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Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viable harvest of monotone bioeconomics models

Preservation and production issues Michel De Lara1 Pedro Gajardo2 H´ ector Ram´ ırez C.3

1CERMICS, Universit´

e de Paris-Est, France

2Departamento de Matem´

atica, Universidad T´ ecnica Federico Santa Mar´ ıa

3Departamento de Ingenier´

ıa Matem´ atica, Universidad de Chile

Sixi` emes Journ´ ees Franco-Chiliennes d’Optimisation May 20th, 2008 Universit´ e du Sud Toulon-Var

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Outline

The core model for fishery management decisions The model Some questions about sustainability of landings Discrete time viability issues Monotonicity properties Viability kernel properties Maximum sustainable yield Minimal viable feedback The Patagonian toothfish (L´ egine australe) Questions & Answers

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Outline

The core model for fishery management decisions The model Some questions about sustainability of landings Discrete time viability issues Monotonicity properties Viability kernel properties Maximum sustainable yield Minimal viable feedback The Patagonian toothfish (L´ egine australe) Questions & Answers

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Harvested fish population age structured model

Example

Typical stock-recruitment relationship:

◮ Constant: ϕ(B) = R. ◮ Linear: ϕ(B) = RB. ◮ Beverton-Holt: ϕ(B) = B α+βB. ◮ Ricker: ϕ(B) = αBe−βB.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Harvested fish population age structured model

Example

Typical stock-recruitment relationship:

◮ Constant: ϕ(B) = R. ◮ Linear: ϕ(B) = RB. ◮ Beverton-Holt: ϕ(B) = B α+βB. ◮ Ricker: ϕ(B) = αBe−βB.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Harvested fish population age structured model

Example

Typical stock-recruitment relationship:

◮ Constant: ϕ(B) = R. ◮ Linear: ϕ(B) = RB. ◮ Beverton-Holt: ϕ(B) = B α+βB. ◮ Ricker: ϕ(B) = αBe−βB.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Harvested fish population age structured model

Example

Typical stock-recruitment relationship:

◮ Constant: ϕ(B) = R. ◮ Linear: ϕ(B) = RB. ◮ Beverton-Holt: ϕ(B) = B α+βB. ◮ Ricker: ϕ(B) = αBe−βB.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

The harvest term

The exploitation is described by catch-at-age Ca and yield Y, both defined for a given vector of abundance N and a given control λ : Ca

• N, λ
• =

λFa λFa + Ma

• 1 − e−(Ma+λFa)

Na The production in term of biomass is : Y

• N, λ
• =

A

• a=1

wa Ca(N, λ)

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Questions

◮ Given a desirable level of landings (tons), what are the

vectors of abundances N = (Na)a=1,..,A (initial conditions) for which one can always harvest at least that quantity?

◮ What levels of catch (landings) are non sustainable? ◮ Given an abundance at age N = (Na)a=1,..,A What is the

maximum sustainable yield starting from N respecting preservation constraints?

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Questions

◮ Given a desirable level of landings (tons), what are the

vectors of abundances N = (Na)a=1,..,A (initial conditions) for which one can always harvest at least that quantity?

◮ What levels of catch (landings) are non sustainable? ◮ Given an abundance at age N = (Na)a=1,..,A What is the

maximum sustainable yield starting from N respecting preservation constraints?

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Questions

◮ Given a desirable level of landings (tons), what are the

vectors of abundances N = (Na)a=1,..,A (initial conditions) for which one can always harvest at least that quantity?

◮ What levels of catch (landings) are non sustainable? ◮ Given an abundance at age N = (Na)a=1,..,A What is the

maximum sustainable yield starting from N respecting preservation constraints?

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Outline

The core model for fishery management decisions The model Some questions about sustainability of landings Discrete time viability issues Monotonicity properties Viability kernel properties Maximum sustainable yield Minimal viable feedback The Patagonian toothfish (L´ egine australe) Questions & Answers

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Discrete time control system

Let us consider a nonlinear control system described in discrete time by the difference equation N(t + 1) = g(N(t), λ(t)), ∀t ∈ N, N0 given, where

◮ The state variable N(t) belongs to the state space X ⊆ I

RnX.

◮ The control variable λ(t) is an element of the control set

U ⊆ I RnU.

◮ The dynamics g maps X × U into X.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Discrete time control system

Let us consider a nonlinear control system described in discrete time by the difference equation N(t + 1) = g(N(t), λ(t)), ∀t ∈ N, N0 given, where

◮ The state variable N(t) belongs to the state space X ⊆ I

RnX.

◮ The control variable λ(t) is an element of the control set

U ⊆ I RnU.

◮ The dynamics g maps X × U into X.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Discrete time control system

Let us consider a nonlinear control system described in discrete time by the difference equation N(t + 1) = g(N(t), λ(t)), ∀t ∈ N, N0 given, where

◮ The state variable N(t) belongs to the state space X ⊆ I

RnX.

◮ The control variable λ(t) is an element of the control set

U ⊆ I RnU.

◮ The dynamics g maps X × U into X.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Discrete time control system

Let us consider a nonlinear control system described in discrete time by the difference equation N(t + 1) = g(N(t), λ(t)), ∀t ∈ N, N0 given, where

◮ The state variable N(t) belongs to the state space X ⊆ I

RnX.

◮ The control variable λ(t) is an element of the control set

U ⊆ I RnU.

◮ The dynamics g maps X × U into X.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Desirable configurations

A decision maker describes desirable configurations of the system through a set D ⊂ X × U termed the desirable set (N(t), λ(t)) ∈ D, ∀t ∈ N, where D includes both system states and controls constraints.

Example

◮ Dprotect := {(N, λ) : N ≥ ¯

N}

◮ Dyield := {(N, λ) : Y(N, λ) ≥ ymin,

S S B(N) ≥ Blim}

◮ DICES := {(N, λ) : S

S B(N) ≥ Blim, F(λ) ≤ Flim}

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Desirable configurations

A decision maker describes desirable configurations of the system through a set D ⊂ X × U termed the desirable set (N(t), λ(t)) ∈ D, ∀t ∈ N, where D includes both system states and controls constraints.

Example

◮ Dprotect := {(N, λ) : N ≥ ¯

N}

◮ Dyield := {(N, λ) : Y(N, λ) ≥ ymin,

S S B(N) ≥ Blim}

◮ DICES := {(N, λ) : S

S B(N) ≥ Blim, F(λ) ≤ Flim}

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Desirable configurations

A decision maker describes desirable configurations of the system through a set D ⊂ X × U termed the desirable set (N(t), λ(t)) ∈ D, ∀t ∈ N, where D includes both system states and controls constraints.

Example

◮ Dprotect := {(N, λ) : N ≥ ¯

N}

◮ Dyield := {(N, λ) : Y(N, λ) ≥ ymin,

S S B(N) ≥ Blim}

◮ DICES := {(N, λ) : S

S B(N) ≥ Blim, F(λ) ≤ Flim}

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Desirable configurations

A decision maker describes desirable configurations of the system through a set D ⊂ X × U termed the desirable set (N(t), λ(t)) ∈ D, ∀t ∈ N, where D includes both system states and controls constraints.

Example

◮ Dprotect := {(N, λ) : N ≥ ¯

N}

◮ Dyield := {(N, λ) : Y(N, λ) ≥ ymin,

S S B(N) ≥ Blim}

◮ DICES := {(N, λ) : S

S B(N) ≥ Blim, F(λ) ≤ Flim}

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viability domains and viability kernel

Definition

◮ V ⊂ X is a Viability Domain if for all N ∈ V there exists

λ ∈ U such that (N, λ) ∈ D and g(N, λ) ∈ V.

◮ Viability kernel

V(g, D) =        N0 ∈ X : there exist λ(0), λ(1), λ(2), ... N(0), N(1), N(2), ...such that N(0) = N0 N(t + 1) = g(N(t), λ(t)) and (N(t), λ(t)) ∈ D

Goals

◮ Determine or approximate the viability kernel V(g, D)

for a given dynamics g and a given desirable set D.

◮ Determine when a given set V is a viability domain.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viability domains and viability kernel

Definition

◮ V ⊂ X is a Viability Domain if for all N ∈ V there exists

λ ∈ U such that (N, λ) ∈ D and g(N, λ) ∈ V.

◮ Viability kernel

V(g, D) =        N0 ∈ X : there exist λ(0), λ(1), λ(2), ... N(0), N(1), N(2), ...such that N(0) = N0 N(t + 1) = g(N(t), λ(t)) and (N(t), λ(t)) ∈ D

Goals

◮ Determine or approximate the viability kernel V(g, D)

for a given dynamics g and a given desirable set D.

◮ Determine when a given set V is a viability domain.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viability domains and viability kernel

Definition

◮ V ⊂ X is a Viability Domain if for all N ∈ V there exists

λ ∈ U such that (N, λ) ∈ D and g(N, λ) ∈ V.

◮ Viability kernel

V(g, D) =        N0 ∈ X : there exist λ(0), λ(1), λ(2), ... N(0), N(1), N(2), ...such that N(0) = N0 N(t + 1) = g(N(t), λ(t)) and (N(t), λ(t)) ∈ D

Goals

◮ Determine or approximate the viability kernel V(g, D)

for a given dynamics g and a given desirable set D.

◮ Determine when a given set V is a viability domain.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viability domains and viability kernel

Definition

◮ V ⊂ X is a Viability Domain if for all N ∈ V there exists

λ ∈ U such that (N, λ) ∈ D and g(N, λ) ∈ V.

◮ Viability kernel

V(g, D) =        N0 ∈ X : there exist λ(0), λ(1), λ(2), ... N(0), N(1), N(2), ...such that N(0) = N0 N(t + 1) = g(N(t), λ(t)) and (N(t), λ(t)) ∈ D

Goals

◮ Determine or approximate the viability kernel V(g, D)

for a given dynamics g and a given desirable set D.

◮ Determine when a given set V is a viability domain.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viability domains and viability kernel

Definition

◮ V ⊂ X is a Viability Domain if for all N ∈ V there exists

λ ∈ U such that (N, λ) ∈ D and g(N, λ) ∈ V.

◮ Viability kernel

V(g, D) =        N0 ∈ X : there exist λ(0), λ(1), λ(2), ... N(0), N(1), N(2), ...such that N(0) = N0 N(t + 1) = g(N(t), λ(t)) and (N(t), λ(t)) ∈ D

Goals

◮ Determine or approximate the viability kernel V(g, D)

for a given dynamics g and a given desirable set D.

◮ Determine when a given set V is a viability domain.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Monotonicity properties on the dynamics

Definition

We say that the dynamics g : X × U → X is a monotone bioeconomic dynamics if g is increasing with respect to the state i.e. ∀(N, N′, λ) ∈ X × X × U , N′ ≥ N ⇒ g(N′, λ) ≥ g(N, λ), and is decreasing with respect to the control i.e. ∀(N, λ, λ′) ∈ X × U × U , λ′ ≥ λ ⇒ g(N, λ′) ≤ g(N, λ).

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Monotonicity properties on the dynamics

Definition

We say that the dynamics g : X × U → X is a monotone bioeconomic dynamics if g is increasing with respect to the state i.e. ∀(N, N′, λ) ∈ X × X × U , N′ ≥ N ⇒ g(N′, λ) ≥ g(N, λ), and is decreasing with respect to the control i.e. ∀(N, λ, λ′) ∈ X × U × U , λ′ ≥ λ ⇒ g(N, λ′) ≤ g(N, λ).

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Production and preservation desirable sets

Definition

A desirable set D is said to be a production desirable set if D is increasing w.r.t. both the state and to the control, that is ∀ λ, λ′ ∈ U, N, N′ ∈ X s.t. N′ ≥ N, λ′ ≥ λ if (N, λ) ∈ D then (N′, λ′) ∈ D.

Example

Dyield = {(N, λ) | Y(N, λ) ≥ ymin}, where Y : X × U − → I R is increasing w.r.t. both variables (state and control).

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Production and preservation desirable sets

Definition

A desirable set D is said to be a production desirable set if D is increasing w.r.t. both the state and to the control, that is ∀ λ, λ′ ∈ U, N, N′ ∈ X s.t. N′ ≥ N, λ′ ≥ λ if (N, λ) ∈ D then (N′, λ′) ∈ D.

Example

Dyield = {(N, λ) | Y(N, λ) ≥ ymin}, where Y : X × U − → I R is increasing w.r.t. both variables (state and control).

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Production and preservation desirable sets

Definition

A desirable set D is said to be a preservation desirable set if D is increasing w.r.t. the state and decreasing w.r.t. the control: ∀ λ, λ′ ∈ U, N, N′ ∈ X s.t. N′ ≥ N, λ′ ≤ λ if (N, λ) ∈ D then (N′, λ′) ∈ D.

Example

Dprotect = {(N, λ) ∈ X × U | D(N, λ) ≥ d♭}, where D : X × U − → I R is increasing w.r.t. the state but decreasing w.r.t. the control.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Production and preservation desirable sets

Definition

A desirable set D is said to be a preservation desirable set if D is increasing w.r.t. the state and decreasing w.r.t. the control: ∀ λ, λ′ ∈ U, N, N′ ∈ X s.t. N′ ≥ N, λ′ ≤ λ if (N, λ) ∈ D then (N′, λ′) ∈ D.

Example

Dprotect = {(N, λ) ∈ X × U | D(N, λ) ≥ d♭}, where D : X × U − → I R is increasing w.r.t. the state but decreasing w.r.t. the control.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Outline

The core model for fishery management decisions The model Some questions about sustainability of landings Discrete time viability issues Monotonicity properties Viability kernel properties Maximum sustainable yield Minimal viable feedback The Patagonian toothfish (L´ egine australe) Questions & Answers

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viability kernels estimates

Assume that λ♭ ≤ λ ≤ λ♯ for all λ ∈ U.

Proposition

Suppose that g is an increasing-decreasing dynamics. Then:

◮ If D is a production desirable set, then

• t≥0

{N ∈ X : ((g♯)t(N), λ♭) ∈ D} ⊆ V(g, D) ⊆

• t≥0

{N ∈ X : ((g♭)t(N), λ♯) ∈ D} where g♭(·) = g(·, λ♭) and g♯(·) = g(·, λ♯)

◮ If D is a preservation desirable set, then

V(g, D) =

• t≥0

{N ∈ X : ((g♭)t(N), λ♭) ∈ D}

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viability kernels estimates

Assume that λ♭ ≤ λ ≤ λ♯ for all λ ∈ U.

Proposition

Suppose that g is an increasing-decreasing dynamics. Then:

◮ If D is a production desirable set, then

• t≥0

{N ∈ X : ((g♯)t(N), λ♭) ∈ D} ⊆ V(g, D) ⊆

• t≥0

{N ∈ X : ((g♭)t(N), λ♯) ∈ D} where g♭(·) = g(·, λ♭) and g♯(·) = g(·, λ♯)

◮ If D is a preservation desirable set, then

V(g, D) =

• t≥0

{N ∈ X : ((g♭)t(N), λ♭) ∈ D}

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Viability kernels estimates

Assume that λ♭ ≤ λ ≤ λ♯ for all λ ∈ U.

Proposition

Suppose that g is an increasing-decreasing dynamics. Then:

◮ If D is a production desirable set, then

• t≥0

{N ∈ X : ((g♯)t(N), λ♭) ∈ D} ⊆ V(g, D) ⊆

• t≥0

{N ∈ X : ((g♭)t(N), λ♯) ∈ D} where g♭(·) = g(·, λ♭) and g♯(·) = g(·, λ♯)

◮ If D is a preservation desirable set, then

V(g, D) =

• t≥0

{N ∈ X : ((g♭)t(N), λ♭) ∈ D}

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Maximum sustainable yield

Assume the existence of a steady state N(λ) for the dynamics N → g(N, λ), for all λ ∈ [λ♭, λ♯]. Given a yield function Y : X × U − → I R, we define the maximum sustainable yield by MSY = sup

λ∈[λ♭,λ♯]

Y(N(λ), λ) .

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Maximum sustainable yield

Consider the production desirable set Dyield given by Dyield = {(N, λ) | Y(N, λ) ≥ ymin} .

Proposition

Suppose that the yield function Y : X × U − → I R is increasing with respect both to the state and to the control. Then,

◮

MSY ≥ ymin ⇒ V(g, Dyield) = ∅ .

◮ If the steady state N(λ♭) is globally attractive for the

dynamics g♭, we have V(g, Dyield) = ∅ ⇒ Y

• N(λ♭), λ♯
• ≥ ymin .

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Minimal viable feedback

Define X1 ⊂ X as those states N such that S S B(N) ≥ Blim and there exists λ⋆(N) ∈ [λ♭, λ♯] satisfying Y(N, λ(N)) = ymin.

Proposition

N ∈ V(g, Dyield), with V(g, Dyield) = {(N, λ) : Y(N, λ) ≥ ymin, S S B(N) ≥ Blim}, if and only if the trajectory N(t0) = N , N(t+1) = g

• N(t), λ⋆(N(t))
• ,

t = t0, t0+1, . . . is well defined, namely N(t) ∈ X1.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Minimal viable feedback

Define X1 ⊂ X as those states N such that S S B(N) ≥ Blim and there exists λ⋆(N) ∈ [λ♭, λ♯] satisfying Y(N, λ(N)) = ymin.

Proposition

N ∈ V(g, Dyield), with V(g, Dyield) = {(N, λ) : Y(N, λ) ≥ ymin, S S B(N) ≥ Blim}, if and only if the trajectory N(t0) = N , N(t+1) = g

• N(t), λ⋆(N(t))
• ,

t = t0, t0+1, . . . is well defined, namely N(t) ∈ X1.

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Outline

The core model for fishery management decisions The model Some questions about sustainability of landings Discrete time viability issues Monotonicity properties Viability kernel properties Maximum sustainable yield Minimal viable feedback The Patagonian toothfish (L´ egine australe) Questions & Answers

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Age structured model

◮ the state : N = (Na)a=1,...,A ∈ I

RA

+, the abundances at age ◮ the control : λ the fishing effort multiplier ◮ the dynamics : N(t + 1) = g(N(t), λ(t)) given by

           g1(N, λ) = ϕ(S S B(N)) , ga(N, λ) = e−(Ma−1+λFa−1)Na−1, a = 2, . . . , A − 1 , gA(N, λ) = e−(MA−1+λFA−1)NA−1 + e−(MA+λFA)NA . where

◮ the spawning stock biomass S

S B is defined by S S B(N) :=

A

• a=1

γawaNa

◮ the function ϕ describes the stock-recruitment relationship ◮ Ma is the natural mortality rate of individuals of age a ◮ Fa is the mortality rate of individuals of age a due to

harvesting

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

The harvest term

The exploitation is described by catch-at-age Ca and yield Y, both defined for a given vector of abundance N and a given control λ : Ca

• N, λ
• =

λFa λFa + Ma

• 1 − e−(Ma+λFa)

Na The production in term of biomass is : Y

• N, λ
• =

A

• a=1

wa Ca(N, λ)

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

The Patagonian toothfish (L´ egine australe)1

◮ Abundance at age (state): N = (Na)a=1,..,A

Patagonian toothfish A = 36

◮ Fishing effort multiplier (control): λ ∈ U = [λ♭, λ♯]

Patagonian toothfish λ♭ = 0, λ♯ = 0.3

◮ Stock-recruitment relationship ϕ

Patagonian toothfish ϕ(B) = B α + βB

1Data: CEPES, SUBPESCA, Chile

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

The Patagonian toothfish (L´ egine australe)1

◮ Abundance at age (state): N = (Na)a=1,..,A

Patagonian toothfish A = 36

◮ Fishing effort multiplier (control): λ ∈ U = [λ♭, λ♯]

Patagonian toothfish λ♭ = 0, λ♯ = 0.3

◮ Stock-recruitment relationship ϕ

Patagonian toothfish ϕ(B) = B α + βB

1Data: CEPES, SUBPESCA, Chile

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

The Patagonian toothfish (L´ egine australe)1

◮ Abundance at age (state): N = (Na)a=1,..,A

Patagonian toothfish A = 36

◮ Fishing effort multiplier (control): λ ∈ U = [λ♭, λ♯]

Patagonian toothfish λ♭ = 0, λ♯ = 0.3

◮ Stock-recruitment relationship ϕ

Patagonian toothfish ϕ(B) = B α + βB

1Data: CEPES, SUBPESCA, Chile

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Some questions about sustainability of landings

◮ Given a desirable level of landings (tons), what are the

vectors of abundances N = (Na)a=1,..,A (initial conditions) for which one can always harvest at least that quantity?

◮ What levels of catch (landings) are non sustainable? ◮ Given an abundance at age N = (Na)a=1,..,A What is the

maximum sustainable yield starting from N and satisfying preservation constraints?

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Some questions about sustainability of landings

◮ Given a desirable level of landings (tons), what are the

vectors of abundances N = (Na)a=1,..,A (initial conditions) for which one can always harvest at least that quantity?

◮ What levels of catch (landings) are non sustainable? ◮ Given an abundance at age N = (Na)a=1,..,A What is the

maximum sustainable yield starting from N and satisfying preservation constraints?

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Some questions about sustainability of landings

◮ Given a desirable level of landings (tons), what are the

vectors of abundances N = (Na)a=1,..,A (initial conditions) for which one can always harvest at least that quantity?

◮ What levels of catch (landings) are non sustainable? ◮ Given an abundance at age N = (Na)a=1,..,A What is the

maximum sustainable yield starting from N and satisfying preservation constraints?

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Results

• =

landings = maximum sustainable yield (considering the abundance) = quota of the government regulatory Chilean agency (SUBPESCA) = non sustainable level (independently of the abundance) = maximum sustainable yield in the equilibrium

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Open questions

◮ To consider a vector control λ = (λ1, . . . , λp) ◮ Interaction between species:

◮ Technical interactions ◮ Biological interactions ⇒ to consider a non monotone

dynamics g

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Open questions

◮ To consider a vector control λ = (λ1, . . . , λp) ◮ Interaction between species:

◮ Technical interactions ◮ Biological interactions ⇒ to consider a non monotone

dynamics g

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Open questions

◮ To consider a vector control λ = (λ1, . . . , λp) ◮ Interaction between species:

◮ Technical interactions ◮ Biological interactions ⇒ to consider a non monotone

dynamics g

Viable harvest of monotone bioeconomics models A fishery management model

The model Some questions about sustainability of landings

Discrete time viability issues

Monotonicity properties

Viability kernel properties

Maximum sustainable yield Minimal viable feedback

The Patagonian toothfish

Questions & Answers

Open questions

◮ To consider a vector control λ = (λ1, . . . , λp) ◮ Interaction between species:

◮ Technical interactions ◮ Biological interactions ⇒ to consider a non monotone

dynamics g