Measuring and Modeling Population Changes Homework: p664 #1, p 665 #2, and p 668 #3,4

Measuring and Modeling Population Changes An ecosystem is made up of abiotic (non-living) and biotic (living) resources. Biotic factors tend to vary greatly over time, whereas most abiotic factors vary little over time (some exceptions: temp, water availability, etc...)

Biotic vs. Abiotic

Biotic or abiotic? Amount of sunlight available to a plant Number of predators feeding on a mouse population Amount of food available to a rabbit Availability of water

Carrying Capacity: the maximum number of individuals an ecosystem can support based on resources available. Carrying capacity is dynamic, always changing, since resource levels are never constant.

Fecundity/Biotic Potential: potential number of offspring that can be produced. eg HIGH – starfish, 1 million eggs/year LOW – hippo, 20 births in 45 years

SURVIVORSHIP CURVES General patterns in the survivorship of species

Type I Species have a low mortality rate when young These are slow to reach sexual maturity and produce small numbers of offspring. Have a longer life expectancy. Ex: Large mammals, including humans

Type 2 Intermediate between types I and III Have a uniform risk of mortality over their lifetime. Certain lizards, perching birds, rodents.

Type 3 Very high mortality rate when young Those that reach sexual maturity have a greatly reduced mortality rate Very low average life expectancy (ex. Green Sea Turtle).

How can you relate survivorship to parental care?

POPULATION CHANGE A negative result means population is declining. Population change (%) = [(b + i) – (d+e)] x 100 n b = births, d = deaths, i = immigration, e = emigration, n = initial population size A negative result means population is declining. A positive result means population is growing. In an open population all four factors come into play (i.e. in the wild). In a closed population only births and deaths are a factor (i.e. in a zoo)

POPULATION GROWTH There are 3 main types of population growth. Geometric, Exponential and Logistic.

Geometric Growth In many species deaths occur throughout the year, but births are restricted to a particular period. These populations grow rapidly during breeding season and then decline slowly the rest of the year. Breeding Periods TIME

Geometric Growth Rate The growth rate is a constant (λ) and can be determined using the following equation: λ = N (t + 1) N(t) λ is the fixed growth rate (one year vs next) N is the population size at year (t+1) or (t)

EXAMPLE The initial Puffin population on Gull Island, Newfoundland is 88 000. Over the course of the year they have 33 000 births and 20 000 deaths. a) What is their growth rate?

Answer to a) ANSWER a) N (0) = 88 000 λ = N(t + 1) = 101 000 = 1.15 Therefore the growth rate is 1.15.

To find the population size at any given year, the formula is: N(t) = N(0)λt N(0) is the initial population size

EXAMPLE continued The initial Puffin population on Gull Island, Newfoundland is 88 000. Over the course of the year they have 33 000 births and 20 000 deaths. b) What will the population size be in 10 years at this current growth rate?

Answer to b) From a) growth rate, or λ = 1.15 N(10) = N(0)λ10 * BEDMAS = 88 000 (1.15)10 = 356 009 Therefore the population size will be 356 009 in 10 years.

SAMPLE PROBLEM Each May, harp seals give birth on pack ice off the coast of Newfoundland. In a hypothetical initial population of 2000 harp seals, 950 seal pups are born, and 150 seals die over the course of a year. There is no immigration or emigration. a) Assuming the population is growing geometrically, what will the harp seal population be in 2 years? b) Assuming the same growth rate, what will the harp seal population be in 8 years?

EXPONENTIAL GROWTH A wide variety of species are able to reproduce on a continuous basis. Exponential growth describes populations growing continuously at a fixed rate in a fixed time interval. Since they are growing continuously, biologists are able to determine the instantaneous growth rate, or intrinsic growth rate (r) r = b-d b=per capita birth rate d=per capita death rate

Exponential population growth rate is: dN = rN dt To find the time it takes a population that is reproducing exponentially to double, we use the equation: td = 0.69 r Population Size Intrinsic growth rate Instantaneous growth rate

Sample Problem A population of 2500 yeast cells in a culture tube is growing exponentially. If the intrinsic growth rate is 0.030 per hour, calculate: a) the initial instantaneous growth rate of the yeast population. b) the time it will take for the population to double in size. c) the population size after four doubling periods.

Answer A population of 2500 yeast cells in a culture tube is growing exponentially. If the intrinsic growth rate is 0.030 per hour, calculate: N = 2500 yeast, r = 0.030 /hr dN/dt = rN = 0.030 /hr x 2500 yeast = 75 /hr When the population size is 2500 the instantaneous population growth rate is 75 per hour

Answer A population of 2500 yeast cells in a culture tube is growing exponentially. If the intrinsic growth rate is 0.030 per hour, calculate: N = 2500, r = 0.030 /hr b) td = 0.69/r = 0.69 / 0.03/hr = 23 hours

Answer A population of 2500 yeast cells in a culture tube is growing exponentially. If the intrinsic growth rate is 0.030 per hour, calculate: N = 2500 yeast, r = 0.030 yeast /hr c) the population size after four doubling periods.

LOGISTIC GROWTH The previous two models assume an unlimited resource supply, which is never the case in the real world. When a population is just starting out, resources are plentiful and the population grows rapidly. BUT, as the population grows, resources are being used up and the population nears the ecosystem's carrying capacity.

The growth rate drops and a stable equilibrium exists b/w births and deaths. The population size is now the carrying capacity (K). This is known as a sigmoidal curve.

Logistic Growth Curve A: Population small, increasing slowly B: Population goes through largest increases C: Dynamic equilibrium (at carrying capacity), b=d, no net population increase Pop'n Size (N) Time (t) A. Lag Phase B. Log Phase C. Stationary Phase Carrying Capacity

LOGISTIC GROWTH EQUATION Logistic growth represents the effect of carrying capacity on the growth of a population. It is the most common growth pattern in nature. dN = rmaxN K – N dt K Population size at given time Carrying capacity Max intrinsic growth rate Population growth at a given time

continued... If the population size is close to the carrying capacity, there is virtually no growth (K-N = 0), thus the equation takes into account declining resources as the population increases.

Sample Problem A population of humans on a deserted island is growing continuously. The carrying capacity of that island is 1000 individuals and the maximum growth rate is 0.50. Determine the population growth rates over 5 years if the initial population size is 200.