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Lecture 6

Lecture 6. BSC 417. More models. Logistic growth Overshoot and collapse. Population – a group of individuals of a single species that occupies the same general area. Exponential growth model – the rate of expansion of a population under ideal conditions

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Lecture 6

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  1. Lecture 6 BSC 417

  2. More models • Logistic growth • Overshoot and collapse

  3. Population – a group of individuals of a single species that occupies the same general area. • Exponential growth model – the rate of expansion of a population under ideal conditions • Population-limiting factors – hunting, amount of space suitable for breeding, restricted population growth, food availability • Logistic growth model – idealized population growth slowed by limiting factors as the population size increases • Carrying capacity – the maximum population size that an environment can support at a particular time with no degradation to the habitat

  4. Exponential growth of bacteria

  5. Logistic growth and exponential growth compared

  6. Population Regulation: A Multitude of Forces Density-dependent growth • The logistic equation (p 49) The change in population size over time can be written as: dR(t)/dt = k(t) x R(t) Where k(t) = unconstrained growth rate x (1-R(t)/cc)

  7. Solution to the rate equation • R(t) = cc/(1+Ae^-unconstrained growth rate x t) • A = (cc-R0)/R0 • Steady state when?

  8. Cc-R0 R0 Modeling density-dependent growth • The Logistic Equation (cont.) The next part, can be thought of as the “braking term”, in that it causes the growth rate to slow as population size increases – making it dependent on density. As the population size approaches CC… actual growth rate slows down stable equilibrium at R(t) = CC

  9. carrying capacity Logistic phase: growing at a decreasing rate Exponential phase: growing at an increasing rate Graphing Logistic Growth CC R(t) ● inflection point Time 

  10. Modeling density-dependent growth • Real Populations Real populations do not always behave as smoothly as our graph suggests. Why not? Examples of real populations:

  11. Growth of a population of fur seals

  12. Growth of Yeast Cells Population of yeast cells grown under laboratory conditions: R0 = 10, CC = 700, k = .54, Δt = 20 hours

  13. US Population Prediction: Logistic Logistic model prediction of the US population for the period 1900 – 2050, with initial data taken in 1900: t0 = 1900; R0 = 76.2M; k = 0.017, CC = 661.9

  14. Logistics Growth with Harvesting Harvesting populations, removing members from their environment, is a real-world phenomenon. Assumptions: • Per unit time, each member of the population has an equal chance of being harvested. • In time period dt, expected number of harvests is f*dt*P where f is a harvesting intensity factor.

  15. What does the logistic growth model suggest about real populations in nature? • A population’s growth rate will be small when the population size is either small or large and highest when the population is at an intermediate level relative to the carrying capacity. • Limiting factors make the birth rate decrease, the death rate increase or both • Eventually the population will stabilize at the carrying capacity when the birth rate equals the death rate • These are mathematical models and no population fits either perfectly

  16. Some factors that limit population growth • As density of song sparrows increase, the number of eggs laid decreases because of food shortages • Plants grown under crowded conditions tend to be smaller and less likely to survive • Disease transmission or accumulation of toxic waste products can increase mortality

  17. Continued…… • A predator may capture more of a particular kind of prey as the prey becomes abundant • White-footed mice stop reproducing at a colony size of 30-40 even when food and shelter are provided. Stress? • The graph shows aphids which feed on the phloem sap of plants increase in population in the summer and then die-off in the fall and winter

  18. Continued…. • Some populations remain fairly stable in size close to carrying capacity • Most populations fluctuate as seen at the left • This graph shows song sparrow populations, with periodic catastrophic reductions due to severe winter weather

  19. Boom and bust cycles • Hare cycles may be caused by increasing food shortages during winter caused by overgrazing • They may be due to predator-prey interactions • Cycles could be affected by a combination of food resource limitation and excessive predation • Predators reproduce more slowly than their prey so they always lag behind prey in population growth.

  20. Exponential growth of the human population • Throughout human history parents had many children but only two on average survived to adulthood • Estimates that by 2025 the world will have to double food production, 2/3 of the available fresh water on earth will be in use, 60,000 plant species will be lost to support the population • Issues: overgrazing, rivers running dry, decrease in groundwater, energy?

  21. Human carrying capacity estimates • Ecological footprint with multiple constraints such as food, fuel, water, housing, and waste disposal used. • Calculates current demand on resources by each country in hectares of land per person • World ecological capacity is 1.7 ha per person alive in 1997

  22. How to achieve population stability? • Zero population growth – when birth rates equal death rates • Two ways to reach ZPG. High birth and death rates or low birth and death rates. • Demographic transition is moving from the first to the second. Most developed countries have made the transition • See the demographic transition in Mexico at the left.

  23. Collapse of northern cod fishery • Renewable resource management – harvesting crops without damaging the resource • Maximum sustainable yield – harvest at a level that produces a consistent yield without forcing a population into decline • Can be just as tricky to reduce population sizes

  24. Overshoot and collapse • Nonrenewable resource and a population that depends on it • Population dynamics linked to resource consumption • Resource base affects death rate

  25. Key features • Uses a coupled set of rate equations: one for resource consumption, one for population • In the beginning, when R(t)≈ R(t=0), birth rate is maximum and exponential growth occurs • R(t) always decreases at a rate proportional to the size of P(t) • Both reservoirs need to reach a steady state for the overall system to be in steady state • Achieved as t∞

  26. Examples: famous “overshoot and collapse” theories • Peak oil • Decline in extraction rate reflecting the end of “easy oil” • “Limits to growth”, “Carrying capacity” • Malthusian demography

  27. Peak oil

  28. More examples: • The boy who cried wolf • Stress • Others:

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