Strategies for Sustainable Development Planning of Savanna System Using Optimal Control Model

Transcription

1 Strateies for Sustainable Development Plannin of Savanna System Usin Optimal Control Model Jiabao Guan Multimedia Environmental Simulation Laboratory School of Civil and Environmental Enineerin Georia Institute of Technoloy

2 Contents Introduction Stability Analysis Model and Alorithm Numerical Experiment Discussions Conclusions

3 Introduction Savanna system A savanna system is a complex natural ecosystem that describes competition behaviors between rasses and woody veetation under driven of razin cattle in a semi-arid raneland

4 Introduction Sustainable development The development that meets the needs of the present without compromisin the ability of future enerations to meet their own needs by Brundtland Commission (Brundtland, 987) Associated with economy, socioloy, and environment Junction of profit between present and future

5 Introduction Sustainable development of savanna system meets the increasin demand for present economic profit by raisin cattle protects ecoloical environment of raneland to uarantee promotin needs of future development development sustainability = resilience

6 Introduction Ecoloical resilience The capacity of a system to absorb disturbance and reoranize while underoin chane so as to still retain essentially the same function, structure, identity, and feedbacks (Walker et al., 24)

7 Introduction Resilience of savanna system Resilience of savanna system depends on current state and disturbance Current state is a result from previous manaement strateies and cannot be chaned Stockin rate is an external disturbance which reflects philosophy and tarets of new manaement strateies and is controllable.

8 Introduction Research on savanna system Resilience and stability (Walker, Ludwi et al.). Effects of physical, ecoloical, and economic factor on resilience (Anderies et al.). Stochastic model for rainfall uncertainty to find robust manaement strateies (Janssen et al.).

9 Introduction Sustainable development plannin Resilience thinkin: a perspective for analysis of social-ecoloical systems Optimization for conservation: an outcomeoriented tool to obtain a defensible solution to a well-defined problem (Fisher et al.) Interation of resilience thinkin and optimization for conservation to find stratey for sustainable development plannin

10 Stability Analysis Savanna system d( t) = r dt dw( t) = rw dt ( s c c w) [ α + w( c c w) ] w : density of rass w: density of woody veetation s: stockin rate of cattle others: parameters to describe characteristics of raneland w ww

11 Stability Analysis Two stable equilibrium states Absolute stable state: (, w) = (, ) Conditional stable state: (, w) = ( s s 5.2s +.583, s.39 4s 2 5.2s +.583)

12 Stability Analysis Two stable equilibrium states Absolute stable state Resilience = Development sustainability = Conditional stable state Resilience = f(s) s <.486 (constant stockin rate)

13 Stability Analysis Case: s =.25 r =.5 r w = c =.7 c w = c w = 2 c ww =.3 α =.3 Density of woody veetation (a) s = Density of rass

14 Stability Analysis Case: s =.45 r =.5 r w = c =.7 c w = c w = 2 c ww =.3 α =.3 Density of woody veetation (b) s = Density of rass

15 Stability Analysis Observations Which stable point rass and woody veetation approach is dependent on initial states and stockin rate Stable reion with resilience depends on stockin rate. It shrinks as stockin rate increases and will disappear when s >.486.

16 Model and Alorithm Sustainable development plannin Control the system operation in the area close to second stable point Obtain maximum profit for development of razin cattle without loss of ecoloical resilience of the system

17 Model and Alorithm Optimization model J t f = miaximize rs 2 t d( dt dw( t dt t) ) = = r r w 2 ( t) dt ( s c c w) [ α + w( c c w) ] ( w ) T X ( t ) =, X ( t ) =, f w ( w ) T f f w ww

18 Model and Alorithm Optimal control model J t f = miaximize rs 2 t 2 ( t) dt 2 h e 2 e 2 [ ( t ) ( t )] h [ w( t ) w ( t )] f f 2 w f f d( t) = r dt dw( t) = rw dt ( s c c w) [ α + w( c c w) ] ( w ) T X ( t ) =, w w ww

19 Model and Alorithm Pontryain s optimal principle Hamiltonian function H ( X( t), s( t), λ( t), t) = 2 rs 2 + λ + λ 2 { r ( s c c w) } { r ( a + w( c c w )} w w w ww

20 Model and Alorithm Pontryain s optimal principle Necessary conditions for s to be optimal control ( ) ( ) [ ] + = = ) ( ) ( w c c w r dt t dw w c c s r dt t d ww w w w α ( ) [ ] = + = ) ( 2 ) ( λ λ λ λ w c c r c r t λ w c r w c c s r t λ ww w w w w w w ) ( ) ( ) ( t t r r t s λ = ( ) T w t X, ) ( = ( ) ( ) = = ) ( ) ( ) ( ) ( ) ( ) ( 2 2 f e f f f e f f t w t w h t λ t t h t λ

21 Model and Alorithm Solution of two-point boundary-value problem - radient alorithm s () (t) = s, t solve state equation calculate boundary condition for λ f solve co-state equation backward evaluate H (i) (t)/ s(t), t s (i) (t) = s(t), t s(t)=s (i) (t)+β H (i) (t)/ s(t), t no H(i)/ s γ yes stop

22 Numerical Experiment Conditions Plannin period = years Desired final state ( e (t f ), w e (t f )) = (.69,.7) r =, h = hw = s () (t) =.486 Initial condition: from four sides in phase diaram

23 Numerical Experiment Phase diaram.8 Density of woody veetation Density of rass

24 Numerical Experiment Density of woody veetation Observations Density of rass The system can transfer to the specified desired final state from most of initial states. The system still move toward the desired final state throuh stoppin raisin razin cattle in the raneland in the situation woody veetation dominates rass. Once the system falls in this situation, it will need to take a lon time for recoverin the system resilience.

25 Numerical Experiment Optimal stockin rates Density of woody veetation Density of rass.6 (a) Stockin rate.4.2 Initial state (.,.) (.8,.) (.6,.) (.4,.) (.2,.) Year

26 Numerical Experiment Optimal stockin rates (cont) Density of woody veetation Density of rass.8 (b).6 Stockin rate.4.2 Initial state (.,.) (.,.8) (.,.6) (.,.4) (.,.2) (.,.) Year

27 Numerical Experiment Optimal stockin rates (cont) Density of woody veetation Density of rass Stockin rate (c) Initial state (.,.) (.8,.) (.6,.) (.4,.) (.2,.) (.2,.) Year

28 Numerical Experiment Optimal stockin rates (cont) Density of woody veetation Density of rass Stockin rate (d) Initial state (.2,.) (.2,.2) (.2,.4) (.2,.6) (.2,.8) (.2,.) Year

29 s Numerical Experiment Density of woody veetation Annual averae stockin rates Density of rass (, w ) s (, w ) s (, w ) s (, w ) s (.,.).379 (.,.8).394 (.8,.).55 (.2,.2).66 (.8,.).357 (.,.6).43 (.6,.).488 (.2,.4). (.6,.).323 (.,.4).435 (.4,.).46 (.2,.6). (.4,.).268 (.,.2).458 (.2,.).439 (.2,.8). (.2,.).74 (.,.).56 (.2,.).294 (.2,.).

30 Numerical Experiment Density of woody veetation Observations Density of rass The optimal strateies can obtain maximum economic profit from raisin razin cattle. The optimal strateies can sinificantly improve the resilience of the raneland so that the system operate in a stable reion. Where the initial state is located directly affects operation of the system and economic profit.

31 Discussion Alorithm converence (, w ) = (.5,.5), desired final state (.69,.7) s () (t) = Performance Iteration

32 Discussion Impact of initial stockin rate s () (t) =.486,.4,.55.8 Density of rass.6 Variables.4 Stockin rate.2 s () (t) =.486 s () (t) =.4 s () (t) =.55 Density of woody veetation Year

33 Discussion Impact of penalty coefficients h and hw h =hw =,,,,,.8 Density of rass.6 Variables.4 Stockin rate.2 h = h w =, h = h w =, h = h w =, Density of woody veetation Year

34 Conclusions The initial state of the raneland has an important impact on optimal strateies. When rass is dominant, the system can obtain maximum economic profit without loss of resilience. If the raneland is dominated by woody veetation, the system can still move toward the desired final state throuh controllin stockin rate so that it can restore resilience and re-obtain development sustainability. The optimal strateies can sinificantly improve the resilience of the raneland so that the system operate in a stable reion. If the raneland is almost covered by woody veetation with a lower density of rass such as s <=.2, a best stratey is to stop raisin razin cattle until the system recovers resilience. If we would be still eaer for instant success and quick profits in this situation, the system would inevitably become worse until it loses all resilience. Once the system falls into this situation, the raneland will become a dead land for razin cattle and will difficultly restore its resilience aain.

35 Conclusions The desired final state in plannin period reflects the requirement for development sustainability of the raneland in lon-term. It is not only a terminal point of this plannin period, but also is the initial state for next plannin period. In other words, it is a junction between short-term and lon-term profits. It is always a ood choice to use the second stable equilibrium state as the desired final state. Optimal alorithm can be efficiently applied to obtain optimal strateies throuh selection of initial stockin rate and penalty coefficients. It is a ood uidance to take.486 as initial stockin rate while selection of penalty coefficients should have the penalty terms comparable with oriinal objective function. Sustainability, resilience, economic profit are consistent

36 Thank You!