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

Transcription

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

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 2

3 Why use discrete time models? 3

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) 4

5 Single species models 5

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 Nt+1 = (1 + b d)nt = λnt 6

7 Suppose at the initial time t = 0, N0 = 1 and λ = 2, then N1 N2 N3 N4 N5 = λn0 = λn1 = λn2 = λn3 = λn4 =2 1=2 =2 2=4 =2 4=8 = 2 8 = 16 = 2 16 = = λ N0... = λ3 N0... = λ4 N = λ N0 We can solve the difference equation to give the population level at time t, Nt in terms of the initial population level, N0 Nt = λt N0 Malthus population, when unchecked, increases in a geometric ratio 7

8 Geometric growth Nt = λt N0 8

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! 9

10 Hassell equation The Hassell equation takes into account intraspecific competition Nt+1 R0 N t = (1 + knt )b Under-compensation (0<b<1) Exact compensation (b=1) Over-compensation (1<b) 10

11 Population growth for the Hassell equation Nt+1 R0 N t = (1 + knt )b 11

12 Special case: Beverton-Holt model Beverton-Holt stock recruitment model (1957) is a special case of the Hassell equation (b=1) Nt+1 rnt = 1 + ant Used, originally, in fishery modeling 12

13 Cobweb diagrams Steady State Stability Nt+1 rnt = 1 + ant 13

14 Cobweb diagrams Sterile insect release Adding an Allee effect Extinction is now a stable steady state Nt+1 14 Nt rnt = 1 + ant S + Nt

15 Ricker growth Another model arising from the fisheries literature is the Ricker stock recruitment model (1954, 1958) Nt /K Nt+1 = ant e This is an over-compensatory model which can lead to complicated behavior 15

16 80 Nt /K 70 Nt+1 = ant e Nt a 12 richer behavior Period doubling to chaos in the Ricker growth model 16

17 N t+1 = an t e N t/k 17

18 Age structured models 18

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 19 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 20

21 Multi-species models 21

22 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

23 multi species models Growth Growth Nn Pn die die Nn+1 Pn+1 = F (Nn, Pn ) = G(Nn, Pn ) 23

24 Competition model Discrete time version of the Lokta-Volterra competition model is the Leslie-Gower model (1958) Nt+1 Mt+1 1 Nt = b1 1 + c11 Nt + c12 Mt 1 Mt = b2 1 + c21 Nt + c22 Mt Used to model flour beetle species 24

25 Predator-Prey models Analogous discrete time predator-prey model (with mass action term) Pt+1 Qt+1 rpt spt Qt = 1 + apt = σqt + vpt Qt Displays similar cycles to the continuous version 25

26 Host-Pathogen models An example of a host-pathogen model is the Nicholson and Bailey model (extended) Nn+1 Pn+1 = ann e e bpn = λ(1 e )Nn bnn αpn Many forest insects often display cyclic populations similar to the cycles displayed by these equations 26

27 SIR models Susceptibles Infectives Removed Often used to model with-in season Extended to include other categories such as Latent or Immune St+1 = St βsit It+1 Rt+1 = It + βst It γit = Rt + γit 27

28 Control in discrete time models 28

29 Control methods Controls that add/remove a portion of the population Cutting, harvesting, perscribed burns, insectides etc rnt = 1 + ant Nt+1 rnt (1 αt ) = 1 + ant Nt+1 = rnt e bpt Nt+1 = rnt e bpt Pt+1 = λ(1 e bpt )Nt Pt+1 = λ(1 e bpt )Nt + Qt Nt+1 29

30 Adding control to our models Controls that change the population system Introducing a new species for control, sterile insect release etc Nt+1 Nt+1 Nt+1 rnt = 1 + ant Pt+1 rnt = 1 + ant Nt+1 30 rnt e bpt = 1 + ant = λ(1 e bpt )Nt Nt rnt = 1 + ant S + Nt

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? 31

32 Optimal control theory 32

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 33

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 34

35 Single species control State equations Nt+1 rnt (1 αt ) = 1 + ant Control set 0 αt 1 Objective functional J(α) = " T 1! e t=0 δt rnt Bαt2 αt 1 + ant 35 #

36 how do we find the Pontryagins best control discrete maximum princple strategy? 36

37 Method to find the optimal control We first form the following expression "!T 1 rnt δt 2 e αt Bαt H= t=0 1 + ant # $% rnt +λt Nt+1 (1 αt ) 1 + ant By differentiating this expression, it will provide us with a set of necessary conditions 37

38 adjoint equations r r H δt + λt 1 λt (1 αt ) = e αt 2 2 Nt (1 + ant ) (1 + ant ) Set H Nt =0 Then re-arranging the equation above gives the adjoint equation λt 1 r r δt (1 αt ) e αt = λt 2 (1 + ant ) (1 + ant )2 38

39 Controls rnt rnt H δt =e 2Bα + λt αt 1 + ant 1 + ant Set H =0 αt Then re-arranging the equation above gives the adjoint equation (e δt + λt ) rnt αt = 2B 1 + ant 39

40 Optimality system Forward in time Nt+1 rnt (1 αt ) = 1 + ant r r δt Backward λt 1 = λt (1 αt ) e αt 2 2 (1 + an ) (1 + an ) t t in time Control equation (e δt + λt ) rnt αt = 2B 1 + ant 40

41 One step away! Found conditions that the optimal control must satisfy For the last step, we try to solve using a numerical method 41

42 numerical method Starting guess for control values State equations forward Update controls Adjoint equations backward 42

43 Results B large B small 43

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 44