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Discrete time mathematical models in ecology

Discrete time mathematical models in ecology. Andrew Whittle University of Tennessee Department of Mathematics. Outline. Introduction - Why use discrete-time models? Single species models Geometric model, Hassell equation, Beverton-Holt, Ricker Age structure models Leslie matrices

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Discrete time mathematical models in ecology

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

  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

  3. Why use discrete time models?

  4. Discrete time When are discrete time models appropriate ? • 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)

  5. Single species models

  6. Simple population model Consider a continuously breading population • 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

  7. 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”

  8. Geometric growth

  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!

  10. Hassell equation The Hassell equation takes into account intraspecific competition • Under-compensation (0<b<1) • Exact compensation (b=1) • Over-compensation (1<b)

  11. Population growth for the Hassell equation

  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

  13. Cobweb diagrams “Steady State” “Stability”

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

  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

  16. Nt a richer behavior • Period doubling to chaos in the Ricker growth model

  17. Age structured models

  18. 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 N2t+1 N3t+2 N4t+3 N5t+4

  19. 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

  20. Multi-species models

  21. Multi-species models Single species models can be extended to multi-species • 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.

  22. multi species models Growth Growth Nn Pn die die

  23. Competition model • Discrete time version of the Lokta-Volterra competition model is the Leslie-Gower model (1958) • Used to model flour beetle species

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

  25. 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

  26. SIR models • Often used to model with-in season • Extended to include other categories such as Latent or Immune Susceptibles Infectives Removed

  27. Control in discrete time models

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

  29. Controls that change the population system Introducing a new species for control, sterile insect release etc Adding control to our models

  30. 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?

  31. Optimal control theory

  32. 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

  33. 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

  34. Single species control State equations Control set Objective functional

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

  36. 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

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

  38. Controls Set Then re-arranging the equation above gives the adjoint equation

  39. Optimality system Forward in time Backward in time Control equation

  40. One step away! • Found conditions that the optimal control must satisfy • For the last step, we try to solve using a numerical method

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

  42. Results B large B small

  43. 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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