1 Optimal control for integrodifferencequa tions Andrew Whittle University of Tennessee Department of Mathematics Andrew Whittle University 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

3 space

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

8 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 NPV Infected Larvae

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 fecundity Reduction 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 model We 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 equations forward Adjoint equations backward Update controls

23 results

24 Total Cost, J(p) B,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 B n (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