Population Dynamics. Max Flöttmann and Jannis Uhlendorf. June 12, Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

Size: px

Start display at page:

Download "Population Dynamics. Max Flöttmann and Jannis Uhlendorf. June 12, Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54"

April Dorsey
5 years ago
Views:

1 Population Dynamics Max Flöttmann and Jannis Uhlendorf June 12, 2007 Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

2 1 Discrete Population Models Introduction Example: Fibonacci Sequence Analysing Difference Equations 2 Continuous Population Models Predator-prey Models: Lotka-Volterra Systems Example: Lynx - Snowhoe Hare 3 Catastrophe Model for Fishing The Problem The Model Steady States Phaseplane Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

3 Discrete Population Models Differential equation models require overlap of generations Often no overlap between the generations (e.g. salomon, snowdrops, Octopus Vulgaris ) Difference equation: N t+1 = (Nt ) = N t F (N t ) Discrete time steps Can in general be solved analytically Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

4 Small Example e.g. N t+1 = rn t, r > 0 N t = r t N 0 Two populations Life span: 1 year Bees 100 Bees produce 90 new ones each year r = 0.9 Wasps 100 Wasps produce 101 new ones each year r = 1.01 Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

Fibonacci Sequence Rabbit population Rabbits take one month to

R n+1 = R n + R n 1 Golden ratio: N t N t+1 5 1 2 0.

5 Figure: Rabbit pedigree taken from http://www.math.temple.

5 Fibonacci Sequence Rabbit population Rabbits take one month to mature Each productive pairs bears a new pair Rabbits never die R n+1 = R n + R n 1 Golden ratio: N t N t Golden angle: = φ = Figure: Rabbit pedigree taken from reich/fib/fibo.html Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

6 Fibonacci Sequence: Sunflower head Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

7 Fibonacci Sequence: Sunflower head Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

8 Fibonacci Sequence: Sunflower head Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

9 Further examples Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

10 Analysing difference equations Remember: N t+1 = (Nt ) Steady state if N t+1 = N t Example: (Nt ) = N t exp[r ( ) 1 Nt K ] (Ricker function) K max. capacity r intrinsic growth rate Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

11 Graphical Solution: Cobwebbing Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

12 Steady State Steady state: Intersections of the the curve N t+1 = (Nt ) and the line N t+1 = N t Stable steady state: small pertubation system will fall back to steady state Derivative of (Nt ) in the steady state determines stability Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

13 Examine steady state in the example Linearize around steady state Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

14 Linearization Around Steady State Derivation: df dn t Nt=N Linearization (1. taylor expasion) Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

15 Stability: Graphical Solution This is a (linear) stable steady state Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

16 Stability: Graphical Solution slope > 1 slope < -1-1 < slope < 1 Observation: Steady states with abs. derivative < 1 are stable Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

17 Stability and Summary Stability: Small perturbations vanish Discrete case x t+1 = Ax t : Solution: xt = A t x 0 p(λ)(max. abs. eigenvalue) 1 Stability Continuous case dx dt = Ax: Solution: x(t) = x 0 exp(ta) v(λ)(max. real part of eigenvalues) 0 Stability Populations that reproduce in certain intervals difference equations Steady state: intersection of N t+1 = (Nt ) and N t+1 = N t Fibonacci sequence in plants Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

18 Continuous Population Models Most populations have overlap between generations (e.g Humans) Can to some extend be modeled by ODEs ds dt = (S, t) Good for large populations Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

19 Predator-Prey: Lotka-Volterra Systems Simple predator prey model dn dt dp dt = N(a bp) = P(cN d) a growth rate prey b neg. effect of predator on pey c benefit of prey for pedator d decay rate of predator prey in absence of predator grows unbounded predator reduces preys growth rate without prey the predator decays exponentially Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

20 Predator-Prey: Lotka-Volterra Systems Timecourse Phasediagram Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

21 Example: Lynx - Snowshoe Hare Lynx hunt snow hares Long term data available ( ) Data from fur catch records Assumption: Fixed proportion of the population was caught Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

22 Timecourse Lynx Hare Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

23 Phase Diagram Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

24 Time Course Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

25 Phase Diagram Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

26 Does the hare eat the lynx? Proposal: The hare carries a disease No such disease known Hunting is the disease Data does not represent a fixed proportion Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

27 Summary Non overlaping generations discrete models Stability: max. abs. eigenvalue 1 Difference equations occur in nature Overlap and continuous behaviour ODE models Stability: max. real part of eigenvalue 0 Modeling populations can involve traps Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

28 1 Discrete Population Models Introduction Example: Fibonacci Sequence Analysing Difference Equations 2 Continuous Population Models Predator-prey Models: Lotka-Volterra Systems Example: Lynx - Snowhoe Hare 3 Catastrophe Model for Fishing The Problem The Model Steady States Phaseplane Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

29 Peruvian Anchovies Figure: Fishing is a large economic sector in Peru Max Flo ttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

30 Peruvian Anchovies Figure: Sudden breakdown after many years of fishing. Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

31 Why? Question: Why did this breakdown happen so sudden? Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

32 Why? Approach: Mathematical modeling! Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

33 How to Model the Problem What has to be taken into accont? Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

34 How to Model the Problem What has to be taken into accont? Biology population size growthrate of anchovies amount of harvested fish Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

35 How to Model the Problem What has to be taken into accont? Biology population size growthrate of anchovies amount of harvested fish Economy number of trawlers price for fish costs for fishing Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

36 Biology How can the growthrate of anchovies be modelled? Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

37 Biology How can the growthrate of anchovies be modelled? We could use a simple logistic equation like this: x = rx(1 x K ) (1) Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

38 Biology But: Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

39 Biology But: Anchovies live in huge swarms Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

40 Biology But: Anchovies live in huge swarms The bigger the swarm the better the chances for survival and to find a partner Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

41 Biology But: Anchovies live in huge swarms The bigger the swarm the better the chances for survival and to find a partner So this function fits better: x = rx 2 (1 x K ) (2) , , ,5 5 7, ,5 Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

42 Biology We add a small constant for fishes that can t be caught: x = a + rx 2 (1 x K ) (3) Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

43 Biology We add a small constant for fishes that can t be caught: x = a + rx 2 (1 x K ) (3) Finally we subtract the amount of fish that is harvested: x = a + rx 2 (1 x ) vex (4) K Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

44 Biology We add a small constant for fishes that can t be caught: x = a + rx 2 (1 x K ) (3) Finally we subtract the amount of fish that is harvested: x = a + rx 2 (1 x ) vex (4) K But how is E defined? Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

45 Economy We build a strongly simplified model of Economy: Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

46 Economy We build a strongly simplified model of Economy: an abstract value for fishing effort E Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

47 Economy We build a strongly simplified model of Economy: an abstract value for fishing effort E fish has a price p Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

48 Economy We build a strongly simplified model of Economy: an abstract value for fishing effort E fish has a price p fishing has certain costs c Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

49 Economy We build a strongly simplified model of Economy: an abstract value for fishing effort E fish has a price p fishing has certain costs c v is the fishing yield per inset unit E Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

50 Economy We build a strongly simplified model of Economy: an abstract value for fishing effort E fish has a price p fishing has certain costs c v is the fishing yield per inset unit E This yields the equation: E = αe(pvx c) (5) Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

51 The final Model x = a + rx 2 (1 x ) vex K (6) E = αe(pvx c) (7) a: small constant for remaining fishes r : linear growth factor K: unharvested equilibrium density v: gain per investement p: price for fishes c: cost per inset α: small factor to represent that E is a slow changing variable Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

52 Time Series Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

53 What is this Model good for? 1 We can analyse the behaviour of the model. 2 We can change parameters and get different results. 3 We can predict future developments and try to react before catastrophe happens. Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

54 Steady States 6,4 growthrate 5,6 4,8 4 3,2 2,4 1,6 0,8 population 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 Figure: Stable steadty state at high population Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

55 Steady States 6,4 growthrate 5,6 4,8 4 3,2 2,4 1,6 0,8 population 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 Figure: bifurcation point Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

56 Steady States 6,4 growthrate 5,6 4,8 4 3,2 2,4 1,6 0,8 population 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 Figure: three steady states Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

57 Steady States 6,4 growthrate 5,6 4,8 4 3,2 2,4 1,6 0,8 population 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 Figure: stable steady state at low level Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

58 Phaseplane Figure: The Phaseplane with coloring by gradient Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

59 Hysteresis Figure: hysteresis - a memory of the system Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

60 Low Harvest Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

61 High Harvest Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

62 After Catastrophe Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

63 Regeneration Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

64 There are even worse cases! Figure: new fishes have to be added to the environment Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

65 Maximum Sustainable Yield maximum value for fishing before the breakdown occurs can be predicted, but should be used carefully many different prediction methods with different outcomes parameters are hard to determine at MSY minimal disturbances can lead to breakdown of population Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

66 Separatrix Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

67 El Niño Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

68 Outlook Max Flöttmann and Jannis Uhlendorf () Population Dynamics June 12, / 54

Figure 5: Bifurcation diagram for equation 4 as a function of K. n(t) << 1 then substituting into f(n) we get (using Taylor s theorem)

Figure 5: Bifurcation diagram for equation 4 as a function of K n(t)