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

• 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

Exponential growth of bacteria

Logistic growth and exponential growth compared

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)

Solution to the rate equation

• R(t) = cc/(1+Ae^-unconstrained growth rate x t)• A = (cc-R0)/R0• Steady state when?

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 downstable equilibrium at R(t) = CC

Cc-R0

R0

Graphing Logistic Growth

Time

R(t)

CC

● inflection point

Logistic phase: growing at a decreasing rate

Exponential phase: growing at an increasing rate

carrying capacity

Modeling density-dependent growth

• Real PopulationsReal populations do not always behave as smoothly as our graph suggests.

Why not?

Examples of real populations:

Growth of a population of fur seals

Growth of Yeast Cells

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

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

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.

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

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

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

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

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.

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?

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

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.

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

Overshoot and collapse

• Nonrenewable resource and a population that depends on it

• Population dynamics linked to resource consumption

• Resource base affects death rate

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∞

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

Peak oil

More examples:

• The boy who cried wolf• Stress• Others: