1. Introduction Dynamic System The proposal
Using stochastic Population Viability Analysis
(PVA) to compare sustainable fishing
exploitation strategies
A draft proposal of a PhD project
University of St Andrews
JC Quiroz

2. Introduction Dynamic System The proposal
Outline of the presentation
1 Introduction
Motivation
The Problem
2 Dynamic System
The Population Models
Population Viability Analysis (PVA)
3 The proposal
Some Ideas

3. Introduction Dynamic System The proposal
Motivation
Fisheries management issues are highly dependent of uncertainty:

4. Introduction Dynamic System The proposal
Motivation
Fisheries management issues are highly dependent of uncertainty:
Demographic and environmental stochasticity affecting
population dynamics

5. Introduction Dynamic System The proposal
Motivation
Fisheries management issues are highly dependent of uncertainty:
Demographic and environmental stochasticity affecting
population dynamics
‡ Demographic: stochastic variations in reproduction, survival and
recruitment
‡ Environmental: catchability, fishing efforts, yield levels and
ecosystemic effects

6. Introduction Dynamic System The proposal
Motivation
Fisheries management issues are highly dependent of uncertainty:
Demographic and environmental stochasticity affecting
population dynamics
‡ Demographic: stochastic variations in reproduction, survival and
recruitment
‡ Environmental: catchability, fishing efforts, yield levels and
ecosystemic effects
Conflicts between population conservation and social−economic
priorities

7. Introduction Dynamic System The proposal
Motivation
Fisheries management issues are highly dependent of uncertainty:
Demographic and environmental stochasticity affecting
population dynamics
‡ Demographic: stochastic variations in reproduction, survival and
recruitment
‡ Environmental: catchability, fishing efforts, yield levels and
ecosystemic effects
Conflicts between population conservation and social−economic
priorities
‡ Economic: guaranteed income for fishermen
‡ Social: equity income, employment, legal issues

8. Introduction Dynamic System The proposal
Motivation
Fisheries management issues are highly dependent of uncertainty:
Demographic and environmental stochasticity affecting
population dynamics
‡ Demographic: stochastic variations in reproduction, survival and
recruitment
‡ Environmental: catchability, fishing efforts, yield levels and
ecosystemic effects
Conflicts between population conservation and social−economic
priorities
‡ Economic: guaranteed income for fishermen
‡ Social: equity income, employment, legal issues
In many fisheries, these issues are integrated in a Management
Procedure (MP), which try to explain major sources of uncertainty of
a system.

9. Introduction Dynamic System The proposal
Motivation
According to several authors, the MP is a simulation-tested set of
rules used to determine management actions, in which the
management objetives, fishery data, assessment methods and the
exploitation strategies (i.e., the rules used for decision making) are
pre-specified.

10. Introduction Dynamic System The proposal
Motivation
According to several authors, the MP is a simulation-tested set of
rules used to determine management actions, in which the
management objetives, fishery data, assessment methods and the
exploitation strategies (i.e., the rules used for decision making) are
pre-specified.
For example:
To achieve different management objetive . . .

 sb(t) ≥ α · sb(t = 0), α ∈ {0, 1}

y(t) = msy

Φ := ,
 f (t) < fbrp

y(t) ≥ ylim


11. Introduction Dynamic System The proposal
. . . the MP may use different exploitation strategies
 f (t) = f

y(t)

µ(t) = µ := sb(t)

Ψ := .
 y(t) = y


y(t) = h (n(t), f (t))

12. Introduction Dynamic System The proposal
. . . the MP may use different exploitation strategies
 f (t) = f

y(t)

µ(t) = µ := sb(t)

Ψ := .
 y(t) = y


y(t) = h (n(t), f (t))

13. Introduction Dynamic System The proposal
. . . the MP may use different exploitation strategies
 f (t) = f

y(t)

µ(t) = µ := sb(t)

Ψ := .
 y(t) = y


y(t) = h (n(t), f (t))
These exploitation strategies are tested by simulations to ensure that
they are reasonably robust in terms of expected catch and the
population risk.

14. Introduction Dynamic System The proposal
The Problem
According to different exploitation strategies used and the
management objetives, several MP’s may be developed to satisfy the
multi-criteria decision problem that underlying fisheries management.

15. Introduction Dynamic System The proposal
The Problem
According to different exploitation strategies used and the
management objetives, several MP’s may be developed to satisfy the
multi-criteria decision problem that underlying fisheries management.

16. Introduction Dynamic System The proposal
The Problem
Therefore, before defining the MP to be applied, is necessary
compare different potential MP’s and rank them according to their
ability to achieve the management objectives.

17. Introduction Dynamic System The proposal
The Problem
Therefore, before defining the MP to be applied, is necessary
compare different potential MP’s and rank them according to their
ability to achieve the management objectives.
Consequently, the question is: How can we do this? ...
taking into account that in fisheries science there is not clear
consensus 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 .

18. Introduction Dynamic System The proposal
The Population Models
n0 , t = t0 = 1
n(t) =
g (t, n(t − 1), ω(t − 1), ε(t − 1)) , t = 2, . . . , T
n0 is the initial state for the time t = t0 = 1

19. Introduction Dynamic System The proposal
The Population Models
n0 , t = t0 = 1
n(t) =
g (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) is a state vector representing the biomass/abundance of a single
specie or a vector of abundance at ages

20. Introduction Dynamic System The proposal
The Population Models
n0 , t = t0 = 1
n(t) =
g (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) is a state vector representing the biomass/abundance of a single
specie or a vector of abundance at ages
ω(t) is the control vector representing the projected catch/effort or any
management strategy

21. Introduction Dynamic System The proposal
The Population Models
n0 , t = t0 = 1
n(t) =
g (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) is a state vector representing the biomass/abundance of a single
specie or a vector of abundance at ages
ω(t) is the control vector representing the projected catch/effort or any
management strategy
ε(t) denotes the uncertainty in the population at each time t, which is
caused by stochasticity in the population dynamics due to random
effects in the demography and environmental fluctuations

22. Introduction Dynamic System The proposal
The Population Models
n0 , t = t0 = 1
n(t) =
g (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) is a state vector representing the biomass/abundance of a single
specie or a vector of abundance at ages
ω(t) is the control vector representing the projected catch/effort or any
management strategy
ε(t) denotes the uncertainty in the population at each time t, which is
caused by stochasticity in the population dynamics due to random
effects in the demography and environmental fluctuations
g(·) is the population dynamics described by age or size-structured
models, surplus-production models, logistic growth models, etc. The
sequence g(t, n(t)|n(t − 1), θ) represents a state-space process, where θ
is a vector of parameters

23. Introduction Dynamic System The proposal
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) ∈ R2 (predator-prey system)
n(t) ∈ Rn (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
+

24. Introduction Dynamic System The proposal
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) ∈ R2 (predator-prey system)
n(t) ∈ Rn (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
+

25. Introduction Dynamic System The proposal
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) ∈ R2 (predator-prey system)
n(t) ∈ Rn (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
+
In the case when the population dynamics is deterministic, ε(t) = 0,
the control of the g(·) system, is driven only by selecting an unique
sequence of decision rules ω ∗ (·) = (ω ∗ (t0 ), · · · , ω ∗ (T − 1)), resulting in
a single realisation (∗ ) of sequential states n(·).

26. Introduction Dynamic System The proposal
Population Viability Analysis (PVA)
When uncertainties affect population dynamics, ε(t) = 0, the control
vector, ω(t), can be defined as a mapping, ω : K × N → W, where the
ˆ
decision rule contain a feed-back control:
ω(t) = ω (t, n(t)).
ˆ

27. Introduction Dynamic System The proposal
Population Viability Analysis (PVA)
When uncertainties affect population dynamics, ε(t) = 0, the control
vector, ω(t), can be defined as a mapping, ω : K × N → W, where the
ˆ
decision rule contain a feed-back control:
ω(t) = ω (t, n(t)).
ˆ
In this case, a sequence of decision ω(·) may result in several
sequential states n(·), depending of the realisation of uncertainty. In
such case, a sequence of uncertainty as,
ε(·) := {ε(t0 ), . . . , ε(T − 1)} ∈ E × · · · × E,
can define as a set of scenarios:
E := E T −t0 .

28. Introduction Dynamic System The proposal
Stochastic PVA
If we assume that the set E is drawn from a probability distribution
P, 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, the
viability probability associated with the initial time t0 , the initial state
n(t0 ) and the exploitation strategy ω is denoted as,
ˆ
P[Eω ,t0 ,n(t0 ) ].
ˆ

29. Introduction Dynamic System The proposal
Stochastic PVA
If we assume that the set E is drawn from a probability distribution
P, 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, the
viability probability associated with the initial time t0 , the initial state
n(t0 ) and the exploitation strategy ω is denoted as,
ˆ
P[Eω ,t0 ,n(t0 ) ].
ˆ
If we now consider j-functions that represent the indicators of the
management objetives, as a mapping Ij : K × N × W → R, we may
define:
Ij (t, n(t), ω(t)) ıj ,
where ıj are thresholds or reference points, ı1 ∈ R, . . . , ıJ ∈ R,
associated with the management objetives.

30. Introduction Dynamic System The proposal
Viability probability of a exploitation strategy
For any exploitation strategy ω , initial state n0 and initial time t0 , let
ˆ
define the set of viable scenarios as:
n(t0 ) = n0
 

 

n(t) = g(t, n(t − 1), ω(t − 1), ε(t − 1)) 

 

 

 
ω(t) = ω (t, n(t))
ˆ

 

Eω,t0 ,n(t0 ) := ε(·) ∈ E
ˆ

 Ij (t, n(t), ω(t)) ıj 

 
j = 1, . . . , J

 


 

 
t = t0 , . . . , T
 

31. Introduction Dynamic System The proposal
Viability probability of a exploitation strategy
For any exploitation strategy ω , initial state n0 and initial time t0 , let
ˆ
define the set of viable scenarios as:
n(t0 ) = n0
 

 

n(t) = g(t, n(t − 1), ω(t − 1), ε(t − 1)) 

 

 

 
ω(t) = ω (t, n(t))
ˆ

 

Eω,t0 ,n(t0 ) := ε(·) ∈ E
ˆ

 Ij (t, n(t), ω(t)) ıj 

 
j = 1, . . . , J

 


 

 
t = t0 , . . . , T
 
An scenario ε(·) is not viable under decision rules ω (·), if whatever
ˆ
state n(·) or control ω(·) trajectories generated by g(·) not satisfy the
state and control constraints imposed by Ij .
In terms to compare different exploitation strategies, a ω is considered
ˆ
better if the corresponding set of viable scenarios is ”larger”.

32. Introduction Dynamic System The proposal
Viability probability of a exploitation strategy
The viability probability space is a triplet (E, H, P), where H is a
σ-algebra on E, because g(·), Ij and all different exploitation
strategies ω (·) are measurables.
ˆ

33. Introduction Dynamic System The proposal
Viability probability of a exploitation strategy
The viability probability space is a triplet (E, H, P), where H is a
σ-algebra on E, because g(·), Ij and all different exploitation
strategies ω (·) are measurables.
ˆ
Therefore, it is possible to rank different MP’s according to their
viability probability for any set of thresholds or reference points ıj , by
define:
n(t0 ) = n0
 

 

n(t) = g(t, n(t − 1), ω(t − 1), ε(t − 1)) 

 

 

 
ω(t) = ω (t, n(t))
ˆ

 

M(ˆ , ıi , . . . , ıJ ) := P ε(·) ∈ E
ω

 Ij (t, n(t), ω(t)) ıj 

 
j = 1, . . . , J

 


 

 
t = t0 , . . . , T
 

34. Introduction Dynamic System The proposal
Viability probability of a exploitation strategy
The probability can be drawn by numeric algorith such as Monte
Carlo simulations, thus the marginal variation of viability probability,
∂
M (ˆ , ıi , . . . , ıJ ) = 0
ω
∂ıJ
can be calculated to ranking MP’s with respect to their ability to
achieve a set of sustainability management objetives.

35. Introduction Dynamic System The proposal
Some Ideas
Using the conceptual framework exposed here, I propose to explore
the distributional properties of the viability probability P, using the
stochastic viability analysis by compare differents management
procedures. The species selected for this analysis can be the southern
hake and toothfish fished in Chile.

36. Introduction Dynamic System The proposal
Some Ideas
Using the conceptual framework exposed here, I propose to explore
the distributional properties of the viability probability P, using the
stochastic viability analysis by compare differents management
procedures. The species selected for this analysis can be the southern
hake and toothfish fished in Chile.
Specific objectives:
Incorporing managements objetives into the different decision
rules
Clarifying the diferences between objetives and decision rules
Explore the conflicts between conservation and economic
objetives
Explore the consistence on the management objetives with
sustainable exploitation
Explore the properties of viability probability density in southern
hake and toothfish fishery

37. Introduction Dynamic System The proposal
The toothfish case
100 run
ε → CVcpue = 0,25
imperfect information → CP U E(t) = h(n(t), ε(t))
y(t)
ω→
ˆ sb(t) = rule (t, n(t))
Ij → P(sbproj sbact ) 0,10

38. Introduction Dynamic System The proposal
The toothfish case
100 run
ε → CVcpue = 0,25
imperfect information → CP U E(t) = h(n(t), ε(t))
y(t)
ω→
ˆ sb(t) = rule (t, n(t))
Ij → P(sbproj sbact ) 0,10

39. Introduction Dynamic System The proposal
Thanks