Whittle Discretetimew

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Discrete time

mathematicalmodels in ecology

Andrew WhittleUniversity of Tennessee

Department of Mathematics


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Outline

• Introduction - Why use discrete-time models?

• Single species models

➡ Geometric model, Hassell equation, Beverton-Holt, Ricker• Age structure models

➡ Leslie matrices

• Non-linear multi species models

➡ Competition, Predator-Prey, Host-Parasitiod, SIR

• Control and optimal control of discrete models

➡ Application for single species harvesting problem


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Why use discretetime models


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Discrete time

• Populations with discrete non-overlappinggenerations (many insects and plants)

• Reproduce at specific time intervals or timesof the year

• Populations censused at intervals (meteredmodels)

When are discrete time models appropriate ?


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"ingle speciesmodels


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"imple populationmodel

• Let Nt be the population level at census time t

• Let d be the probability that an individualdies between censuses

• Let b be the average number of births per

individual between censusesThen

Consider a continuously breading population


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Suppose at the initial time t = 0, N0 = 1 and λ = 2, then

We can solve the difference equation to give thepopulation level at time t, Nt in terms of the initial

population level, N0

Malthus “population, when unchecked, increases in a

geometric ratio”


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#eometric growth


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$ntraspeci%ccompetition

• No competition - Population grows uncheckedi.e. geometric growth

• Contest competition - “Capitalist competition”all individuals compete for resources, the onesthat get them survive, the others die!

• Scramble competition - “Socialist competition”individuals divide resources equally amongthemselves, so all survive or all die!


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&assell e'uation

• Under-compensation (0


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(opulation growth forthe &assell e'uation


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"pecial case)*everton+&olt model

• Beverton-Holt stock recruitment model(1957) is a special case of the Hassell

equation (b=1)

• Used, originally, in fishery modeling


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,o-we- diagrams

“Steady State”

“Stability”


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,o-we- diagrams

• Sterile insect

release

• Adding an

Allee effect

• Extinction isnow a stable

steady state


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.ic/er growth

• Another model arising from the fisheriesliterature is the Ricker stock recruitment

model (1954, 1958)

• This is an over-compensatory model whichcan lead to complicated behavior


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richer -ehavior Period doubling to chaos in theRicker growth model

a

Nt


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Age structuredmodels


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Age structured models

• A population may be divided up into separate discreteage classes

• At each time step a certain proportion of the population

may survive and enter the next age class

• Individuals in the first age class originate byreproduction from individuals from other age classes

• Individuals in the last age class may survive and remainin that age class

N1t N2t+1 N3t+2 N4t+3 N5t+4


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2eslie matrices

• Leslie matrix (1945, 1948)

• Leslie matrices are linear so the population level of

the species, as a whole, will either grow or decay• Often, not always, populations tend to a stable agedistribution


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Multi+speciesmodels


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Multi+species models

• Competition: Two or more species compete againsteach other for resources.

• Predator-Prey: Where one population depends onthe other for survival (usually for food).

• Host-Pathogen: Modeling a pathogen that is specific

to a particular host.• SIR (Compartment model): Modeling the numberof individuals in a particular class (or compartment).For example, susceptibles, infecteds, removed.

Single species models can be extended to multi-species


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multi species models

NNnn PPnn

die die

Growth Growth


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,ompetition model

• Discrete time version of the Lokta-Volterra

competition model is the Leslie-Gower model

(1958)

• Used to model flour beetle species


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(redator+(rey models

• Analogous discrete time predator-preymodel (with mass action term)

• Displays similar cycles to thecontinuous version


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&ost+(athogen models

An example of a host-pathogen model is theNicholson and Bailey model (extended)

Many forest insects often display cyclic populationssimilar to the cycles displayed by these equations


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"$. models

Susceptibles Infectives Removed

• Often used to model with-in season• Extended to include other categories such asLatent or Immune


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,ontrol in discretetime models


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,ontrol methods

• Controls that add/remove a portion of thepopulation

Cutting, harvesting, perscribed burns,insectides etc

Addi t l t


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Adding control to ourmodels

• Controls that change the population system

Introducing a new species for control, sterileinsect release etc


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We could test lots of different scenarios and see

which is the best.

&ow do we decided what is

the -est control strategy

Is there a better way?

However, this may be teadius and time

consuming work.


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Optimal controltheory


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Optimal control

• We first add a control to the populationmodel

• Restrict the control to the control set• Form a objective function that we wish toeither minimize or maximize

• The state equations (with control), control setand the objective function form what is calledthe bioeconomic model


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45ample

• We consider a population of a crop which haseconomic importance

• We assume that the population of the cropgrows with Beverton-Holt growth dynamics

• There is a cost associated to harvesting the

crop• We wish to harvest the crop, maximizingprofit


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"ingle species control

State equations

Objective functional

Control set


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how do we %nd the

-est controlstrategy

(ontryagins

discrete ma5imumprincple

Method to %nd the


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Method to %nd theoptimal control

• We first form the following expression

• By differentiating this expression, it will provideus with a set of necessary conditions


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ad7oint e'uations

Set

Then re-arranging the equation above gives the adjointequation


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,ontrols

Set

Then re-arranging the equation above gives the adjoint

equation


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Optimality system

Forwardin time

Backwardin time

Controlequation


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One step away8

• Found conditions that the optimal control mustsatisfy

• For the last step, we try to solve using anumerical method


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numerical method

• Starting guess for control values

State equations

forward

Adjoint equations backward

Updatecontrols


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.esults

B smallB large


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"ummary

• Introduced discrete time population models

• Single species models, age-structured models

• Multi species models

• Adding control to discrete time models

• Forming an optimal control problem using a bioeconomic model

• Analyzed a model for crop harvesting

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Whittle Discretetimew