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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