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Overview of population growth: New Concepts: discrete continuous density independent Geometric Exponential density dependent Discrete Logistic Logistic - Stability - DI (non-regulating) vs. DD (regulating) growth - equilibrium Variability in growth (1) Individual variation in births and deaths (2) Environmental (extrinsic variability) (3) Intrinsic variability How do populations grow – a derivation of geometric growth Growth rate (r) = birth rate – death rate (express as per individual) N1 = N0 + rN0 N0 = initial population density (time = 0) N1 = population density 1 year later (time =1) How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 + rN0 = N0 (1 + r) How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 + rN0 = N0 (1 + r) N2 = N1 + rN1 = N1 (1 + r) How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 + rN0 = N0 (1 + r) N2 = N1 + rN1 = N1 (1 + r) Can we rewrite N2 in terms of N0 ??? How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 + rN0 = N0 (1 + r) substitute N2 = N1 + rN1 = N1 (1 + r) How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 + rN0 = N0 (1 + r) substitute N2 = N1 + rN1 = N1 (1 + r) rewrite: N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2 How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 + rN0 = N0 (1 + r) substitute N2 = N1 + rN1 = N1 (1 + r) N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2 or } Nt = N0 (1 + r)t = , finite rate of increase Discrete (geometric) growth 5 Nt = N0 t = finite rate of increase N 4 3 1 2 time Continuous (exponential) growth 5 Nt = N0ert N 4 3 1 2 time r = intrinsic growth rate Continuous (exponential) growth 5 population growth rate N 4 3 1 per capita growth rate dN = rN; dt 1 dN = r N dt 2 time dN dt Read as change in N (density) over change in time. 1 dN = r N dt 1 dN N dt Y N = b + mX Per capita growth is constant and independent of N Comparison Discrete Continuous Nt = N0t Nt = N0ert Where: = er Increasing: > 1 Decreasing: < 1 r = ln r>0 r<0 Every time-step (e.g., generation) None Compounded instantaneously Applications: Populations w/ discrete breeding season No breeding season - at any time there are individuals in all stages of reproduction Examples: Most temperate vertebrates and plants Humans, bacteria, protozoa Often intractable; simulations Mathematically convenient Time lag: Mathematics: Geometric (or close to it) growth in wildebeest population of the Serengeti following Rinderpest inoculation Exponential growth in the total human population The Take Home Message: Simplest expression of population growth: 1 parameter, e.g., r = intrinsic growth rate Population grows geometrically/exponentially, but the Per capita growth rate is constant First Law of Ecology: All populations possess the capacity to grow exponentially Exponential/geometric growth is a model to which we build on Overview of population growth: New Concepts: discrete continuous density independent Geometric X Exponential X density dependent Discrete Logistic Logistic - Stability - DI (non-regulating) vs. DD (regulating) growth - equilibrium Variability in growth (1) Individual variation in births and deaths (2) Environmental (extrinsic variability) (3) Intrinsic variability Variability in space In time Variability in space Source-sink structure In time Variability in space Source-sink structure (arithmetic) Source-sink structure with the rescue effect In time Variability in space Source-sink structure (arithmetic) Source-sink structure with the rescue effect In time (geometric) G < A G declines with increasing variance Variability in space Source-sink structure (arithmetic) Source-sink structure with the rescue effect In time (geometric) G < A G declines with increasing variance (arith & geom) Increase the number of subpopulations increases the growth rate (to a point), and slows the time to extinction Temporal variability reduces population growth rates Cure – populations decoupled with respect to variability, but coupled with respect to sharing individuals