1 discrete time mathematical models in ecology andrew whittle university of tennessee department of...
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Discrete time mathematical
models in ecology
Discrete time mathematical
models in ecologyAndrew Whittle
University of TennesseeDepartment of Mathematics
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 discrete time models?
Why use discrete time models?
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Discrete time
• Populations with discrete non-overlapping generations (many insects and plants)
• Reproduce at specific time intervals or times of the year
• Populations censused at intervals (metered models)
When are discrete time models appropriate ?
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Single species models
Single species models
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Simple population model
• Let Nt be the population level at census time t
• Let d be the probability that an individual dies between censuses
• Let b be the average number of births per individual between censuses
Then
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 the population 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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Geometric growth
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Intraspecific competition
• No competition - Population grows unchecked i.e. geometric growth
• Contest competition - “Capitalist competition” all individuals compete for resources, the ones that get them survive, the others die!
• Scramble competition - “Socialist competition” individuals divide resources equally among themselves, so all survive or all die!
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Hassell equation
• Under-compensation (0<b<1)
• Exact compensation (b=1)
• Over-compensation (1<b)
The Hassell equation takes into account intraspecific competition
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Population growth for the Hassell equation
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Special case: Beverton-Holt 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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Cobweb diagrams
“Steady State”
“Stability”
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Cobweb diagrams
• Sterile insect release
• Adding an Allee effect
• Extinction is now a stable steady state
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Ricker growth
• Another model arising from the fisheries literature is the Ricker stock recruitment model (1954, 1958)
• This is an over-compensatory model which can lead to complicated behavior
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richer behavior Period doubling to chaos in the Ricker growth model
a
Nt
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Age structured models
Age structured models
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Age structured models
• A population may be divided up into separate discrete age 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 by reproduction from individuals from other age classes
• Individuals in the last age class may survive and remain in that age class
N1t N2
t+1 N3t+2 N4
t+3 N5t+4
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Leslie 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 age distribution
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Multi-species models
Multi-species models
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Multi-species models
• Competition: Two or more species compete against each other for resources.
• Predator-Prey: Where one population depends on the other for survival (usually for food).
• Host-Pathogen: Modeling a pathogen that is specific to a particular host.
• SIR (Compartment model): Modeling the number of 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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Competition 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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Predator-Prey models
• Analogous discrete time predator-prey model (with mass action term)
• Displays similar cycles to the continuous version
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Host-Pathogen models
An example of a host-pathogen model is the Nicholson and Bailey model (extended)
Many forest insects often display cyclic populations similar to the cycles displayed by these equations
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SIR models
Susceptibles Infectives Removed
• Often used to model with-in season
• Extended to include other categories such as Latent or Immune
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Control in discrete time models
Control in discrete time models
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Control methods• Controls that add/remove a portion of
the population
Cutting, harvesting, perscribed burns, insectides etc
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Adding control to our models
• Controls that change the population system
Introducing a new species for control, sterile insect release etc
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We could test lots of different scenarios and see which is the best.
How do we decided what is the best control strategy?
Is there a better way?
However, this may be teadius and time consuming work.
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Optimal control theory
Optimal control theory
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Optimal control
• We first add a control to the population model
• Restrict the control to the control set
• Form a objective function that we wish to either minimize or maximize
• The state equations (with control), control set and the objective function form what is called the bioeconomic model
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Example
• We consider a population of a crop which has economic importance
• We assume that the population of the crop grows with Beverton-Holt growth dynamics
• There is a cost associated to harvesting the crop
• We wish to harvest the crop, maximizing profit
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Single species controlState equations
Objective functional
Control set
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how do we find the best control strategy?
how do we find the best control strategy?
Pontryaginsdiscrete maximum
princple
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Method to find the optimal control
• We first form the following expression
• By differentiating this expression, it will provide us with a set of necessary conditions
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adjoint equations
Set
Then re-arranging the equation above gives the adjoint equation
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Controls
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 away!
• Found conditions that the optimal control must satisfy
• For the last step, we try to solve using a numerical method
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numerical method• Starting guess for control values
State equationsforward
Adjoint equationsbackward
Updatecontrols
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Results
B smallB large
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Summary• 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