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Population Growth1. Reproduction and survival in population ecology2. Exponential growtha. Exponential model- Nx+1 = Nx + b – d + i – eo Let i=e=0- Nx+1 = Nx + b – d Nx+1 - Nx = b – d ΔN/Δt = b – d dN/dt = b – d o Let b and d e expressed per capita- dN/dt = (b-d)No let (b-d)=r; r is the intrinsic rate of natural increase- dN/dt = rNo J-shaped curveb. Integral form of model- Nt = N0 . ertc. Calculating r- r can be calculated from life-table and fecundity schedule data- fecundity scheduleo Fx = total number of offspring produced during stage or interval xo mx = number of offspring produced per individual during stage or age interval x mx = Fx/Nxo R0 = net reproductive rate; per capita offspring per individual of original cohort R0 = Σ(Fx)/N0 = Σ(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 = Σ(x . lx . mx)/R03. Assessment of modela. Examples from real populations—handout- Most lab and natural examples s-shaped curves- Pheasants nearly exponential- Humans exponentialb. Is the exponential growth model wrong or just incomplete?- Population crasheso E.g. willows with rabbitso E.g. wildebeests and rinderpest (virus)- Post-crash population growtho Initially nearly exponentialo But growth slows, yielding an s-shaped curve- * model incomplete—potential to grow exponentially but not for longc. the meaning of r, the intrinsic rate of natural increase- r = individuals per individual per unit time—units: time-1d. possible modifications of the model4. 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 parameters- Nt = N0 . ert--Stochastic models permit chance variation in r a. Random variation in r - Let R0 = 4 - Heads: 0 Tails: 8- Reproductive bout1 2 3 4Female A 0 0 8 0Female B 0 8 8 8Female C 0 8 8 0b. 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 chance aloneo probability extinction > 0o as N decreases, probability extinction increases- when r is zero: certain to go extinct by chance alone, given enough timeo probability extinction=1o expected time to extinction decreases as N decreases- note how important population size is for understanding probabilities of extinction by chance alone6. Population projection for loggerhead sea turtle—see

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