Generalization of age-structured models in theory and practice

Transkript

1 Generalization of age-structured models in theory and practice Stein Ivar Steinshamn,

2 Outline How age-structured models can be generalized. What this generalization can be used for. Results from the generalized model compared to classical agestructured modelling Fornavn Etternavn, 2

3 Potential of the model State of the art: The conceptual model is ready for application. Published in American Journal of Agricultural Economics. Derived some preliminary results in a single-species context. Future potential: Apply to more real-world fisheries Include aspects like demand functions, increasing marginal costs, etc. Simulation versus optimization Include various aspects related to uncertainty Multi-species modelling Fornavn Etternavn, navn@nhh.no 3

4 Basic age-structured modelling: Dynamics Instantaneous change in number of fish: dn dt ( F M ) N( t) Solution: N( t) N 0 e ( F M ) t or N t 1 N t e ( F M ) Fornavn Etternavn, navn@nhh.no 4

5 Basic age-structured modelling: Production Instantaneous catch in number of fish: dc dt F N(t) Total catch: C t t 1 F N( t) dt F F M t N t 1 e ( F M ) Fornavn Etternavn, navn@nhh.no 5

6 Relationship between fishing effort and fishing mortality In the classical model we have Ft qe t where E is fishing effort and q is defined as the catchability coefficient; constant by hypothesis Fornavn Etternavn, navn@nhh.no 6

7 Novel contribution An alternative way to calculate instantaneous catch k is the selectivity parameter and ρ is the density parameter dc dt k E t Density parameter: h N where Fornavn Etternavn, navn@nhh.no 7

8 The density parameter With perfectly schooling species, the density parameter is constant 0 h With uniform distribution 1 h N Fornavn Etternavn, navn@nhh.no 8

9 WHY? Look at the two expressions for instantaneous catch: dc dt F N q E N dc dt k h N E These imply q k h N 1 or F k h N 1 E Fornavn Etternavn, navn@nhh.no 9

10 Deriving the general expressions This general expression for F is substituted back into the basic equations and solved yielding some quite messy and complicated expressions Fornavn Etternavn, navn@nhh.no 10

11 Fornavn Etternavn, 11

12 Numerical optimization model Optimization method: Non-linear programming Objective function: Maximize t p a wa Ca, t t a t c E t subject to C N t C( N, E ), E t 1 t t t n( N t ) plus various additional constraints Fornavn Etternavn, navn@nhh.no 12

13 Results from stylized model Hypothetical fish stock 4 year-classes 10 periods No discounting Fixed recruitment Constant price of catch over time Constant cost of effort over time Sustainability constraint Fornavn Etternavn, navn@nhh.no 13

14 Effort Optimal effort over time for various values of α 4 3,5 3 2,5 2 1,5 α = , Time Fornavn Etternavn, navn@nhh.no 14

15 Effort Optimal effort over time for various values of α 4 3,5 3 2,5 2 1,5 α = 0.0 α = , Time Fornavn Etternavn, navn@nhh.no 15

16 Effort Optimal effort over time for various values of α 4 3,5 3 2,5 2 1,5 α = 0.0 α = 0.5 α = , Time Fornavn Etternavn, navn@nhh.no 16

17 A model for Arcto-Norwegian herring 17 age-groups, 35 periods Stock - recruitment relationship Realistic price, costs and maturity data Discounting Sustainability constraint Fornavn Etternavn, navn@nhh.no 17

18 Optimal F when α = 0 (schooling fishery) 4 3,5 3 2,5 2 1,5 F 1 0,5 0-0, Fornavn Etternavn, navn@nhh.no 18

19 Optimal F when α = ½ (intermediate) 3 2,5 2 1,5 F 1 0, , Fornavn Etternavn, navn@nhh.no 19

20 Optimal F when α = 1 (uniform distribution) 4 3,5 3 2,5 2 1,5 F 1 0,5 0-0, Fornavn Etternavn, navn@nhh.no 20

21 More realistic recruitment? Actual Estimated Fornavn Etternavn, navn@nhh.no 21

22 The effects of discounting. In this example we use cyclical recruitment Discount rates 0, 5 % and 10 % Fornavn Etternavn, navn@nhh.no 22

23 Perfect schooling, α = 0. 0,9 0,8 0,7 0,6 0,5 0,4 0,3 r = 0.0 r = 0.05 r = 0.1 0,2 0, , Fornavn Etternavn, navn@nhh.no 23

24 Intermediate case, α = 1/2 1,8 1,6 1,4 1,2 1 0,8 0,6 r = 0.0 r = 0.05 r = 0.1 0,4 0, , Fornavn Etternavn, navn@nhh.no 24

25 Uniform distribution, α = 1 0,6 0,5 0,4 0,3 0,2 r=0 r=0.05 r=0.1 0, , Fornavn Etternavn, navn@nhh.no 25

26 Recruitment What about constant recruitment? Zero dicounting Recruitment constant at Fornavn Etternavn, 26

27 Optimal effort when α = 0. 0,5 0,45 0,4 0,35 0,3 0,25 0,2 Serie1 0,15 0,1 0,05 0-0, Fornavn Etternavn, navn@nhh.no 27

28 Optimal effort when α = 1/2. 0,8 0,7 0,6 0,5 0,4 Serie1 0,3 0,2 0, , Fornavn Etternavn, navn@nhh.no 28

29 Optimal effort when α = 1. 0,4 0,35 0,3 0,25 0,2 Serie1 0,15 0,1 0, , Fornavn Etternavn, navn@nhh.no 29

30 Summing up Have generalized the classical age-structured model to take density dependence in the stock properly into account. Can be used both for simulation and optimization. Optimization runs indicate that pulse fishing gradually cease to be optimal as we move from uniformly distributed fish to schooling fish. Discounting contributes also to even out the pulses whereas various assumptions about recruitment have only a small effect Fornavn Etternavn, navn@nhh.no 30