# Population Ecology: Growth & Regulation Photo of introduced Population Ecology: Growth & Regulation Photo of introduced (exotic) rabbits at plague proportions in Australia from Wikimedia Commons. Life Cycle Diagram 4 2 0.8 0.3

0 1 0 2 3 age survival fecundity

seed seedling 1 to 2 yr old adult 2 to 3 yr old adult Life Table (a.k.a. Actuarial Table) Demographic rates often vary with age, size or stage

Cain, Bowman & Hacker (2014), Table 10.3 Life Table (a.k.a. Actuarial Table) Cohort Life Table Fates of individuals in a cohort are followed from birth to death Static Life Table Survival & reproduction of individuals of known age are assessed for a given time period Life Table (a.k.a. Actuarial Table) Sx = Age-specific survival rate; prob. surviving from age x to x+1 lx = Survivorship; proportion surviving from birth (age 0) to age x

Fx = Age-specific fecundity; average number of offspring produced by a female at age x Cain, Bowman & Hacker (2014), Table 10.3 Life Table (a.k.a. Actuarial Table) Population growth from t0 (beginning population size) to t1 (one year later) F1 = 2, so 6 x 2 = 12 F2 = 4, so 24 x 4 = 96 Cain, Bowman & Hacker (2014), Table 10.4

108 offspring Life Table (a.k.a. Actuarial Table) Population growth from t0 (beginning population size) to t1 (one year later) Nt+1 Population growth rate = = Cain, Bowman & Hacker (2014), Table 10.4 Nt =

138 100 = 1.38 Life Table (a.k.a. Actuarial Table) If age-specific survival & fecundity remain constant, the population settles into a stable age distribution and population growth rate 1 = 1.38 11 = 1.32 12 = 1.32

13 = 1.32 etc. = 1.32 Cain, Bowman & Hacker (2014), Fig. 10.8 B Leslie Matrix Age-structured matrix model (L) of population growth parameters Age structure at t+1 Age-specific survival & fecundity

Age structure at t Dominant Eigenvalue of L = Dominant Eigenvector of L = stable age distribution Example of a Leslie matrix from Wikimedia Commons Lefkovitch Matrix Stage-structured matrix model (L) of population growth parameters Stage structure at t+1 Stage-specific survival & fecundity

Stage structure at t Dominant Eigenvalue of L = Dominant Eigenvector of L = stable stage distribution Example of a Lefkovitch matrix adapted from Leslie matrix from Wikimedia Commons Population Age Structure Useful for predicting population growth Age structure for China in 2014 from Wikimedia Commons; China implemented a one-child policy in 1960s Survivorship Curves

Which is most likely to characterize an r-selected species? K-selected species? Cain, Bowman & Hacker (2014), Fig. 10.5 Exponential Growth Geometric growth when reprod. occurs at regular time intervals Population grows by a constant proportion in each time step

Nt+1 = Nt Nt = tN0 = Geometric population growth rate or Per capita finite rate of increase Cain, Bowman & Hacker (2014), Fig. 10.10

Exponential Growth Exponential growth when reproduction occurs continuously Reproducing is not synchronous in discrete time periods dN = rN dt N(t) = N(0)ert r= Exponential growth rate

or Per capita intrinsic rate of increase Cain, Bowman & Hacker (2014), Fig. 10.10 Exponential Growth Geometric Nt = tN0 Exponential N(t) = N(0)ert = ertN(0)

= er r = ln() Exponential decline / decay Cain, Bowman & Hacker (2014), Fig. 10.11 Constant population size Exponential

growth The Fundamental Law of Population Ecology Peter Turchin A population will grow exponentially as long as the environment experienced by all individuals in the population remains constant. In other words, as long as the amount of resources necessary for survival & reproduction continues expanding indefinitely as the population expands. Original idea from Turchin (2001) Oikos Laws of Thermodynamics

1st Law of Thermodynamics Law of Conservation of Energy Related to Law of Conservation of Mass E=mc2 Sun Earth Bio-geochemical processes Image of Carnot engine from Wikimedia Commons

Limited Scope for Population Increase No population can increase in size forever. Number of particles in the universe Quote from Cain, Bowman & Hacker (2014), pg. 227 < 10 < 100 Limits to Exponential Growth Density independent

Density-independent factors can limit population size Density dependent Cain, Bowman & Hacker (2014), Fig. 10.14 Limits to Exponential Growth Density independent Density-independent

factors can limit population size Density dependent Density-dependent factors can regulate population size Cain, Bowman & Hacker (2014), Fig. 10.14 Logistic Growth

r = Intrinsic Rate of Increase K = Carrying Capacity Cain, Bowman & Hacker (2014), Fig. 10.18 r- vs. K-selection r = Intrinsic Rate of Increase K = Carrying Capacity

Cain, Bowman & Hacker (2014), Fig. 10.18