Discrete time mathematical models in ecology Thank the person introducing you. 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 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

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

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)

Single species models

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

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”

Geometric growth

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!

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)

Population growth for the Hassell equation

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

Cobweb diagrams “Steady State” “Stability”

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

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

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

Age structured models

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

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

Multi-species models

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

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

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

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

Host-Pathogen models An example of a host-pathogen model is the Nicholson and Bailey model (extended) Nicholson Bailey model 1935-7, 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

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

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

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

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

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?

Optimal control theory

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

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

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

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.

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.

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

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

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

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

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

Results See the different scales B large B small

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