Overview of population growth: discrete continuous density independent Geometric Exponential Discrete Logistic density dependent Logistic New Concepts: Stability DI (non-regulating) vs. DD (regulating) growth equilibrium Variability in growth Individual variation in births and deaths Environmental (extrinsic variability) 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 N = finite rate of increase 4 3 2 1 time

Continuous (exponential) growth 5 Nt = N0ert N r = intrinsic growth rate 4 3 2 1 time

Continuous (exponential) growth 5 population growth rate per capita growth rate dN dt 1 dN N dt N = r = rN; 4 3 2 1 dN dt Read as change in N (density) over change in time. time 1 dN N dt = r 1 dN N dt Y = b + mX Per capita growth is constant and independent of N N

Comparison Discrete Continuous Nt = N0t Nt = N0ert Where:  = er r = ln  Increasing: Decreasing: > 1 r > 0  < 1 r < 0 None Compounded instantaneously Every time-step (e.g., generation) Time lag: No breeding season - at any time there are individuals in all stages of reproduction Populations w/ discrete breeding season Applications: Most temperate vertebrates and plants Examples: Humans, bacteria, protozoa Often intractable; simulations Mathematics: Mathematically convenient

Geometric (or close to it) growth in wildebeest population of the Serengeti following Rinderpest inoculation

Exponential growth in the total human population

Exponential/geometric growth is a model 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: discrete continuous density independent Geometric Exponential Discrete Logistic X X density dependent Logistic New Concepts: Stability DI (non-regulating) vs. DD (regulating) growth equilibrium Variability in growth Individual variation in births and deaths Environmental (extrinsic variability) Intrinsic variability

Variability in space In time No migration migration

Variability in space In time Source-sink structure No migration migration

Variability in space In time Source-sink structure No migration (arithmetic) Source-sink structure with the rescue effect migration

Variability in space In time (geometric) G < A G declines with increasing variance Source-sink structure No migration (arithmetic) Source-sink structure with the rescue effect migration

Source-sink structure No migration Variability in space In time (geometric) G < A G declines with increasing variance Source-sink structure No migration (arith & geom) Increase the number of subpopulations increases the growth rate (to a point), and slows the time to extinction (arithmetic) Source-sink structure with the rescue effect migration Temporal variability reduces population growth rates Cure – populations decoupled with respect to variability, but coupled with respect to sharing individuals