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Population limitation: history & backgroundPowerPoint Presentation

Population limitation: history & background

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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

- 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

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

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

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

r

K

Population size (N)

Graphic of logistic modelper capita growth rate

{(dN/dt)*(1/N)}

Exponential model

r

r

{(dN/dt)*(1/N)}

Population size (N)

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.

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

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).

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

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...

Example of logistic growth: yeast population growth in lab density-dependence?

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

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.)

Density-dependent fecundity in Population growth rate declines away from exponential, approaching constant population because of limited resources!Daphnia pulex, lab cultures

Density-dependent survival probability in Population growth rate declines away from exponential, approaching constant population because of limited resources!Daphnia pulex, lab cultures

Density-dependent Lambda for populations of Population growth rate declines away from exponential, approaching constant population because of limited resources!Daphnia pulex, lab cultures: Note linear decline in lambda with density!

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

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

Density-dependence of plant dry weight in flax ( grain beetle (Linum) plants in greenhouse

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

Example of one year’s thrips population sizes (1932) versus density-dependent (= regulatory) population limitation?

78% of variability in versus density-dependent (= regulatory) population limitation?Thrips imaginis population (just prior to peak abundance in December) attributable to weather variables (e.g., rainfall in Sept. & Oct.)

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

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

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.

Conclusions: density-independence?

- 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

Acknowledgements: density-independence?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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