Viable harvest of monotone bioeconomics A fishery management model - PowerPoint PPT Presentation

Viable harvest of monotone bioeconomics models Viable harvest of monotone bioeconomics A fishery management model The model models Some questions about sustainability of landings Preservation and production issues Discrete time viability issues Monotonicity properties Viability kernel properties Michel De Lara 1 Pedro Gajardo 2 ırez C. 3 H´ ector Ram´ Maximum sustainable yield Minimal viable feedback The Patagonian 1 CERMICS, Universit´ e de Paris-Est, France toothfish 2 Departamento de Matem´ atica, Universidad T´ ecnica Federico Santa Mar´ ıa Questions & Answers 3 Departamento 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 Outline monotone bioeconomics models The core model for fishery management decisions A fishery management model The model The model Some questions about Some questions about sustainability of landings sustainability of landings Discrete time viability issues Discrete time viability issues Monotonicity properties Viability kernel Monotonicity properties properties Maximum sustainable yield Minimal viable feedback Viability kernel properties The Patagonian toothfish Maximum sustainable yield Questions & Answers Minimal viable feedback The Patagonian toothfish (L´ egine australe) Questions & Answers

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

Viable harvest of Age structured model monotone bioeconomics models R A ◮ the state : N = ( N a ) a = 1 ,..., A ∈ I + , the abundances at age ◮ the control : λ the fishing effort multiplier A fishery ◮ the dynamics : N ( t + 1 ) = g ( N ( t ) , λ ( t )) given by management model The model Some questions about  g 1 ( N , λ ) = ϕ ( S S B ( N )) , sustainability of landings   Discrete time   viability issues  e − ( M a − 1 + λ F a − 1 ) N a − 1 , g a ( N , λ ) = a = 2 , . . . , A − 1 , Monotonicity properties Viability kernel   properties  e − ( M A − 1 + λ F A − 1 ) N A − 1 + e − ( M A + λ F A ) N A .  g A ( N , λ ) =  Maximum sustainable yield Minimal viable feedback where The Patagonian toothfish ◮ the spawning stock biomass S S B is defined by Questions & Answers A � B ( N ) := S S γ a w a N a a = 1 ◮ the function ϕ describes the stock-recruitment relationship ◮ M a is the natural mortality rate of individuals of age a ◮ F a is the mortality rate of individuals of age a due to harvesting

Viable harvest of Age structured model monotone bioeconomics models R A ◮ the state : N = ( N a ) a = 1 ,..., A ∈ I + , the abundances at age ◮ the control : λ the fishing effort multiplier A fishery ◮ the dynamics : N ( t + 1 ) = g ( N ( t ) , λ ( t )) given by management model The model Some questions about  g 1 ( N , λ ) = ϕ ( S S B ( N )) , sustainability of landings   Discrete time   viability issues  e − ( M a − 1 + λ F a − 1 ) N a − 1 , g a ( N , λ ) = a = 2 , . . . , A − 1 , Monotonicity properties Viability kernel   properties  e − ( M A − 1 + λ F A − 1 ) N A − 1 + e − ( M A + λ F A ) N A .  g A ( N , λ ) =  Maximum sustainable yield Minimal viable feedback where The Patagonian toothfish ◮ the spawning stock biomass S S B is defined by Questions & Answers A � B ( N ) := S S γ a w a N a a = 1 ◮ the function ϕ describes the stock-recruitment relationship ◮ M a is the natural mortality rate of individuals of age a ◮ F a is the mortality rate of individuals of age a due to harvesting

Viable harvest of Age structured model monotone bioeconomics models R A ◮ the state : N = ( N a ) a = 1 ,..., A ∈ I + , the abundances at age ◮ the control : λ the fishing effort multiplier A fishery ◮ the dynamics : N ( t + 1 ) = g ( N ( t ) , λ ( t )) given by management model The model Some questions about  g 1 ( N , λ ) = ϕ ( S S B ( N )) , sustainability of landings   Discrete time   viability issues  e − ( M a − 1 + λ F a − 1 ) N a − 1 , g a ( N , λ ) = a = 2 , . . . , A − 1 , Monotonicity properties Viability kernel   properties  e − ( M A − 1 + λ F A − 1 ) N A − 1 + e − ( M A + λ F A ) N A .  g A ( N , λ ) =  Maximum sustainable yield Minimal viable feedback where The Patagonian toothfish ◮ the spawning stock biomass S S B is defined by Questions & Answers A � B ( N ) := S S γ a w a N a a = 1 ◮ the function ϕ describes the stock-recruitment relationship ◮ M a is the natural mortality rate of individuals of age a ◮ F a is the mortality rate of individuals of age a due to harvesting

Viable harvest of Age structured model monotone bioeconomics models R A ◮ the state : N = ( N a ) a = 1 ,..., A ∈ I + , the abundances at age ◮ the control : λ the fishing effort multiplier A fishery ◮ the dynamics : N ( t + 1 ) = g ( N ( t ) , λ ( t )) given by management model The model Some questions about  g 1 ( N , λ ) = ϕ ( S S B ( N )) , sustainability of landings   Discrete time   viability issues  e − ( M a − 1 + λ F a − 1 ) N a − 1 , g a ( N , λ ) = a = 2 , . . . , A − 1 , Monotonicity properties Viability kernel   properties  e − ( M A − 1 + λ F A − 1 ) N A − 1 + e − ( M A + λ F A ) N A .  g A ( N , λ ) =  Maximum sustainable yield Minimal viable feedback where The Patagonian toothfish ◮ the spawning stock biomass S S B is defined by Questions & Answers A � B ( N ) := S S γ a w a N a a = 1 ◮ the function ϕ describes the stock-recruitment relationship ◮ M a is the natural mortality rate of individuals of age a ◮ F a is the mortality rate of individuals of age a due to harvesting

Viable harvest of Age structured model monotone bioeconomics models R A ◮ the state : N = ( N a ) a = 1 ,..., A ∈ I + , the abundances at age ◮ the control : λ the fishing effort multiplier A fishery ◮ the dynamics : N ( t + 1 ) = g ( N ( t ) , λ ( t )) given by management model The model Some questions about  g 1 ( N , λ ) = ϕ ( S S B ( N )) , sustainability of landings   Discrete time   viability issues  e − ( M a − 1 + λ F a − 1 ) N a − 1 , g a ( N , λ ) = a = 2 , . . . , A − 1 , Monotonicity properties Viability kernel   properties  e − ( M A − 1 + λ F A − 1 ) N A − 1 + e − ( M A + λ F A ) N A .  g A ( N , λ ) =  Maximum sustainable yield Minimal viable feedback where The Patagonian toothfish ◮ the spawning stock biomass S S B is defined by Questions & Answers A � B ( N ) := S S γ a w a N a a = 1 ◮ the function ϕ describes the stock-recruitment relationship ◮ M a is the natural mortality rate of individuals of age a ◮ F a is the mortality rate of individuals of age a due to harvesting

Viable harvest of Age structured model monotone bioeconomics models R A ◮ the state : N = ( N a ) a = 1 ,..., A ∈ I + , the abundances at age ◮ the control : λ the fishing effort multiplier A fishery ◮ the dynamics : N ( t + 1 ) = g ( N ( t ) , λ ( t )) given by management model The model Some questions about  g 1 ( N , λ ) = ϕ ( S S B ( N )) , sustainability of landings   Discrete time   viability issues  e − ( M a − 1 + λ F a − 1 ) N a − 1 , g a ( N , λ ) = a = 2 , . . . , A − 1 , Monotonicity properties Viability kernel   properties  e − ( M A − 1 + λ F A − 1 ) N A − 1 + e − ( M A + λ F A ) N A .  g A ( N , λ ) =  Maximum sustainable yield Minimal viable feedback where The Patagonian toothfish ◮ the spawning stock biomass S S B is defined by Questions & Answers A � B ( N ) := S S γ a w a N a a = 1 ◮ the function ϕ describes the stock-recruitment relationship ◮ M a is the natural mortality rate of individuals of age a ◮ F a is the mortality rate of individuals of age a due to harvesting