Transcript Discrete time mathematical models in ecology

Discrete time
mathematical models in
ecology
Andrew Whittle
University of Tennessee
Department of Mathematics
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Outline
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Introduction - Why use discrete-time models?
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Single species models
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Age structure models
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Leslie matrices
Non-linear multi species models
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Geometric model, Hassell equation, Beverton-Holt,
Ricker
Competition, Predator-Prey, Host-Parasitiod, SIR
Control and optimal control of discrete models
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Application for single species harvesting problem
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Why use discrete time
models?
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Discrete time
When are discrete time models appropriate ?
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Populations with discrete non-overlapping
generations (many insects and plants)
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Reproduce at specific time intervals or times
of the year
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Populations censused at intervals (metered
models)
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Single species models
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Simple population model
Consider a continuously breading population
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Let Nt be the population level at census time t
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Let d be the probability that an individual dies
between censuses
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Let b be the average number of births per
individual between censuses
Then
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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
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No competition - Population grows unchecked
i.e. geometric growth
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Contest competition - “Capitalist competition” all
individuals compete for resources, the ones that
get them survive, the others die!
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Scramble competition - “Socialist competition”
individuals divide resources equally among
themselves, so all survive or all die!
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Hassell equation
The Hassell equation takes into account intraspecific
competition
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Under-compensation (0<b<1)
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Exact compensation (b=1)
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Over-compensation (1<b)
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Population growth for the
Hassell equation
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Special case:
Beverton-Holt model
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Beverton-Holt stock recruitment model
(1957) is a special case of the Hassell
equation (b=1)
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Used, originally, in fishery modeling
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Cobweb diagrams
“Steady State”
“Stability”
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Cobweb diagrams
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Sterile insect
release
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Adding an Allee
effect
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Extinction is
now a stable
steady state
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Ricker growth
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Another model arising from the fisheries
literature is the Ricker stock recruitment
model (1954, 1958)
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This is an over-compensatory model which
can lead to complicated behavior
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Nt
a
Period doubling to chaos in the
Ricker growth model
richer behavior
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Age structured
models
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Age structured models
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A population may be divided up into separate discrete
age classes
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At each time step a certain proportion of the population
may survive and enter the next age class
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Individuals in the first age class originate by
reproduction from individuals from other age classes
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Individuals in the last age class may survive and remain
in that age class
N1t
N2t+1
N3t+2
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N4t+3
N5t+4
Leslie matrices
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Leslie matrix (1945, 1948)
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Leslie matrices are linear so the population level of
the species, as a whole, will either grow or decay
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Often, not always, populations tend to a stable age
distribution
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Multi-species models
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Multi-species models
Single species models can be extended to multi-species
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Competition: Two or more species compete
against each other for resources.
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Predator-Prey: Where one population depends on
the other for survival (usually for food).
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Host-Pathogen: Modeling a pathogen that is
specific to a particular host.
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SIR (Compartment model): Modeling the number
of individuals in a particular class (or compartment).
For example, susceptibles, infecteds, removed.
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multi species models
Growth
Growth
Nn
Pn
die
die
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Competition model
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Discrete time version of the Lokta-Volterra
competition model is the Leslie-Gower model
(1958)
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Used to model flour beetle species
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Predator-Prey models
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Analogous discrete time predator-prey
model (with mass action term)
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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
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Often used to model with-in season
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Extended to include other categories such as
Latent or Immune
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Control in discrete time
models
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Control methods
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Controls that add/remove a portion of the
population
 Cutting, harvesting, perscribed burns,
insectides etc
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Adding control to our
models
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Controls that change the population system
 Introducing a new species for control, sterile
insect release etc
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How do we decided what is the
best control strategy?
We could test lots of different scenarios and see
which is the best.
However, this may be teadius and time
consuming work.
Is there a better way?
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Optimal control theory
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Optimal control
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We first add a control to the population model
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Restrict the control to the control set
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Form a objective function that we wish to
either minimize or maximize
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The state equations (with control), control set
and the objective function form what is called
the bioeconomic model
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Example
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We consider a population of a crop which has
economic importance
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We assume that the population of the crop
grows with Beverton-Holt growth dynamics
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There is a cost associated to harvesting the
crop
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We wish to harvest the crop, maximizing profit
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Single species control
State equations
Control set
Objective functional
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Pontryagins
how
do
we
find
the
discrete maximum
best control
strategy?
princple
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Method to find the optimal
control
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We first form the following expression
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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
Forward
in time
Backward
in time
Control
equation
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One step away!
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Found conditions that the optimal control must
satisfy
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For the last step, we try to solve using a
numerical method
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numerical method
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Starting guess for control values
State equations
forward
Update
controls
Adjoint equations
backward
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Results
B large
B small
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Summary
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Introduced discrete time population models
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Single species models, age-structured models
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Multi species models
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Adding control to discrete time models
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Forming an optimal control problem using a
bioeconomic model
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Analyzed a model for crop harvesting
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