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Overview of population growth:

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

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Overview of population growth:

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  1. 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

  2. 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)

  3. How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 +rN0= N0 (1 + r)

  4. How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 +rN0 = N0 (1 + r) N2 = N1 +rN1 = N1 (1 + r)

  5. 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 ???

  6. How do populations grow? Growth rate (r) = birth rate – death rate N1 = N0 +rN0 = N0 (1 + r) substitute N2 = N1 +rN1 = N1 (1 + r)

  7. 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

  8. 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

  9. Discrete (geometric) growth 5 Nt = N0t N • = finite rate of increase 4 3 2 1 time

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

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

  12. Comparison DiscreteContinuous 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

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

  14. Exponential growth in the total human population

  15. 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

  16. 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

  17. Variability in space In time No migration migration

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

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

  20. 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

  21. 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

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