# Population-Projection Models Population-Projection Models 100000000 Stage 1 Stage 2 Stage 3 Number of female frogs 10000000 1000000 100000 10000 1000 100

10 1 0 5 10 15 Years 20 25

Individuals in a pop. are not created equal. Age classes <1.5 (calves) 1.5 2.5 3.5 4.5 Stage classes Fawns Button bucks Spikes Branch-antlered

Common frog (Rana temporaria) Anatomy of a population-projection matrix (M) PJUVJUV AD PJUV JUV AD The reproductive contribution of each stage to the next time step. Stage-specific survival rates.

A female-based matrix model for the common frog with 3 stage classes: pre-juvenile (PJUV), juvenile (JUV), and adult (AD). PJUVJUV AD PJUV JUV AD Mean number of eggs a female frog produces annually = 650. 0.08 x 650 = 52 0.43 x 650 = 279.5 PJUVJUV AD

PJUV JUV AD Annual survival of JUV females = 0.25 + 0.08 = 0.33. Population-size vector (n) n(t) = Multiplying M by n(t) x M

= x n(t) = n(t+1) = Resultant vector Projecting a matrix through time M n(t) n(t+1) =

= N2013= 3848 x = = N2014= 2093 x =

x = N2015= 5812 Projecting the matrix for 25 years Number of female frogs 25000000 Stage 1 Stage 2

Stage 3 20000000 15000000 10000000 5000000 0 0 5

10 15 Years 20 25 Change the y-axis to a logarithmic scale 100000000 Stage 1 Stage 2

Stage 3 Number of female frogs 10000000 1000000 100000 10000 1000 100 10 1 0 5

10 15 Years 20 25 Now you do it Consider a population with 4 age cohorts (0-3) and age-specific survival rates of 0.50, 0.65, 0.85, and 0.40, respectively. Assume that the age 0 cohort consists of immature (non-breeding) individuals

and that reproductive rates for the other 3 cohorts are age specific: 0.1, 20, and 150. Using a population-projection matrix, project and graph each cohort in the population for 10 years starting with an initial age-specific population size of 1000, 500, 200, and 85 for age cohorts 0-3, respectively. 0 1 2 3

0 0 0.1 20 150 1 0.50 0 2 3 0

0 0 0.65 0 0 0 0 0.85 0.40 1000

X 500 200 = ? 85 Project and graph each cohort in the population for 10 years.