Population Growth 1 Reproduction and survival in population ecology 2 Exponential growth a Exponential model Nx 1 Nx b d i e o Let i e 0 Nx 1 Nx b d Nx 1 Nx b d N t b d dN dt b d dN dt b d N o Let b and d e expressed per capita o let b d r r is the intrinsic rate of natural increase dN dt rN o J shaped curve b Integral form of model Nt N0 ert c Calculating r r can be calculated from life table and fecundity schedule data fecundity schedule o Fx total number of offspring produced during stage or o mx number of offspring produced per individual during interval x stage or age interval x mx Fx Nx o R0 net reproductive rate per capita offspring per individual of original cohort R0 Fx N0 lx mx R 1 population stays same R 1 population size increases R 1 population size decreases Obtaining r o Nt N0 ert r ln R0 T T mean generation time average length of time between the birth of a mother and the birth of her offspring T x lx mx R0 3 Assessment of model a Examples from real populations handout Most lab and natural examples s shaped curves Pheasants nearly exponential Humans exponential b Is the exponential growth model wrong or just incomplete Population crashes o E g willows with rabbits o E g wildebeests and rinderpest virus Post crash population growth o Initially nearly exponential o But growth slows yielding an s shaped curve model incomplete potential to grow exponentially but not for long c the meaning of r the intrinsic rate of natural increase r individuals per individual per unit time units time 1 d possible modifications of the model 4 Logistic models to be covered in Competition 5 Stochastic growth models Deterministic models predict an exact numerical outcome Always yields the same outcome given the same starting Stochastic models permit chance variation in r a Random variation in r parameters Nt N0 ert Let R0 4 Heads 0 Tails 8 Reproductive bout Female A Female B Female C 1 0 0 0 2 0 8 8 3 8 8 8 4 0 8 0 b probability of extinction under stochastic growth is a function of population size equation probability extinction d0 b0 N when r is positive non zero chance of becoming extinct by o probability extinction 0 o as N decreases probability extinction increases when r is zero certain to go extinct by chance alone given chance alone enough time o probability extinction 1 o expected time to extinction decreases as N decreases note how important population size is for understanding probabilities of extinction by chance alone 6 Population projection for loggerhead sea turtle see handout