Overview of Lecture: Ecology: Biosphere & Populations Read: Text ch 52 & 53 Bullet Points: •! population ecology •! limiting factors •! spatial patterns •! dynamic demographics •! geometric pop growth •! looking back •! exponential growth •! logistic growth •! maximum sustainable yield •! deterministic chaos •! the human footprint10 Molecules 9 Organelles 8 Cells 7 Tissues 6 Organs & organ systems 5 Organisms 4 Communities of populations living together & exchanging resources form Ecosystems 3 2 1 the Biosphere Not as tightly integrated as an organism, but more tightly integrated than a community of populations.We begin with the current distribution map for each tree species, to characterize and define the occupied (or occupiable) volume of environmental space by statistically cataloging all combinations of env conditions which are habitable by this species. To predict a new distribution for the species under altered environmental conditions, we re-cluster using all cells from a map ... the altered spatial distribution now habitable by this species under the new env conditions will be shown ... {assuming what about time, luck & adaptation?}If we know abundance & area (& a little probability theory) we can construct expected random distribution of a) nearest neighbor distances, or b) number indiv’s in small sample quadrats (like expected chocolate chips per cookie if Poisson) Random distributions rarely persist in nature – ecological interactions. {consider seating pattern in a bar: might be some clumping & some hyperdispersal} Potential processes include aggregation at patchy resources and/or social attraction to each other If there are too many short nearest neighbor distances, or the variance in # indiv’s per quadrat is too high, then reject the null model that the distribution is random. Potential processes include resource competition and aggression. If there is too little variation in nearest neighbor distances, or in # indiv’s per quadrat, then reject the null model.In general, if you look at a big enough scale, species tend to be clumped but if you look at a small enough scale they tend to be hyperdispersed. At continental scale, breeding birds are clumped in regions, biomes, habitats (grasslands, pastures) Within habitats, aggregate at profitable patches (heavily grazed) Within patches, territories uniform – hyperdispersed Within territories, behaviors ‘aggregated’ over microhabitatsBegin w/ simple binary fission in bacteria.Then N0 = 1 Suppose we start w/ 1 bacteria at time t = 0: N0 = 1, and at each unit of time the bacteria undergo binary fission and the number of bacteria doubles. {obviously something is missing from this simple model of geometric growth} 2 N0 = N1 = 2 = 21 2 N1 = N2 = 4 = 22 2 N2 = N3 = 8 = 23 2(60/20) = 23 = 8 per hr 2(3!!24) = 272 = ? "The mathematics of uncontrolled growth are frightening. ... in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth."March 15 - HOW POPULATIONS CAN GROW Populations can grow very rapidly. … if a simple geometric growth rate is assumed (which was the assumption made by Charles Darwin in relation to his imagined “struggle for existence” in nature), it would only take about 1100 years—assuming 35 years per generation— to develop that present world population of six billion people. … All of which indicates that the evolutionary scenario, which assumes that human populations have been on the earth for about a million years, is absurd. The whole universe could not hold all the people! What is wrong with this story? Did Darwin assume unlimited geometric growth with constant ! ? What is missing from the geometric growth model? From reading Malthus (Text ch 22) Darwin concluded that geometric pop growth results in a relentless “struggle for existence” as demand for resources exceeds the supply.Nt-1 Nt "t IN = births + immigration (ignore for now) OUT = deaths + emigration (ignore for now){note the overshoot & damped oscillation toward carrying capacity}A simple way to model that is to let r = rmax (1 - N/K)N If you want to find the pop size N* where dN/dt is at maximum (growing fastest), take the derivative of (dN/dt ) with respect to N, set it equal to 0 & solve for N* K/2 max dN/dt r=0 @N = K @N=0: r = rmax NWe’d like to see a map of N(t), to visualize the population dynamics predicted by the logistic model. Unfortunately, it is ‘difficult’ to solve the equation [ dN/dt = [rmax (1 - N/K)] N ] for N(t) = an explicit function f(K, rmax,N0,t). at small µµ (2.0), nice logistic growth {a simplifying quirk: approaches K/2} However, as µµ gets larger (4.8), strange things begin to appear; all chaos breaks loose!µµ = 2.0: asymptote to 1 stable pt µµ = 4.8: ‘chaotic meandering’ µµ = 2.8: damped oscillations µµ = 3.2: stable limit cycle w/ 2 pts µµ = 3.5: stable limit cycles w/ 4 pts X X X Xhttp://theoryx8.uwinnipeg.ca/fractals/Fractal_chaos.html What is Chaos? … in the 1960's, a meteorologist named Edward Lorenz … was attempting to simulate weather patterns in a mathematical model. These patterns did not follow any "predictable" evolution as the simulation progressed, he eventually realized that his model was extremely sensitive to his starting conditions; & slight variations in numerical precision … http://www.amazon.com Gleick's "Chaos" will change the way you look at the world. This is as much a testament to Gleick's powerful prose as it is to the profound implications of chaos theoryThe Lake Mi perch pop has small proportion of females and too few young age classes to replace older age classes as they die off. {fishing restrictions ! recent recovery of perch pop} Most multicellular organisms grow for a while, then they reproduce, if they survive, if they are female. People all around Lake Michigan are asking questions about the nine-year decline in yellow perch populations. (1990s) http://www.seagrant.wisc.edu/communications/publications/One-pagers/yellowperchFactSheet.html … few young perch are surviving to adulthood. … the average age of the population is increasing quickly. … the