Introduction to Populations #8: Exponential Growth, Geometric Growth

Population: Group of SAME species individuals that live in SAME area at SAME time and interact with each other Note: - Difficult/impossible to portray population w/ absolute precision b/c blurry (don’t know where every individual goes and interacts with) Example: - All 40 sockeye salmon populations feed in SAME Pacific Ocean at SAME time - But spawn in DIFFERENT locations = considered separate populations - Breeding location can determine population

Exponential Growth: Continous reproduction - Becomes straight line if plotted on log scale - Populations can grow exponentially when conditions favourable - Density independent - Does NOT apply when out of resources Example: - Humans Anti-Example: - Daphnia observed population growth rate (r_obs) decreases as density increases

Geometric Growth: Discrete-time periods - Becomes straight line if plotted on log scale - Populations grow geometrically when reproduction occurs at regular time intervals - Density independent Example: - Annual plants

Geometric Growth Equations: - N_t+1= (𝞴)(N_t) - N_t = (𝞴^t)(N_0) Variables: - N_t = population size after t generations/periods (ex. years) - N_t+1 = population size 1 year/period after t generations/periods - 𝞴 = geometric population growth rate - N_0 = initial population size

Population Growth Rate (𝞴): - Ratio of population size in year t+1 (N_t+1) to population size in year t (N_t) - 𝞴 = (N_t+1)/(N_t) Note: 𝞴 does not have to same at every time period Population Increasing: - 𝞴 > 1 Population Decreasing: - 𝞴 < 1 Population at EQ: - 𝞴 = 1

Exponential Growth Equations: - dN/dt = rN - N(t) = N(0)e^(rt) Variables: - N(t) = population size each instant time t - N(0) = initial population size - dN/dt = rate of change in population size each instant time t - r = exponential growth rate Note: - r = 0 when population stable - When population starts very small (ex. after run of bad years for Australian mice/after new species introduction), initial population growth phase may be very close to exponential (B > D) - Evolutionary/technological innovations can also lead to exponential population growth (B > D) (ex. Evolution of antibiotic resistance in microbes)

Populations Size Change: - N_t+1 = N_t + (B - D) + (I - E) Variables: - B = Births - D = Deaths - I = Immigration (enter another habitat to live in permanently) - E = Emigration (leave habitat to live in another) Example: - Greater precipitate = greater crop yield = more rate births = increased rat population size