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Population limitation: history & background

Population limitation: history & background. Both geometric, exponential growth rare in nature--because populations are limited (by amount of resources, by predators, parasites, competitors, etc.)

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Population limitation: history & background

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  1. Population limitation: history & background • Both geometric, exponential growth rare in nature--because populations are limited (by amount of resources, by predators, parasites, competitors, etc.) • Thomas Malthus understood this idea, expressed in his 1798 book (“An essay on the principle of population as it affects the future improvement of society”) • Darwin picked up on Malthus’s idea in his theory of evolution by Natural Selection • Population limitation involves any factor that keeps a population from growing

  2. Density-independent limiting factors are not proportional to population size (e.g., catastrophic weather events) • Density-dependent limitation is proportional to population size, and has a special name: population regulation

  3. Population regulation • Implicit in the concept of regulation is intraspecific competition • decreased per capita growth, reproduction, and/or survival within a population or species due to interactions among individuals over limited resources • Strength of intraspecific competition proportional to population size

  4. Yeast begin growing exponentially...

  5. Modeling population regulation • Assumptions of logistic model • Relationship between density and rate of increase is linear • Effect of density on rate of increase is instantaneous (we’ll relax this assumption later) • Environment is constant (i.e., r is constant, as is K = carrying capacity) • All individuals are identical (i.e., no sexes, ages, etc.) • No immigration, emigration, predation, parasitism, interspecific competition, etc. • Purpose of such a heuristic, deterministic model is to include just the essential idea of regulation, and nothing else

  6. Logistic model of population regulation • First, an “intuitive” mathematical derivation of logistic population model • Also known as sigmoid population growth model • Developed by Pearl and Reed, based on work of Verhulst & others (early 1900’s) • Let dN/dt = r(N)N • This is just the exponential model, except that r is now a function of N (= population size) • Specifically, r declines with population size • Define r(N) as r*(1-(N/K)); notice that this function r(N) goes from r as N-->0, to 0 as N-->K; • K defined as population carrying capacity. • Model in full: dN/dt = r*N*(1-(N/K)) = r*N*(K-N)/K

  7. r(N) r K Population size (N) Graphic of logistic model per capita growth rate {(dN/dt)*(1/N)} Exponential model r r {(dN/dt)*(1/N)} Population size (N)

  8. How does logistic model behave? • Here’s the model again: dN/dt = r*N*(K-N)/K • When N approaches K, right-hand expression ((K-N)/K) approaches 0. Thus, dN/dt approaches 0, which means that N does not change with time: Population is stable! • Alternatively, when N approaches 0, right-hand expression ((K-N)/K) approaches 1. Thus dN/dt approaches r*N*1, i.e., dN/dt is approximately equal to r*N: Population grows exponentially! • Graph of N versus time (t) is sigmoidal in shape, starting out like exponential growth, but approaching a line with slope of zero.

  9. Logistic population growth in Lynx; recall that r = b - d! • Solution to logistic model (involves solving a differential equation, using methods of differential calculus): • N(t) = K/(1 + b*e-rt), where b = [K-N(0)]/N(0) • This equation can be used to plot N vs. t

  10. More about behavior of logistic population model: • How does the slope of the logistic curve (N as a function of t) vary with N? This can be seen intuitively--goes from 0 (at low N) to maximum (at intermediate N), back to 0 at N = K (i.e., hump-shaped curve, with maximum at N = K/2).

  11. Per capita rate Per capita rate Per capita rate N N N What exactly does “regulation” mean? • Regulation means the tendency for a population to remain dynamically stable, no matter where it starts (assuming it is non-zero) • Thus N approaches K, the carrying capacity, from both N < K, and N > K • K is thus an “equilibrium point” of the model, because of negative feedback on r as N gets larger • We can show this idea of dynamic equilibrium graphically: deaths deaths deaths births births births K K K

  12. How do ecologists test for population regulation, density-dependence? • Laboratory: Study population growth in controlled, simple environment with limited resources • Look for evidence for carrying capacity (population stays at, or returns to fixed abundance) • Field: look for evidence of density-dependence of demographic variables Let’s look at some evidence of these three types...

  13. Example of logistic growth: yeast population growth in lab

  14. Sheep population on island of Tasmania leveled off after initial exponential growth

  15. Ringed necked pheasants on Protection Island again: Population growth rate declines away from exponential, approaching constant population because of limited resources!(from G.E. Hutchinson, 1978, An Introduction to Population Ecology, Yale University Press.)

  16. Density-dependent fecundity in Daphnia pulex, lab cultures

  17. Density-dependent survival probability in Daphnia pulex, lab cultures

  18. Density-dependent Lambda for populations of Daphnia pulex, lab cultures: Note linear decline in lambda with density!

  19. Regulatory density-dependence in Mandarte Island (British Columbia) song sparrow population (Melospiza melodia): (a) size of “floater” = non-territorial individuals, (b) no. young fledged per female, (c) proportion of juveniles surviving one year

  20. Density-dependence of larval migration and mortality in grain beetle (Rhizopertha dominica)

  21. Density-dependence of plant dry weight in flax (Linum) plants in greenhouse

  22. Returning to real world, how important are density-dependent versus density-dependent (= regulatory) population limitation? • Controversy erupted among ecologists in 1950’s on relative importance of these two kinds of limitation • Andrewartha and Birch challenged primacy, and even necessity of, density-dependence in population limitation • “The distribution and abundance of animals” (1954) • Work based primarily on Thrips imaginis (rose pest) • Their argument: Weather alone is sufficient to control (regulate?) these insect populations

  23. Example of one year’s thrips population sizes (1932)

  24. 78% of variability in Thrips imaginis population (just prior to peak abundance in December) attributable to weather variables (e.g., rainfall in Sept. & Oct.)

  25. Other ecologists championed primacy of density-dependence in populations • Scientists in this “density-dependence school”: Lotka, Gause, Nicholson, David Lack • Lack’s (1954) book particularly influential: “The natural regulation of animal numbers” • These scientists argued that even in the kinds of insects that Andrewartha and Birch studied, density-dependence is important • Fred Smith (1961) pointed out that even in Andrewartha and Birch’s Thrips imaginis data, population change is density-dependent

  26. Density-dependence inThrips imaginis: Change in population size from November to subsequent October decreases with increasing size of previous October population

  27. Resolution of debate on density-dependence versus density-independence? • Ecologists today recognize that the dichotomized positions of scientists in 1950’s were unnecessarily extreme • Most, if not all, populations are limited at least to some extent by density-dependent factors • Density-independent factors are also usually important (e.g., weather, disease) • Weather does not act just in density-independent way (e.g., proportionately more individuals occupy refuges in smaller population) • Thus DD and DI factors interact in complex ways • We still do not understand regulation in most pops.

  28. Conclusions: • In nature, most populations are limited by resources, predators, etc. • We developed a model for population growth in a limited environment, using linear decrease in r with population size--the logistic population model • Logistic population growth is also known as sigmoid growth, because population approaches an equilibrium size (K) in an s-shaped manner • Lots of examples of limitation, regulation, density-dependence seen in nature • Debate about prominence of density-dependent versus density-independent factors in nature resolved: both are important, but in complex ways

  29. Acknowledgements: Most illustrations for this lecture from R.E. Ricklefs and G.L. Miller. 2000. Ecology, 4th Edition. W.H. Freeman and Company, New York.

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