PhD Proposal St. Andrews University

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PhD Proposal St. Andrews University

  1. 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. 2. Introduction Dynamic System The proposalOutline of the presentation 1 Introduction Motivation The Problem 2 Dynamic System The Population Models Population Viability Analysis (PVA) 3 The proposal Some Ideas
  3. 3. Introduction Dynamic System The proposalMotivation Fisheries management issues are highly dependent of uncertainty:
  4. 4. Introduction Dynamic System The proposalMotivation Fisheries management issues are highly dependent of uncertainty: Demographic and environmental stochasticity affecting population dynamics
  5. 5. Introduction Dynamic System The proposalMotivation 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. 6. Introduction Dynamic System The proposalMotivation 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. 7. Introduction Dynamic System The proposalMotivation 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. 8. Introduction Dynamic System The proposalMotivation 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. 9. Introduction Dynamic System The proposalMotivation 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. 10. Introduction Dynamic System The proposalMotivation 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. 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. 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. 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. 14. Introduction Dynamic System The proposalThe 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. 15. Introduction Dynamic System The proposalThe 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. 16. Introduction Dynamic System The proposalThe 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. 17. Introduction Dynamic System The proposalThe 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. 18. Introduction Dynamic System The proposalThe 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. 19. Introduction Dynamic System The proposalThe 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. 20. Introduction Dynamic System The proposalThe 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. 21. Introduction Dynamic System The proposalThe 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. 22. Introduction Dynamic System The proposalThe 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. 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. 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. 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. 26. Introduction Dynamic System The proposalPopulation 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. 27. Introduction Dynamic System The proposalPopulation 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. 28. Introduction Dynamic System The proposalStochastic 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. 29. Introduction Dynamic System The proposalStochastic 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. 30. Introduction Dynamic System The proposalViability 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. 31. Introduction Dynamic System The proposalViability 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. 32. Introduction Dynamic System The proposalViability 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. 33. Introduction Dynamic System The proposalViability 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. 34. Introduction Dynamic System The proposalViability 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. 35. Introduction Dynamic System The proposalSome 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. 36. Introduction Dynamic System The proposalSome 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. 37. Introduction Dynamic System The proposalThe 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. 38. Introduction Dynamic System The proposalThe 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. 39. Introduction Dynamic System The proposal Thanks

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