# Role of Models in Population Biology Understanding Predicting Implications for empirical work 0 50 100 150 01020304050 Days Density Rotifers (after Halbach.

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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

Pheasants

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

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.

Time Series

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

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

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

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

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

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. )

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

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

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

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

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))

Discrete vs continuous growth

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

Model Assumptions

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

Logistic Growth Curve

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

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

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

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

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:

Cobwebbing N0N0 N1N1 N1N1 N2N2 N2N2 N3N3

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

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

r=1.8

r=2.2

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

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

Model assumptions

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)

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)

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