Optimal control of single species biological population

Transcription

1 Optimal control of single species biological population B. S. Goh Research Institute, Curtin University Sarawak Malaysia Biologists divide species into two groups. K-selected:- like humans, monkeys, whales, kangaroos, etc r-selected:- Most fish, prawns, krill, trees, grass, etc Require very different mathematical models. Ref: B S Goh, Management and Analysis of Biological Populations Elsevier Press, New York,

2 Management of K-selected:- whales, kangaroos, etc The most advanced work, without optimal control theory, was done by scientists working with the International Whaling Commission, IWC. Completed by Dr P Hammond resigned as Chair, Scientific Committee, IWC IWC was originally set up to maximize yield from harvesting not conservation per se. Management of r-selected:- most fish, turtle population have high fecundity.e.g. carp more than a million eggs/yr. The most advanced work, without optimal control theory, was done by Beverton-Holt by 1957 in their study of the plaice/flounder in the North Sea, UK. 2

3 Fishery Management (Goh,1969) System: dn / dt = rn( K N)/ K u( t) Logistic model Maths model which is supposed?? to describe fish population dynamics. Initially: N(0) = a This says at time t = 0 you can measure how many fish there are in the sea. Constraint: 0 ut () q The harvest each year is limited by quota q. Terminally: NT ( ) b Planning period [ 0,T ] and at least b fishes remain at T. Objective: Maximize, T u () t dt 0 Maximize total catch in period [ 0,T ]. 3

4 Hamiltonian: H ( N, p, u) = u + p[ rn( K N)/ K u] Costate equation: dp / dt = H / N = ( pr / K)( K 2 N) Singular control: Bang-bang controls: H/ u= 1 pt () 0 d[ H / u] = ( pr / K)( K 2 N) 0 N = K /2 u = rk /4 u = u max if H/ u= 1 pt () > 0 u = 0 if H/ u= 1 pt () < 0 s 4

5 1 u = q = 0.4 Population Level u = 0.25 = MSY u = Remaining Time Optimal FEEDBACK policy in harvest of a single K-species population. 5

6 1.2 u* = 0.2 umax= 0.5 Population level u-msy = 0.25 robust policies Time Robust stable practical harvesting policies Better PRACTICAL Policy as it is robust/stable 6

7 Critical analysis of Modeling of Biological Populations. Biologists divide species into two groups. r-selected. Fish, trees, grass, krill, prawn, rice, mosquitoes, HIGH FECUNDITY. Ex: Asian carp > 1,000,000 eggs/yr. Krill > 10,000 eggs per year krill Asian Carp, USA 7

8 K-selected. LOW FECUNDITY Humans, whales, monkeys, cows, tigers, birds, bacteria, Minke whale. 10 tons, 8 meters Skull of Blue whale 19ft. Blue whale up to 33m and 180 tons 8

9 K-selected species Initial Logistic model: dn / dt = rn( K N)/ K = g( N) Approx. Whale model: Nt () = no of sexually mature females. Nt ( + 1) = snt ( ) + RNt [ ( 12)] Model R Allen, CSIRO Fisheries Chief, Cronulla, NSW. No of female whales at time (t+1) equals survivals from last year plus females born 13 yrs ago. 9

10 Maths of r- selected species. Defn: Recruitment = No of juvenile fish each year. Very variable. HIGH FECUNDITY : Hard to model completely the population. What can we do? What is the realistic management policy? DO: Optimal size limit. Catch fish above a certain age. North Pacific Tuna Recruitment, DATA:. 10

11 Year class model: dn / dt = ( M + u) N for t t R M = natural mortality rate, u = fishing effort control variable Initially: Nt ( R) = R Terminally: NT ( LS ) > 0 Constraint: 0 u F Objective: T LS max. W ()() t u t N() t dt t R 11

12 Hamiltonian: H (, t N, p, u) = uw () t N p( M + u) N Costate equation: dp / dt = H / N = uw () t + p( M + u) Optimality conditions: Singular control: if H / u < 0, u = 0 if H / u > 0, u = F H/ u= ( Wt () pn ) 0 d[ H / u]/ dt = ( dw / dt MW ) N 0 Implies no singular control. 12

13 Optimal Solution (Goh, 1977): u * = 0 for t t < t R * u* = F for t* t tls t t LS W ( t) > FW ( s)exp[ ( M + F)( s t)] ds for t* t tls Length at t* is the optimal size limit More robust method to determine t* 13

14 Weight Weight /Costate Costate variable 200 t* defines optimal size limit Age 14

15 First ever photographs of the Antarctic sub-species in Australian waters, WA. The WA Research Institute wants to find out how this area relates to Australia s Blue whale population which doesn t seem to be able to stage a comeback at the moment unlike their prolific cousins, the humpbacks. The Australian 9 Feb 2013 Present population 2000<N<3000 down from 230,000 (IWC). 15

16 Whale populations in Antarctica. Australia vs Japan Harvest of Blue whales started in 1878 with explosive harpoon. Original population about 230,000. About 330,000 harvested. Minke whales increased when Blue whales were decimated. Age of sexual maturity declined significantly. IWC data says there more than 500,000 Southern Minkes. Blue and Minke whales both feed mainly on krill. British scientists (Atkinson et. Al.) in 2004 reported in Nature that krill biomass has declined by more than 50% since the 1970s due to global warming. What to do??? Difficulties of Maths Modeling! Ask a farmer!! Cull the Minke? as precaution to save Blue whale?? Dr P Hammond resigned as Chair of Scientific Committee, IWC, 93 16

17 Blue whales decimated 1.2 MINKE Whales Minke whales culled Equilibrium Black trajectories Natural dynamics BLUE Whales 17

18 Blue whales: dx / dt = x(4 2x 2 y) Minke whales: dy / dt = y(3 x 2 y) Heavy harvesting of Blue whales from Equilibrium: dx / dt = x(4 2x 2 y u) u = 0.5 dy / dt = y(3 x 2 y) Heavy harvesting of Minke whales with few initial Blue whales and high Minke population: dx / dt = x(4 2x 2 y) dy / dt = y(3 x 2 y v) v = 0.5 This Qualitative Analysis on speeding Blue Whale recovery. Insensitive to parameters!! 18

19 Graphical analysis of two controlled interacting species. Goh ( Book 1980). Culling of Minke increases total speed at current state P Minke Whales Culling the Minke whales increases TOTAL rate of two competing species at P Blue Whales 19

20 Thank you B. S. Goh Research Institute Curtin University Sarawak Miri, Sarawak East Malaysia Dated: 11 Feb 2013, SPOM, Newcastle 20