Optimal Harvesting with Coupled Population and Price Dynamics Floyd - PowerPoint PPT Presentation

Optimal Harvesting with Coupled Population and Price Dynamics Floyd B. Hanson ∗ Laboratory for Advanced Computing University of Illinois at Chicago and Dennis Ryan Division of Science and Mathematics McKendree College SIAM 50th Anniversary Meeting 08-12 July 2002 in Philadelphia MS26 Control Applications in Mathematical Biology ∗ Work supported in part by National Science Foundation Computational Mathematics Program under grants DMS-93-01107, DMS-96-26692, DMS-99-73231, DMS-02-07081. Hanson and Ryan — 1 — UIC and McKendree

Overview 1. Noninflationary, Deterministic Model. 2. Inflationary, Stochastic Control Model. 3. Numerical Approximations. 4. Numerical Results. 5. Conclusions. Hanson and Ryan — 2 — UIC and McKendree

Outline of Abstract • Optimal Control of Stochastic Resource in Continuous Time. • Model Effects of Large Random Price Fluctuations. • Influence of Continuous growth and Jump Stochastic Noise. • Computational Stochastic Dynamic Programming. • Pronounced Effect of Inflationary Prices on Optimal Return. Hanson and Ryan — 3 — UIC and McKendree

Part 1. Noninflationary, Deterministic Model: Introduction. 1.1. Ordinary Differential Equation (ODE): • Nonlinear (Logistic) Dynamics: d X( s ) = [ r 1 X( s )(1 − X( s ) /K ) − H( s )] ds, 0 < t < s < T. • Initial Conditions: X (0) = x 0 ; 0 < t < T • State Variable (Resource Size): X ( t ) = [ X i ( t )] 1 × 1 ; • BiLinear Control-State Dynamics Assumption (for Resource Harvesting): H ( t ) = q U ( t ) X ( t ) ; • q = Efficiency (Catchability) Coefficient; Hanson and Ryan — 4 — UIC and McKendree

• Control Variable (Harvesting Effort): U( t ) = [ U i { X( t ) , t } ] 1 × 1 , U min ≤ U( t ) ≤ U max < ∞ ; • Growth Parameters: r 1 = Resource Intrinsic Growth Rate; K = Environment Carrying (Saturation) Capacity. Hanson and Ryan — 5 — UIC and McKendree

1.2. Quadratic Performance Index: � T e − δ ( s − t ) [ pq U( s )X( s ) − c (U( s ))] ds , V (X , U , t ) = t where • V ( x , u , t ) = Current Value of Future Resources (i.e., exp( δt ) times Present Value); • T = Time Horizon ( T ≥ t ); • δ = Nominal Discount Rate (NOT adjusted for inflation); • p = Price of Resource per Unit Harvest Rate ; • c ( u ) = c 1 u + c 2 u 2 = Quadratic Costs (Assume Increasing, Convex Quadratic Costs: c 1 > 0 and c 2 > 0); • Instantaneous Net Return: R ( x , u ) = pq ux − c ( u ) . Hanson and Ryan — 6 — UIC and McKendree

1.3. Deterministic Dynamic Programming: • Optimization Goal = Maximize Total Return: v ∗ (x , t ) = V (x , u ∗ , t ) = max [ V (x , u , t )] ; u • PDE of Deterministic Dynamic Programming: v ∗ t (x , t ) + r 1 x(1 − x /K )v ∗ x (x , t ) − δ v ∗ (x , t ) + S ∗ (x , t ) = 0; • Control Switching Term: S ∗ (x , t ) = max p − v ∗ h“ ” q ux − c 1 u − c 2 u 2 i x (x , t ) ; u • Regular (Unconstrained) Control: u R (x , t ) = ( p − v ∗ x (x , t )) q x − c 1 , c 2 > 0; 2 · c 2 Hanson and Ryan — 7 — UIC and McKendree

• Optimal (Constrained) Control:   U max , U max ≤ u R (x , t )     u ∗ (x , t ) =   ; u R (x , t ) , U min ≤ u R (x , t ) ≤ U max     U min , u R (x , t ) ≤ U min   v ∗ ( x , T ) = 0; • Final Boundary Condition: • Extinction Natural Boundary Condition: v ∗ (0 , t ) = − ( c 1 + c 2 U min ) U min “ 1 − e − δ ( T − t ) ” , δ > 0 . δ Hanson and Ryan — 8 — UIC and McKendree

Part 2. Inflationary, Stochastic Control Model 2.1. Stochastic Dynamics Equation (SDE (1)): • Nonlinear Dynamics with Gaussian (G) and Poisson (Z) Noise: d X( s ) = [ r 1 X( s )(1 − X( s ) /K ) − H( s )] ds n � + σ 1 X( s ) dW 1 ( s ) + X( s ) a j dZ j ( s, f j ) , j =1 X( t ) = x , • Initial Conditions: X (0) = x 0 , t 0 < t < s < T ; • Gaussian (Wiener) Noise (Zero Mean and Normalized): E [ dW 1 ( t )] = 0 , V ar [ dW 1 ( t )] = dt σ 1 ≤ 0 ; Hanson and Ryan — 9 — UIC and McKendree

• Poisson (Jump) Noise: E [ dZ j ( t, f j )] = f j dt , V ar [ dZ j ( t, f j )] = f j dt , 1 ≤ j ≤ n , where f j = Jump Rate and a j = Jump Amplitude Coefficient ( − 1 < a j ); • Independent (Uncorrelated) Processes Assumption: Cov [ dW 1 ( t ) , dZ j ( t, f j )] = 0 , Cov [ dZ j ( t, f j ) , dZ j ′ ( t, f j ′ )] = δ j,j ′ f j dt ; Hanson and Ryan — 10 — UIC and McKendree

2.2. Inflationary Factor Model: • Nonlinear Supply–Demand Model Relation: � p 0 � P( t ) = H ( t ) + p 1 Y( t ) , * P ( t ) · H ( t ) = Gross Return on Harvest; * p 0 = Supply–Demand Price Coefficient; * p 1 = Constant Price per Unit Harvest; * Y ( t ) = Fluctuating Inflationary Factor; • Linear Fluctuating Inflationary Factor SDE (2): m X d Y( s ) = r 2 Y( s ) ds + σ 2 Y( s ) dW 2 ( s ) + Y( s ) b j dQ j ( s ; g j ) , j =1 * Y ( t ) = y ; * r 2 = Annual Rate of Inflation without Fluctuations; * g j = j th component of Inflationary Jump Rate; * b j = j th component of Jump Amplitude Coefficient; Hanson and Ryan — 11 — UIC and McKendree

• Inflationary Gaussian (Wiener) Noise: E [ dW 2 ( t )] = 0 , V ar [ dW 2 ( t )] = dt σ 2 ≤ 0 ; • Inflationary Poisson (Jump) Noise: E [ dQ j ( t, g j )] = g j dt , V ar [ dQ j ( t, g j )] = g j dt , 1 ≤ j ≤ m ; • Independent (Uncorrelated) Processes Assumption: Cov [ dW 2 ( t ) , dQ j ( t, g j )] = 0 , Cov [ dQ j ( t, g j ) , dQ j ′ ( t, g j ′ )] = δ j,j ′ g j dt ; Hanson and Ryan — 12 — UIC and McKendree

5 4 P, Price (US Dollars per Kilogram) 3 2 1 1940 1950 1960 1970 1980 Year Figure 1: Pacific halibut prices in USdollars per kilogram for each year from 1935 to 1985 (Raw Data: IPHC 1984 and 1985 Annual Reports). Hanson and Ryan — 13 — UIC and McKendree

40 35 30 H, Catch (Million Kilograms) 25 20 15 10 5 0 1940 1950 1960 1970 1980 Year Figure 2: U.S.-Canadian catch in millions of kilograms for each year from 1935 to 1985 (Raw Data: IPHC 1984 and 1985 Annual Reports). Hanson and Ryan — 14 — UIC and McKendree

4 P, Price (US Dollars per Kilogram) 3 2 1 10 15 20 25 30 35 H, Catch (Million Kilograms) Figure 3: Pacific halibut price in USdollars per kilogram versus catch in millions of kilograms for years from 1935 to 1985. Linear regression for price times catch as a function of catch from 1980 to 1985 displayed as smooth hyperbolic price curve. (Raw Data: IPHC 1984 and 1985 Annual Reports). Hanson and Ryan — 15 — UIC and McKendree

2.3. Mean Quadratic Performance Index: »Z T e − δ ( s − t ) [( p 0 + p 1 q U( s )X( s ))Y( s ) V (x , y , u , t ) = E t – − c (U( s ))] ds | X( t ) = x , Y( t ) = y , U( t ) = u , • V ( x , y , u , t ) = Expected Current Value of Future Resources (i.e., exp( δt ) times Present Value); • { x , y } = 2-Dim State of Inflationary Stochastic Dynamics ; • T = Time Horizon ( T ≥ t ); • δ = Nominal Discount Rate (NOT adjusted for inflation); Hanson and Ryan — 16 — UIC and McKendree

2.4. Stochastic Dynamic Programming: • Optimization Goal = Maximize Total Return: v ∗ (x , y , t ) = V (x , y , u ∗ , t ) = max � � V (x , y , u , t ) ; u • PDE of Stochastic Dynamic Programming: v ∗ t (x , y , t ) + r 1 x(1 − x /K )v ∗ x (x , y , t ) − δ v ∗ (x , y , t ) 0 = σ 2 1 x 2 X v ∗ v ∗ ` − v ∗ (x , y , t ) ˆ ´ ˜ + xx + f j (1 + a j )x , y , t 2 j y + σ 2 2 y 2 r 2 yv ∗ v ∗ X v ∗ (x , (1 + b j )y , t ) − v ∗ (x , y , t ) ˆ ˜ + yy + g j 2 j S ∗ (x , y , t ) , + by General Itˆ o Chain Rule; • Control Switching Term: x (x , y , t ) ´ q ux − c 1 u − c 2 u 2 ˜ ; S ∗ (x , y , t ) = max ˆ p 0 y + ` p 1 y − v ∗ u Hanson and Ryan — 17 — UIC and McKendree

2.4.1. More Stochastic Dynamic Programming: • Regular (Unconstrained) Control: u R (x , y , t ) = ( p 1 y − v ∗ x (x , y , t )) q x − c 1 , c 2 > 0; 2 c 2 • Optimal (Constrained) Control: 8 9 U max , U max ≤ u R (x , y , t ) > > > > u ∗ (x , y , t ) = < = ; u R (x , y , t ) , U min ≤ u R (x , y , t ) ≤ U max > > > > U min , u R (x , y , t ) ≤ U min : ; v ∗ ( x , y , T ) = 0; • Final Boundary Condition: • Extinction Natural Boundary Condition*: v ∗ (0 , 0 , t ) = − ( c 1 + c 2 U min ) U min “ 1 − e − δ ( T − t ) ” , δ > 0 . δ * see Kushner and Dupuis (1992) for proper handling of stochastic reflecting boundary conditions. Hanson and Ryan — 18 — UIC and McKendree

Part 3. Numerical Approximations 3.1 Basic Hybrid Numerical Procedures. • Extrapolated, Predictor-Corrector for Nonlinear Iteration. • Crank-Nicolson Implicit for 2nd Order in Time and State. • Modifications for Poisson Functional Terms. • Modifications for Optimization in Switching Term. Hanson and Ryan — 19 — UIC and McKendree