Presentation on theme: "Discrete time mathematical models in ecology"— Presentation transcript:
1 Discrete time mathematical models in ecology Thank the person introducing you.Andrew WhittleUniversity of TennesseeDepartment of Mathematics
2 Outline Introduction - Why use discrete-time models? Single species modelsGeometric model, Hassell equation, Beverton-Holt, RickerAge structure modelsLeslie matricesNon-linear multi species modelsCompetition, Predator-Prey, Host-Parasitiod, SIRControl and optimal control of discrete modelsApplication for single species harvesting problemGive my interpretation of when to use discrete time modelsDescribe 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 yearPopulations censused at intervals (metered models)
6 Simple population model Consider a continuously breading populationLet Nt be the population level at census time tLet d be the probability that an individual dies between censusesLet b be the average number of births per individual between censuseswait 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, thenWe can solve the difference equation to give the population level at time t, Nt in terms of the initial population level, N0Malthus “population, when unchecked, increases in a geometric ratio”
9 Intraspecific competition No competition - Population grows unchecked i.e. geometric growthContest 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 competitionUnder-compensation (0<b<1)Exact compensation (b=1)Over-compensation (1<b)
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 modelingExplain each one.Explain the figure.Slow wait for next slide - bifurcation diagram
14 Cobweb diagrams Sterile insect release Adding an Allee effect Extinction is now a stable steady statewait for arrows!!!!!
15 Ricker growthAnother 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 behaviorhave to press enter 3 timesSlow wait for next slide - bifurcation diagram
16 NtK=10aricher behaviorPeriod doubling to chaos in the Ricker growth model
19 Age structured modelsA population may be divided up into separate discrete age classesAt each time step a certain proportion of the population may survive and enter the next age classIndividuals in the first age class originate by reproduction from individuals from other age classesIndividuals in the last age class may survive and remain in that age classN1tN2t+1N3t+2N4t+3N5t+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 decayOften, not always, populations tend to a stable age distribution
22 Single species models can be extended to multi-species Multi-species modelsSingle species models can be extended to multi-speciesCompetition: 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 modelsPredator Prey models - Host Pathogen modeldiedie
24 Competition modelDiscrete time version of the Lokta-Volterra competition model is the Leslie-Gower model (1958)Used to model flour beetle species
25 Predator-Prey modelsAnalogous discrete time predator-prey model (with mass action term)Displays similar cycles to the continuous version
26 Host-Pathogen modelsAn example of a host-pathogen model is the Nicholson and Bailey model (extended)Nicholson Bailey model , extended by May 1970’sPoisson distribution - describes the prob of random occurrencesMany forest insects often display cyclic populations similar to the cycles displayed by these equations
27 SIR models Often used to model with-in season SusceptiblesInfectivesRemovedOften used to model with-in seasonExtended to include other categories such as Latent or ImmuneModels 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 etcadding control as a variable
30 Adding control to our models Controls that change the population systemIntroducing a new species for control, sterile insect release etcsterile 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?
33 Optimal control We first add a control to the population model Restrict the control to the control setForm a objective function that we wish to either minimize or maximizeThe state equations (with control), control set and the objective function form what is called the bioeconomic model
34 ExampleWe consider a population of a crop which has economic importanceWe assume that the population of the crop grows with Beverton-Holt growth dynamicsThere is a cost associated to harvesting the cropWe wish to harvest the crop, maximizing profitBest to describe this using an example
35 Single species control State equationsControl setMention discounting factorObjective functional
36 how do we find the best control strategy? Pontryaginsdiscrete maximum princplehow 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 expressionBy differentiating this expression, it will provide us with a set of necessary conditionsState equations are added in and multiplied by the adjoint variablesDifferentiate w.r.t. to all variables.Differentiating w.r.t. the adjoints gives the state equations back again.
38 adjoint equationsSetThen re-arranging the equation above gives the adjoint equationDifferentiating w.r.t. the states gives the adjoint equations that go backward in time
39 ControlsSetDiff. wrt to the control we obtain the control equationThen 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 systemStarting conditions - initial population levelsTerminal conditions - adjointsControlequation
41 One step away! Found conditions that the optimal control must satisfy For the last step, we try to solve using a numerical methodStarted not knowing
42 numerical method Starting guess for control values State equations forwardIterative methodDifferent ways to update the controlsUpdatecontrolsAdjoint equationsbackward
44 Summary Introduced discrete time population models Single species models, age-structured modelsMulti species modelsAdding control to discrete time modelsForming an optimal control problem using a bioeconomic modelAnalyzed a model for crop harvesting