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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