PhD Proposal St. Andrews University

Introduction Dynamic System The proposal

Using stochastic Population Viability Analysis(PVA) to compare sustainable fishing

exploitation strategies

A draft proposal of a PhD projectUniversity of St Andrews

JC Quiroz


Outline of the presentation

1 IntroductionMotivationThe Problem

2 Dynamic SystemThe Population ModelsPopulation Viability Analysis (PVA)

3 The proposalSome Ideas


Motivation

Fisheries management issues are highly dependent of uncertainty:


Motivation


Demographic and environmental stochasticity affectingpopulation dynamics


Motivation



‡ Demographic: stochastic variations in reproduction, survival andrecruitment

‡ Environmental: catchability, fishing efforts, yield levels and

ecosystemic effects


Motivation





ecosystemic effects

Conflicts between population conservation and social−economicpriorities


Motivation





ecosystemic effects


‡ Economic: guaranteed income for fishermen

‡ Social: equity income, employment, legal issues


Motivation





ecosystemic effects


‡ Economic: guaranteed income for fishermen

‡ Social: equity income, employment, legal issues

In many fisheries, these issues are integrated in a ManagementProcedure (MP), which try to explain major sources of uncertainty ofa system.


Motivation

According to several authors, the MP is a simulation-tested set ofrules used to determine management actions, in which themanagement objetives, fishery data, assessment methods and theexploitation strategies (i.e., the rules used for decision making) arepre-specified.


Motivation

According to several authors, the MP is a simulation-tested set ofrules used to determine management actions, in which themanagement objetives, fishery data, assessment methods and theexploitation strategies (i.e., the rules used for decision making) arepre-specified.

For example:

To achieve different management objetive . . .

Φ :=

sb(t) ≥ α · sb(t = 0), α ∈ {0, 1}y(t) = msy

f(t) < fbrpy(t) ≥ ylim

,


. . . the MP may use different exploitation strategies

Ψ :=

f(t) = f

µ(t) = µ := y(t)sb(t)

y(t) = y

y(t) = h (n(t), f(t))

.



Ψ :=

f(t) = f

µ(t) = µ := y(t)sb(t)

y(t) = y

y(t) = h (n(t), f(t))

.



Ψ :=

f(t) = f

µ(t) = µ := y(t)sb(t)

y(t) = y

y(t) = h (n(t), f(t))

.

These exploitation strategies are tested by simulations to ensure thatthey are reasonably robust in terms of expected catch and thepopulation risk.


The Problem

According to different exploitation strategies used and themanagement objetives, several MP’s may be developed to satisfy themulti-criteria decision problem that underlying fisheries management.


The Problem

According to different exploitation strategies used and themanagement objetives, several MP’s may be developed to satisfy themulti-criteria decision problem that underlying fisheries management.


The Problem

Therefore, before defining the MP to be applied, is necessarycompare different potential MP’s and rank them according to theirability to achieve the management objectives.


The Problem

Therefore, before defining the MP to be applied, is necessarycompare different potential MP’s and rank them according to theirability to achieve the management objectives.

Consequently, the question is: How can we do this? ...

taking into account that in fisheries science there is not clearconsensus in the way to compare different potential MP’s

In this proposal, the stochastic Population Viability Analysis (PVA)is suggested as a relevant method to deal with the MP’s comparison .


The Population Models

n(t) =

{

n0, t = t0 = 1g (t, n(t− 1), ω(t− 1), ε(t− 1)) , t = 2, . . . , T

n0 is the initial state for the time t = t0 = 1



n(t) =

{

n0, t = t0 = 1g (t, n(t− 1), ω(t− 1), ε(t− 1)) , t = 2, . . . , T


n(t) is a state vector representing the biomass/abundance of a singlespecie or a vector of abundance at ages



n(t) =

{

n0, t = t0 = 1g (t, n(t− 1), ω(t− 1), ε(t− 1)) , t = 2, . . . , T



ω(t) is the control vector representing the projected catch/effort or anymanagement strategy



n(t) =

{

n0, t = t0 = 1g (t, n(t− 1), ω(t− 1), ε(t− 1)) , t = 2, . . . , T




ε(t) denotes the uncertainty in the population at each time t, which iscaused by stochasticity in the population dynamics due to randomeffects in the demography and environmental fluctuations



n(t) =

{

n0, t = t0 = 1g (t, n(t− 1), ω(t− 1), ε(t− 1)) , t = 2, . . . , T




ε(t) denotes the uncertainty in the population at each time t, which iscaused by stochasticity in the population dynamics due to randomeffects in the demography and environmental fluctuations

g(·) is the population dynamics described by age or size-structuredmodels, surplus-production models, logistic growth models, etc. Thesequence g(t, n(t)|n(t− 1), θ) represents a state-space process, where θ

is a vector of parameters


Using mathematical notation:

time t ∈ K := N, t = {t0, . . . , T}

state n(t) ∈ N := Rn+

n(t) ∈ R (annual abundance of a single specie)n(t) ∈ R

2 (predator-prey system)n(t) ∈ R

n (abundance at n-age)

control ω(t) ∈ W := R+

uncertainty ε(t) ∈ E := R

dynamic g(n(t)|n(t− 1)) ∈ D :={

N× Rn+ × R+ × R

}



time t ∈ K := N, t = {t0, . . . , T}








N× Rn+ × R+ × R

}



time t ∈ K := N, t = {t0, . . . , T}








N× Rn+ × R+ × R

}

In the case when the population dynamics is deterministic, ε(t) = 0,the control of the g(·) system, is driven only by selecting an uniquesequence of decision rules ω∗(·) = (ω∗(t0), · · · , ω∗(T − 1)), resulting ina single realisation (∗) of sequential states n(·).


Population Viability Analysis (PVA)

When uncertainties affect population dynamics, ε(t) 6= 0, the controlvector, ω(t), can be defined as a mapping, ω̂ : K ×N → W , where thedecision rule contain a feed-back control:

ω(t) = ω̂(t, n(t)).


Population Viability Analysis (PVA)

When uncertainties affect population dynamics, ε(t) 6= 0, the controlvector, ω(t), can be defined as a mapping, ω̂ : K ×N → W , where thedecision rule contain a feed-back control:

ω(t) = ω̂(t, n(t)).

In this case, a sequence of decision ω(·) may result in severalsequential states n(·), depending of the realisation of uncertainty. Insuch case, a sequence of uncertainty as,

ε(·) := {ε(t0), . . . , ε(T − 1)} ∈ E × · · · × E ,

can define as a set of scenarios:

E := ET−t0 .


Stochastic PVA

If we assume that the set E is drawn from a probability distributionP, then ε(·) should be interpreted as a sequence of random variables,{ε(t0), . . . , ε(T − 1)}, independent and identically distributed.

Therefore, let ε(·) be a random variables with values in E := R, theviability probability associated with the initial time t0, the initial staten(t0) and the exploitation strategy ω̂ is denoted as,

P[Eω̂,t0,n(t0)].


Stochastic PVA

If we assume that the set E is drawn from a probability distributionP, then ε(·) should be interpreted as a sequence of random variables,{ε(t0), . . . , ε(T − 1)}, independent and identically distributed.

Therefore, let ε(·) be a random variables with values in E := R, theviability probability associated with the initial time t0, the initial staten(t0) and the exploitation strategy ω̂ is denoted as,

P[Eω̂,t0,n(t0)].

If we now consider j-functions that represent the indicators of themanagement objetives, as a mapping Ij : K ×N ×W → R, we maydefine:

Ij(t, n(t), ω(t)) > ıj ,

where ıj are thresholds or reference points, ı1 ∈ R, . . . , ıJ ∈ R,associated with the management objetives.


Viability probability of a exploitation strategy

For any exploitation strategy ω̂, initial state n0 and initial time t0, letdefine the set of viable scenarios as:

Eω̂,t0,n(t0) :=

ε(·) ∈ E

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

n(t0) = n0

n(t) = g(t, n(t− 1), ω(t− 1), ε(t− 1))

ω(t) = ω̂(t, n(t))

Ij(t, n(t), ω(t)) > ıj

j = 1, . . . , J

t = t0, . . . , T



For any exploitation strategy ω̂, initial state n0 and initial time t0, letdefine the set of viable scenarios as:

Eω̂,t0,n(t0) :=

ε(·) ∈ E

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

n(t0) = n0

n(t) = g(t, n(t− 1), ω(t− 1), ε(t− 1))

ω(t) = ω̂(t, n(t))


j = 1, . . . , J

t = t0, . . . , T

An scenario ε(·) is not viable under decision rules ω̂(·), if whateverstate n(·) or control ω(·) trajectories generated by g(·) not satisfy thestate and control constraints imposed by Ij .

In terms to compare different exploitation strategies, a ω̂ is consideredbetter if the corresponding set of viable scenarios is ”larger”.



The viability probability space is a triplet (E,H,P), where H is aσ-algebra on E, because g(·), Ij and all different exploitationstrategies ω̂(·) are measurables.



The viability probability space is a triplet (E,H,P), where H is aσ-algebra on E, because g(·), Ij and all different exploitationstrategies ω̂(·) are measurables.

Therefore, it is possible to rank different MP’s according to theirviability probability for any set of thresholds or reference points ıj, bydefine:

M(ω̂, ıi, . . . , ıJ) := P

ε(·) ∈ E

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

n(t0) = n0

n(t) = g(t, n(t− 1), ω(t− 1), ε(t− 1))

ω(t) = ω̂(t, n(t))


j = 1, . . . , J

t = t0, . . . , T



The probability can be drawn by numeric algorith such as MonteCarlo simulations, thus the marginal variation of viability probability,

∂

∂ıJM (ω̂, ıi, . . . , ıJ) = 0

can be calculated to ranking MP’s with respect to their ability toachieve a set of sustainability management objetives.


Some Ideas

Using the conceptual framework exposed here, I propose to explorethe distributional properties of the viability probability P, using thestochastic viability analysis by compare differents managementprocedures. The species selected for this analysis can be the southernhake and toothfish fished in Chile.


Some Ideas

Using the conceptual framework exposed here, I propose to explorethe distributional properties of the viability probability P, using thestochastic viability analysis by compare differents managementprocedures. The species selected for this analysis can be the southernhake and toothfish fished in Chile.

Specific objectives:

Incorporing managements objetives into the different decisionrules

Clarifying the diferences between objetives and decision rules

Explore the conflicts between conservation and economicobjetives

Explore the consistence on the management objetives withsustainable exploitation

Explore the properties of viability probability density in southernhake and toothfish fishery


The toothfish case

100 run

ε → CVcpue = 0,25

imperfect information → CPUE(t) = h(n(t), ε(t))

ω̂ → y(t)sb(t) = rule (t, n(t))

Ij → P(sbproj 6 sbact) 6 0,10


The toothfish case

100 run

ε → CVcpue = 0,25

imperfect information → CPUE(t) = h(n(t), ε(t))

ω̂ → y(t)sb(t) = rule (t, n(t))

Ij → P(sbproj 6 sbact) 6 0,10


Thanks

PhD Proposal St. Andrews University

Technology

f yt t

dierent management objetive

population conservation

management actions

management objectives

recruitment environmental

dependent of uncertainty

f t fbrp yt ylim