1. Role of Models in Population Biology Understanding Predicting Implications for empirical work 0 50 100 150 01020304050 Days Density Rotifers (after Halbach 1979) 0 50 100 150 01020304050 Days Density 15 o C 25 o C

2. Pheasants

3. Data collection Regional and statewide abundance estimates Predicting quality of coming pheasant season 2003: 1.36 pheasants per 100 miles 2004: 1.75 pheasants per 100 miles 29% increase

4. Commission's upland game program manager, Scott Taylor Nebraska's pheasant population is at its highest level since 1995, according to rural mail carrier surveys conducted in the spring and summer by the Nebraska Game and Parks Commission. The surveys indicate the population is improved over last year and substantially improved over the previous five-year average.

5. Time Series

6. Exponential growth (1)Linear change in K model (2) Linear change in K with switch (3)

7. Akaikes Information Criterion AIC = -2L + 2k L = log likelihood ~ probability the model is correct, given the data k = number of parameters in the model Low AIC is better

8. Exponential growth (1)Linear change in K model (2) Linear change in K with switch (3) Akaikes Information Criterion kAIC 12273 24314 36296

9. A simple (discrete) population model N t+1 = N t + B - D + I – E pop next time period pop. this time period births deaths immigrants emigrants Change in abundance N t+1 - N t = B - D + I - E N = B - D + I - E

10. Basic closed population model Simplifying assumption: no movement between populations Abundance N t+1 = N t + B – D Change in abundance N = B – D if B > D then population increase if B< D then population decrease if B = D then stable population

11. Basic discrete model Let number of births and deaths depend on population size: i.e. B = bN D = dN Hence N t+1 = N t + (b – d) N t Let r = b – d N t+1 = N t + r N t = (1 + r) N t = R N t R is the finite rate of increase (a.k.a. )

12. Varying 0 500 1000 1500 2000 2500 3000 3500 012345678910 time (t) population size N(t) 1.0 1.2 0.8

13. Continuous growth Recall that N=(b-d)N t As t => 0

14. Population growth vs abundance Where N 0 is the initial population size N t is the population size at time t r is the per capita growth rate

15. Different Names for r Instantaneous rate of increase Intrinsic rate of increase Malthusian parameter Per capita rate of population increase over short time interval r = b – d

16. Geometric Population Growth time [t] Population size N(t) 0.02 0.00 -0.002 r-values: time [t] Logarithm of population size Ln(N(t))

17. Discrete vs continuous growth

18. Discrete growth across a time interval; birth and death discrete realistic; hard to generalize results Continuous instantaneous growth; birth and death continuous process abstraction; easier to generalize results Ln( ) = r = e r r > 0 > 1 r = 0 = 1 r < 0 0 < < 1

19. Model Assumptions

20. Closed population Constant b and d Population grows exponentially indefinitely No genetic structure No age or size structure Species exist as single panmictic population unstructured (random-mating) populations

21. Pheasants of Protection Island (Data from Lack 1967) 1937 8 birds 1938 30 birds R = 30/8 = 3.75 r = ln(3.75) = 1.3217

22. Density dependence Population growth is affected by limited resources food breeding sites territories Density dependent factors influence birth and death rates b and d are not constant Logistic population growth b b Allee effect

23. Effects of crowding: decreased reproduction Effect of density on fecundity in the Great Tit Parus major (Lack 1966)

24. Density dependence in song sparrows ( after Arcese and Smith 1988, Smith et al. 1991 ) Surviving young per female Proportion of floaters among males Proportion of surviving juveniles

25. Density Dependence in Death Rate Increased crowding increased death rate d = d 0 + c N d 0 : death rate under uncrowded conditions c : measures strength of density dependence N : current population size

26. Density Dependence in Birth Rate Increased crowding decreased birth rate b = b 0 – a N b 0 : birth rate under uncrowded conditions a : measures strength of density dependence N : current population size

27. Population size (N) Rate b0b0 d0d0 Death rate (d) Birth rate (b)

28. Interpretation of Carrying Capacity K Defined as the level of abundance above which the population tends to decline It is population specific: depends on available resources and space (quality and quantity), abundance of predators and competitors It is an abstraction: crude summary of interactions of a population with environment Units are in number of individuals supported by environment

29. Stable Equilibrium N = K b = d N > K population declines towards K N < K population increases towards K Population size (N) Rate b0b0 d0d0 K Death rate (d) Birth rate (b)

30. Carrying capacity, K Unused proportion of carrying capacity 14% of area is white or unused

31. K=1001 – N/KN t+1 N = 71 - 7/100 = 0.93 rN (0.93) population growths fast unused proportion of carrying capacity

32. K=1001 – N/KN t+1 N = 71 - 7/100 = 0.93 rN (0.93) population growths fast N = 981 - 98/100 = 0.02 rN (0.02) population growths slowly unused proportion of carrying capacity

33. K=1001 – N t /KN t+1 N = 71 - 7/100 = 0.93 rN (0.93) population growths fast N = 981 - 98/100 = 0.02 rN (0.02) population growths slowly N = 105 1 - 105/100 = -0.05 rN (-0.05) population declines unused proportion of carrying capacity

34. Logistic Growth Curve

35. Population growth rate Logistic Growth K N = K/2 Exponential Growth Per capita growth rate K rmrm

36. Time Lag (t) Sometimes individuals do not immediately adjust their growth and reproduction when resources change, and these delays can affect population dynamics Causes Seasonal availability of resources Growth responses of prey populations Age and size structure of consumer populations

37. Time Lag Delay differential equation Important parameters: Length of time lag, Response time = 1/r r controls population growths 0 < r < 0.368 population increases smoothly to K 0.368 < r < 1.570 damped oscillations r > 1.570 stable limit cycle

38. medium r damped oscillations small r Period ( 4 Amplitude (increases with r ) large r stable limit cycle

39. Discrete time density- dependent models Discrete population growth model has built- in time lag of length 1.0 complex behavior Different ways to incorporate density dependence 1. 2. 3. Ricker model:

40. Cobwebbing N0N0 N1N1 N1N1 N2N2 N2N2 N3N3

41. Discrete time density- dependent models In (1) and (3) dynamics depend solely on r r < 2.0 population approaches K 2.0 < r < 2.449 2-point limit cycle 2.449 < r < 2.57 4-point limit cycle r > 2.57 chaos

42. 0 20 40 60 80 100 120 140 05101520 Population size (N) r = 1.8 Population approaches K 0 20 40 60 80 100 120 140 05101520 r = 2.4 Population size (N) 2-point limit cycle 0 20 40 60 80 100 120 140 05101520 Time (t) r = 2.5 Population size (N) 4-point limit cycle

43. r=1.8

44. r=2.2

45. Chaotic behavior r = 2.8 0 20 40 60 80 100 120 140 05101520 Time (t) Population size (N) N 0 = 50 N 0 = 51

46. Predicting the future NtNt N t+1 Ricker model with K = 100 and r = 3.5 NtNt N t+10

47. Model assumptions

48. Increased density causes a (linear) decline in population growth rate No variability in carrying capacity All other assumptions same as for exponential growth (assume single, panmictic population, no age/stage structure)

49. Discrete Logistic Substitute b and d in N t+1 = RN t = (1+b - d) N t N t+1 = N t +[b 0 -aN t - d 0 -cN t ]N t Rearrange and replace K = b 0 -d 0 / (a+c)