mathbb {B}(y,z,\varepsilon _{y},\varepsilon _{z})}(\upsilon _{y},\upsilon _{z})$.

It turns out that, for all t = t ₀,…,T, the robust viability value function V _t is the indicator function $1_{\mathbb {V}iab^{R}(t)}$ of the robust viability kernel $\mathbb {V}iab^{R}(t)$ (see [6]). The sketch of the proof is as follows by backward induction.

By Eq. 22, we have that $V_{T}=1_{\mathbb {A}}=1_{\mathbb {V}iab^{R}(T)}$. Now, assume that V _{t + 1} = $1_{\mathbb {V}iab^{R}(t+1)}$. When the operation $\min _{(\varepsilon _{y},\varepsilon _{z}) \in \mathbb {S}(t)}$ is performed in Eq. 22, the result is 1 if, and only if, for all uncertainties $(\varepsilon _{y},\varepsilon _{z}) \in \mathbb {S}(t)$, we have both $1_{\mathbb {B}(y,z,\varepsilon _{y},\varepsilon _{z})} (\upsilon _{y},\upsilon _{z}) = 1$ and $1_{\mathbb {V}iab^{R}(t)} \left (G(y,z,\upsilon _{y},\upsilon _{z},\varepsilon _{y},\varepsilon _{z}) \right ) = 1$, that is, both efforts (υ _y , υ _z ) satisfy minimal production requirements (8) and the images G(y, z, υ _y , υ _z , ε _y , ε _z ) by the dynamics G belong to the viability kernel Vi a b ^R(t). Then, the operation $\max _{(\upsilon _{y},\upsilon _{z}) \in [0, 1]^{2}}$ yields 1 if, and only if, there is at least one control (υ _y , υ _z )—indeed achieved by continuity of the dynamics G in Eq. 23—such that (8) is satisfied and G(y, z, υ _y , υ _z , ε _y , ε _z )∈Vi a b ^R(t). The term $1_{\mathbb {A}}(y,z)=1$ if, and only if, the conservation objectives (7) are satisfied. To end, we obtain that V _t (y, z) = 1 if, and only if, there exists at least one control (υ _y , υ _z ) such that the conservation objectives (7) and the production requirements (8) are satisfied, and that the images G(y, z, υ _y , υ _z , ε _y , ε _z ) by the dynamics G belong to the viability kernel Vi a b ^R(t) for all uncertainties $(\varepsilon _{y},\varepsilon _{z}) \in \mathbb {S}(t)$. By a simple extension of the results in [6] and [10], we have just characterized $\mathbb {V}iab^{R}(t)$.

2.2 2.2 Numerical Resolution of the Dynamic Programming Equation

Now, we expose how we proceed to find the robust viability kernel numerically thanks to the dynamic programming (22).

We discretize biomass, harvesting effort, and uncertainty values. A top loop for time steps embraces two nested loops for state variables y and z respectively. Next, loops over uncertainties nested in loops over harvesting efforts allow us to obtain the set of images associated with a biomass couple (some of these steps are actually done through matrix computing). Images for target constraints that are not satisfied are set equal to zero. We then project these images on the value function grid of the previous period through linear interpolation. At given efforts, we retain the minimum value obtained over all uncertainty couples. Then, we retain the highest value produced by an effort couple among all tested. It is this value that is multiplied with the value function of the current time period at the location of the biomass couple at stake. The robust viability kernel is defined as the set of grid points where the value function is equal to 1. This implies that biomass couples for which, at a date t, all images do not fall between four 1 in the interpolation are excluded from the robust viability kernel (in the sense that we provide robustness with respect to grid approximation).