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1 Discrete time mathematical models in ecology Andrew Whittle University of Tennessee Department of Mathematics Andrew Whittle University of Tennessee Department of Mathematics

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2 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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3 Why use discrete time models?

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4 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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5 Single species models

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6 Simple population model Let N t 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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7 Suppose at the initial time t = 0, N 0 = 1 and λ = 2, then We can solve the difference equation to give the population level at time t, N t in terms of the initial population level, N 0 Malthus population, when unchecked, increases in a geometric ratio

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8 Geometric growth

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9 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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10 Hassell equation Under-compensation (0**
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11 Population growth for the Hassell equation

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12 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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13 Cobweb diagrams Steady State Stability

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14 Cobweb diagrams Sterile insect release Adding an Allee effect Extinction is now a stable steady state

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15 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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16 richer behavior Period doubling to chaos in the Ricker growth model a NtNt

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18 Age structured models

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19 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 N1tN1t N 2 t+1 N 3 t+2 N 4 t+3 N 5 t+4

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20 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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21 Multi-species models

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22 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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23 multi species models NnNnNnNn PnPnPnPn die Growth

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24 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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25 Predator-Prey models Analogous discrete time predator-prey model (with mass action term) Displays similar cycles to the continuous version

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26 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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27 SIR models SusceptiblesInfectivesRemoved Often used to model with-in season Extended to include other categories such as Latent or Immune

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28 Control in discrete time models

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29 Control methods Controls that add/remove a portion of the population Cutting, harvesting, perscribed burns, insectides etc

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30 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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31 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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32 Optimal control theory

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33 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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34 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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35 Single species control State equations Objective functional Control set

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36 how do we find the best control strategy? Pontryagins discrete maximum princple

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37 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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38 adjoint equations Set Then re-arranging the equation above gives the adjoint equation

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39 Controls Set Then re-arranging the equation above gives the adjoint equation

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40 Optimality system Forward in time Backward in time Control equation

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41 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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42 numerical method Starting guess for control values State equations forward Adjoint equations backward Update controls

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43 Results B smallB large

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44 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

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