Specific Model

• Write on board• Flow:

– James dispersal kernel plus common larval pool

• Fish:– Discrete-time Logistic Growth, r & K shared by all– Multiplicative Kelp

• Fishing:– Linear Economic Payoffs– Marginal profit depending on distance from port

Flow: e.g. kernel patch 17 (sum<1)

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

Destination

% L

andi

ng

Larvae Leaving Patch 17

Flow: whole kernel + CLP(.005)

Destination Patch

Sou

rce

Pat

chDispersal + Larval Pool

5 10 15 20 25 30 35 40 45

5

10

15

20

25

30

35

40

45 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fish: Growth with Kelp

0 500 1000 15000

500

1000

1500

2000

2500

xt

x t+1

Lowest Kelp

Highest Kelp

Steady State Line

Issues to discuss

1. Interior vs. corner

2. Variability

3. GIS display

escapement

2 4 6

2

4

6

0

100

200

300

adults

2 4 6

2

4

6 100

200

300

profit

2 4 6

2

4

6 100

200

300

400marginal profit

2 4 6

2

4

6

1

2

3

good larvae

2 4 6

2

4

6 0.4

0.6

0.8

kelp

2 4 6

2

4

6 1.21.41.6

1.82

Code• clear all• close all• • load Kij; D=K_matrix; D=D+.005 % IxI matrix D from row to column though• %need not sum to 1• load kelp_percent; kvec1=kelp_percent; % Ix1 vector kvec (kelp by patch)• f=find(isnan(kvec1)); kvec1(f)=1;• a=1; kvec = a+((2-a)/18)*kvec1;• • bvec=[3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;2;2;2;2;2;2;2;2;2;2;2;2;1;1;1;1;1;1;1;1;1;1;1;1;1;1];%gives Ix1 vector bvec (marginal profit by patch), , • • I=length(bvec);• dr = .05;• delta = 1/(1+dr);• r=.5;• K=1000;• • bar(D(17,:))• xlabel('Destination')• ylabel('% Landing')• title('Larvae Leaving Patch 17')• figure• • imagesc(D),colorbar• title('Dispersal + Larval Pool')• xlabel('Destination Patch')• ylabel('Source Patch')• figure• • ev=[0:round(1.5*K)];• xv1=min(kvec)*(ev+r*ev.*(1-ev/K));• xv2=max(kvec)*(ev+r*ev.*(1-ev/K));• plot(ev,xv1,'k-',ev,xv2,'k-',ev,ev,'k-')• xlabel('x_t')• ylabel('x_{t+1}')• figure• • • s1 = D*bvec;• C=bvec./(delta.*kvec.*s1); %this is the constant that f' must equal, it is a vector by patch• • estar = (K*(1+r-C))./(2*r); estar=max(0,estar);• • s2=kvec.*(r*estar.*(1-estar/K))+estar;• xstar = (s2'*D)';• • hstar=xstar-estar;• f=find(hstar<0);• if isempty(f),disp('Harvests Positive'),else,disp('Negative Harvest'),estar(f)=xstar(f),hstar=xstar-estar;end• • profstar = bvec.*hstar;• • PVprofit = sum(profstar)*delta/(1-delta);• • [estar xstar profstar]• PVprofit• • • E=reshape(estar,7,7);• X=reshape(xstar,7,7);• P=reshape(profstar,7,7);• B=reshape(bvec,7,7);• DD=reshape((sum(D'))',7,7);• KK=reshape(kvec,7,7);• • subplot(3,2,1)• imagesc(E)• title('escapement')• colorbar• subplot(3,2,2)• imagesc(X)• title('adults')• colorbar• subplot(3,2,3)• imagesc(P)• title('profit')• colorbar• subplot(3,2,4)• imagesc(B)• title('marginal profit')• colorbar• subplot(3,2,5)• imagesc(DD)• title('good larvae')• colorbar• subplot(3,2,6)• imagesc(KK)• title('kelp')• colorbar• • • • • • •