1. Discrete time mathematical models in ecology
Thank the person introducing you.
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
Give my interpretation of when to use discrete time models
Describe some very simple models

3. Why use discrete time models?
Or if we rephrase the question ...

4. 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
wait for lambda to appear!!!!
Then

7. Malthus “population, when unchecked, increases in a geometric ratio”
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 Under-compensation (0<b<1)
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
Explain each one.
Explain the figure.
Slow wait for next slide - bifurcation diagram

13. Cobweb diagrams
“Steady State”
“Stability”

14. Cobweb diagrams Sterile insect release Adding an Allee effect
Extinction is now a stable steady state
wait for arrows!!!!!

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
have to press enter 3 times
Slow wait for next slide - bifurcation diagram

16. Nt
K=10 a
richer behavior
Period doubling to chaos in the Ricker growth model

18. Age structured models

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
N1t
N2t+1
N3t+2
N4t+3
N5t+4

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

21. Multi-species models

22. Single species models can be extended to multi-species
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.
4 typical general models for discrete time multi species models

23. multi species models Growth Growth Nn Pn die die
We can extend the single species model to multiple interacting species.
Competition models
Predator Prey models - Host Pathogen model
die
die

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

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

26. Host-Pathogen models
An example of a host-pathogen model is the Nicholson and Bailey model (extended)
Nicholson Bailey model , extended by May 1970’s
Poisson distribution - describes the prob of random occurrences
Many forest insects often display cyclic populations similar to the cycles displayed by these equations

27. SIR models Often used to model with-in season
Susceptibles
Infectives
Removed
Often used to model with-in season
Extended to include other categories such as Latent or Immune
Models within season - more on this in later talk

28. Control in discrete time models
So we have looked at lots of different classical discrete time models which can be adapted for particular species or systems of species

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

30. Adding control to our models
Controls that change the population system
Introducing a new species for control, sterile insect release etc
sterile insect - adding control as function of population variable + control variable

31. 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?

32. Optimal control theory

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

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
Best to describe this using an example

35. Single species control
State equations
Control set
Mention discounting factor
Objective functional

36. how do we find the best control strategy?
Pontryagins
discrete maximum princple
how do we find the best control strategy?
There is a trick we can use.

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
State equations are added in and multiplied by the adjoint variables
Differentiate w.r.t. to all variables.
Differentiating w.r.t. the adjoints gives the state equations back again.

38. adjoint equations
Set
Then re-arranging the equation above gives the adjoint equation
Differentiating w.r.t. the states gives the adjoint equations that go backward in time

39. Controls
Set
Diff. wrt to the control we obtain the control equation
Then re-arranging the equation above gives the adjoint equation

40. Optimality system Forward in time Backward in time Control equation
We now have created an optimality system
Starting conditions - initial population levels
Terminal conditions - adjoints
Control
equation

41. One step away! Found conditions that the optimal control must satisfy
For the last step, we try to solve using a numerical method
Started not knowing

42. numerical method Starting guess for control values State equations
forward
Iterative method
Different ways to update the controls
Update
controls
Adjoint equations
backward

43. Results
See the different scales
B large
B small

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