11

Optimal control for

integrodifferencequations

Optimal control for

integrodifferencequations

Andrew WhittleUniversity of Tennessee

Department of Mathematics

Andrew WhittleUniversity of Tennessee

Department of Mathematics

2

Outline

• Including space

➡ Cellular Automata, Coupled Lattice maps, Integrodifference equations

• Integrodifference models

➡ Description, dispersal kernels

• Optimal control of Integrodifference models

➡ Set up of the bioeconomic model

➡ Application for gypsy moths

33

spacespace

4

Cellular Automata

5

Coupled lattice maps

6

Integrodifference equations

Integrodifference equations are discrete in one

variable (usually time) and

continuous in another (usually

space)

7

Dispersal data

88

Optimal control for integrodifference

equations

Optimal control for integrodifference

equations

9

Example

• Gypsy moths are a forest pest with cyclic population levels

• Larvae eat the leaves of trees causing extensive defoliation across northeastern US (13 million acres in 1981)

• This leaves the trees weak and vulnerable to disease

• Potential loss of a Oak species

10

life cycle of the gypsy moth

Egg mass

Adult

Larvae

Pupa

NPVInfectedLarvae

11

NPV

• Nuclear Polyhedrosis Virus (NPV) is a naturally occurring virus

• Virus is specific to Gypsy moths and decays from ultraviolet light

12

tannin

Tannin is a chemical produced be plants to defend itself from severe defoliation

Reduction in gypsy moth fecundityReduction in gypsy moth susceptibility to

NPV

13

model for gypsy moths

14

Biocontrol

• Natural enemy or enemies, typically from the intruder’s native region, is introduced to keep the pest under control

• More biocontrol agent (NPV) can be produced (Gypchek)

• Production is labor intensive and therefore costly but it is still used by USDA to fight major outbreaks

15

bioeconomic modelWe form an objective function that we

wish to minimize

where the control belong to the bounded set

16

optimal control of integrodifference equations

We wish to minimize the objective function

subject to the constraints of the state system

We first prove the existence and uniqueness of the optimal control

17

• There is no Pontryagin’s maximum principle for integrodifference equations

• Suzanne, Joshi and Holly developed optimal control theory for integrodifference equations

• This uses ideas from the discrete maximum principle and optimal control of PDE’s

18

Characterization of the optimal control

We take directional derivatives of the objective functional

In order to do this we must first differentiate the

state variables with respect to the control

19

Sensitivities and adjoints

By differentiating the state system we get the sensitivity equations

The sensitivity system is linear. From the sensitivity system we can find the adjoints

20

Finding the adjoint functions allows us to replace the sensitivities in the directional derivative of the objective function

Adjoints

21

Including the control bounds we get the optimal control

By a change of order of integration we have

22

numerical method

• Starting guess for control values

State equationsforward

Adjoint equationsbackward

Updatecontrols

23

results

24

Total Cost, J(p)

B,

J(p)

25

Summary

• Integrodifference equations are a useful tool in modeling populations with discrete non-overlapping generations

• Optimal control of Integrodifference equations is a new growing area and has practical applications

• For gypsy moths, optimal solutions suggest a longer period of application in low gypsy moth density areas

• Critical range of Bn (balancing coefficient) that cause a considerable decrease in Total costs

• More work needs to be done for example, in cost functions for the objective function