Control of a Lake Network Invasion: Bioeconomics

Approach

Alex Potapov, Mark Lewis, and David Finnoff*

Centre for Mathematical Biology, University of Alberta. *Department of Economics and Finance, University of Wyoming

Sea Lamprey

Zebra Mussels

Great Lakes Invasion by Alien Species

Rusty crayfish

Economic Impact:

Average Cost to Control Zebra Mussels by Plant Type as of 1995:Hydroelectric facilities $83,000.Fossil fuel generating facilities $145,000.Drinking water treatment facilities $214,000.Nuclear power plants $822,000.

Damage to Recreation, Change in Ecosystems

Beaches covered by shells, smell, cleared water, less sport fish

Crusty boats…

Zebra mussel spread

20031988

Rusty Crayfish in North America

How do invaders spread?

•To the Great lakes – various ways, mainly by ships in ballast water.

•Within the lake system – naturally

•To land lakes and between them – mainly through fishing and boating equipment.

•Prevention – equipment washing…

•20 researchers from 5 universities: D. Lodge and Gary Lamberti (U Notre Dame), M. Lewis (U Alberta), H. MacIsaac (U Windsor), J. Shogren and D. Finnoff (U Wyoming), Brian Leung (McGill)

•5 year project

•Collaborative project between biologists, economists and mathematicians

http://www.math.ualberta.ca/~mathbio/ISIS

Clark C.W. Mathematical Bioeconomics. The optimal management of renewable resources. 1990.

Van Kooten G. C. and Bulte E. The Economics of Nature, 2000 .

Main idea:

•Model an ecological system as a dynamical system

•Include human activity and costs/benefits

•Determine the optimal harvesting/management via optimal control theory

Analysis using optimal control theory

Invader dynamics + costs/benefits optimization (integrative bioeconomic models)

 

Populationlevel

Dispersal

Introduction Transportation

Prevention/Control Costs/Benefits

Optimization

Expenses

Losses/changes

General invasion model with control

Model includes dynamics of the invader in the lake ui, possible controls, minimization of costs (or maximizing benefits)

min,,

,,,,

iii

jijijjii

i

chuJ

wccuwhuFdt

du

Minimize costs or maximize benefits

Macroscopic model for invasion spread

Invasion is described in terms of proportion of infected lakes

p=NI/N.

Invader propagules are transported from lake to lake by boats

(intensity A1), probability of survival A2, increase in number of

infected lakes t=Np during t is (N–NI)NItA1A2,

NAAApApdt

dp21,1

Invader ControlPrevention effort at infected and uninfected lakes: x and s (effort/lake/time).

Probability of propagule escaping treatment at infected lake is a 1, and at uninfected lake is b 1 .

Washing efficiency 1–a, and 1–b.

Assume effects of two successive prevention treatments are independent: a(x1+x2)=a(x1)a(x2 )

Dynamic equation for proportion of lakes invaded:

)exp()( 1xkxa )exp()( 2sksb

),,()1()exp(

)1( )()(

21

dynamicsontransmissi

nonlineareffort control todue

ratein reduction

control no with rate

infection

psxfppskxkA

ppsbxaAdt

dp

Costs

Invasion cost: g ($/lake/time) – decrease in benefits or increase in costs

Prevention cost: wx at invaded lakes

ws at uninvaded lakes

Total invasion cost/lake:

costs controllossesinvasion

))(1)(()()()( tptswtptxwtgpC sx

Discounting and optimality

Total cost during time interval 0 t T:

Optimal control problem:

minimize J by choosing x(t) and s(t) 0 t T

),,()1()exp(

)1( )()(

)exp()(

,),,( )()(),(

21

dynamicsontransmissi

Nonlineareffort control todue

ratein Reduction

control no with rate

Infection

0

and controls and system of state

on dependingCost gDiscountin

psxfppskxkA

ppsbxaAdt

dp

rtt

dtpsxCttstxJT

sxp

Cost functional

Discounting function

Dynamical equation for proportion of lakes invaded p(t)

(optimization constraints)

Maximum Principle

.0,0|0,0

,0,0|0,0

)0( ,),,(

0)( ,),,,(

,),,(),,(),,,(

lakes invaded ofproportion Initial

0

dynamicsGrowth

invaded lakesproportion

of changeof Rate

horizon timeFinite

dynamicsNonlinear

with of Change

gDiscountin

priceshadow

of changeof Rate

dynamicsGrowth

priceShadowfunctionCost

nHamiltonialueCurrent va

sHsH

xHxH

pppsxfdt

dp

Tpsxgp

Hr

dt

d

psxfpsxCpsxH

ss

xx

TpH

Goal: maximize H (Hamiltonian)

Dynamical equation for shadow price t) with terminal condition

Dynamical equation for proportion of lakes invaded p(t) with initial condition

Optimality (max in x, s) conditions at any 0t T

Optimality conditions

0,0or01211

x

HxppAekpw

x

H skxkx

0,0or011 212

s

HsppAekpw

s

H skxks

Three types of control

1. Donor control

2. Recipient control

3. No control

Sppspxx ,0 ),,( **

Sppxpss ,0 ),,( **

psx SW ,0

Non-overlapping control regions

),0,0( ),,,0,0(

)),,(,0( ),,),(,0(

),0),,(( ),,,0),,((

**

**

pfdt

dppg

dt

d

ppsfdt

dpppsg

dt

d

ppxfdt

dpppxg

dt

d

0 ),,( ** spxx ),( ,0 ** pssx

0 ,0 ** sx

x-control

s-control

No control

The current value Hamiltonian H is maximized by x=x*, s=s*

Finish here at time t=T, p=pe

Start here at time t=0g

fdp

d When there is no discounting (r=0), solution can be calculated analytically from

p0

Terminal time specifies optimal trajectory

pe pe pe 1

Proportion infected lakes

Shadow price

Donor control Recipient control

Proportion infected lakes

Terminal time

T

TT

T small: No control

T intermediate:Donor, then No control

T large: Donor, then Recipient then No control

For any given T, there exists and optimal trajectory

Shadow price

Proportion infected lakes

Proportion infected lakes

Control costs

Donor control Recipient control

Recipient controlDonor control

Two different phase plane representations (p- plane, control-p plane)

Control costs—proportion infected phase plane

Shadow price—proportion infected phase plane

Donor control Recipient control

Shadow price

Proportion infected lakes

Donor control Recipient control

Proportion infected lakes

Control costs

Solid line: No discounting (solution is calculated analytically)

Dashed line: With discounting (solution must be calculated numerically)

Effect of the discount rate

Outcomes with and without discounting

Proportion infected lakes

Proportion infected lakes

Control levels

Control levels

Without discounting

With discounting

Control efficiency k=k1=k2 is varied. Thick solid — x(t), thick dashed — s(t), thin

solid — p(t), thin dashed — uncontrolled p(t), A=1, p0=0.3, g=0.5, r=0, T=50,.

No control is optimal

Conclusions-1

•We can delay invasion but not stop it.•Goal is to delay invasion so as to increase net benefit from a bioeconomic perspective.•Problem can be analyzed using phase plane methods.•Three main strategies for controlling invaders: Donor control, recipient control, no control. Switching occurs between strategies as the invasion progresses.•Short (e.g., political) time horizons can yield no control as optimal.•Control strategies are sensitive to discounting. Discounting reduces early investment in control and allows invasion to progress quickly.

Model extension: eradication

0,0

0,1

p

pthppAe

dt

dp skxk sx

costsneradicatiocosts controllosses invasion

))(1)(()()()( thwtptswtptxwtgpC hsx

C is linear in h, bang-bang control: h=0 or h=hmax.

hh wwh

H

atswitching,

Controls in the phase plane

New kind of solution: complete eradication

If we eradicate invader by some moment t1, then for t>t1 there are no losses and no costs.

New formulation: free terminal time, fixed end state p=0, and hence s=0. Different boundary condition

h

h

wt

thtthwtH

1

1111 ,0

min ),,,(1

0 t rt dthsxpCeJ

Variety of solutions: isochrones view

New effect: several locally optimal solutions.

Complete eradication is the optimum only for big enough T.

p0

Isochrone with appropriate T

Isochrone = set of all initial states (p,) such that (T)=0

Beginning of optimal trajectory

Beginning of suboptimal trajectory

Eradication is optimal

Complete eradication trajectory

)1(25.0

05.0and0,3,05.0,3

kT

rrghw

k

h

Terminal value: beyond the control horizon

At t=T the ecosystem remains and still can bring benefits, must have some value V(pe). Then it is necessary optimize cost+terminal value.

Let a system with invasion level p under controls x(t) produces benefits with a rate W(p,x), then we need

How to define VT(p(T))? No agreement on this at present.

txTrTT rt TpVedtxpWe max,

0

Invariant terminal value

Let us define V through infinite horizon problem.

p(0)=p0. Define

00 ,max dtxpWepV rttx

Value = present cost of maximum future benefits under optimal management

Then solution of a finite time horizon T optimal control problem with terminal cost V(pe) coincides with x(t) on (0,T) (x(t) does not depend on T).

Can be formulated in terms of minimizing future costs under

A solution of an infinite-horizon problem (IHP) ends at an invariant set of the dynamical system.

Theorem. Let the solution of IHP {x(t),p(t)} exists and is unique for each p0=p(0) and the corresponding invariant end-state. Then optimal control xT(t) for finite-horizon problem with the terminal value V(p(T)) and the same p0 xT(t)=x(t) on (0,T).

TpVedtxpWe

dtxpWedtxpWepV

TpVedtxpWepV

rTT rt

T

rtT rt

rTT rtT

0

00

00

,

,,

,max

Either xT(t)=x(t), 0<t<T, p(T)=p(T), or a contradiction

TpVedtxpWe

dtxpWedtxpWepV

TpVedtxpWepV

rTT rt

T

rtT rt

rTT rtT

0

00

00

,

,,

,max

Proof: suppose p(T)p(T), then

•VT>V, then x(t) is not optimal

•VT=V, then x(t) is not unique

•VT<V, then x(t) is not optimal

xT(t)=x(t), 0<t<T (optimality principle)

Example: no eradication

22

11k

w

A

rg

k

w

A

rg ss

r=0.01 r=0.07

0.3;5.1;5.0g

Example: with eradication

r=0.01 r=0.03

r=0.10

Complete eradication

No eradication at the end

Optimal trajectory

Suboptimal trajectory

3;05.0;5.0 hwhg

Implications of terminal value for the problem with explicit spatial dependence

min,,

,,,,,1

Tiii

N

ijjjijjii

i

CchuJ

ccuwhuFdt

du

•Optimal control problem – system of 2N equations;

•Infinite-horizon problem – only steady states are important; at small discount – look for the best steady state;

•May be a considerable simplification: first study steady states, then choose a best way to them

Accounting for Allee effect

•Allee effect – population cannot grow at low density

•Cannot be integrated into the macroscopic model

Single lake description

uauuuFwuFdt

dui

i 1,

No external flow, population goes extinct at small u

Weak external flow, w<|Fmin|, population still goes extinct at small u;

Strong external flow, w>|Fmin|, population grows from any u

Allee effect with external flow

Explicit spatial model with Allee effect

Optimal invasion stopping: find optimal spatial controls distribution that keeps flow below critical at uninvaded lakes

We can look for the optimal place to stop the invasion

isixii

Ti

T

irt

jj

kxij

ksi

i

swxwuwW

VdtWeJ

ueBeuFdt

du ji

0

Example:

Linearly ordered lakes, Bij=B (|i–j|)

Numerical solution gives spatial distribution of controls

Bij=B0exp(–|i–j|) Bij=B0/ (1+(|i–j|)2)

Conclusions-2

•Eradication of the invader can make the problem of finding optimal control more complicated and gives new strategies;

• Terminal value through infinite-horizon problem reduces analysis to steady states and trajectories leading to them – a considerable simplification of analysis, especially for high-dimensional problems, + more transparent management recommendations;

• Allee effect allows to stop invasion without eradication; accounting for the terminal value leads to the natural problem of optimal invasion stopping

Acknowledgements

•ISIS project, NSF DEB 02-13698

•NSERC Collaboration Research Opportunity grant. .

References

A.B. Potapov, M.A. Lewis, D.C. Finoff. Optimal Control of Biological Invasions in Lake Networks. Journal of Economic Dynamics and Control, 2005 (submitted).

D.C. Finoff, M.A. Lewis, A.B. Potapov. Optimal Control of Biological Invasions in Lake Networks., 2005 (in preparation).

A.B. Potapov, M.A. Lewis. Optimal Spatial Control of Invasions with Allee Effect., 2005 (in preparation).

Influence of invasion losses per lake g on the optimal control policy

Influence of control time horizon T on the optimal control policy

Influence of initial proportion of infected lakes on the optimal control policy

Influence of discounting rate r on the optimal control policy