Discrete time mathematical models in ecology Andrew Whittle University of Tennessee Department of Mathematics 1 Outline • Introduction - Why use discrete-time models? • Single species models ➡ • Age structure models ➡ • Leslie matrices Non-linear multi species models ➡ • Geometric model, Hassell equation, Beverton-Holt, Ricker Competition, Predator-Prey, Host-Parasitiod, SIR Control and optimal control of discrete models ➡ Application for single species harvesting problem 2 Why use discrete time models? 3 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) 4 Single species models 5 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 6 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” 7 Geometric growth 8 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! 9 Hassell equation The Hassell equation takes into account intraspecific competition • Under-compensation (0<b<1) • Exact compensation (b=1) • Over-compensation (1<b) 10 Population growth for the Hassell equation 11 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 12 Cobweb diagrams “Steady State” “Stability” 13 Cobweb diagrams • Sterile insect release • Adding an Allee effect • Extinction is now a stable steady state 14 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 15 Nt a Period doubling to chaos in the Ricker growth model richer behavior 16 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 19 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 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 Susceptibles Infectives Removed • Often used to model with-in season • Extended to include other categories such as Latent or Immune 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 Adding control to our models • Controls that change the population system Introducing a new species for control, sterile insect release etc 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 how do we find the discrete maximum best control strategy? princple 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 44