Measuring and Modelling Population Changes

Population Dynamics Carrying Capacity: Maximum number of organisms that can be sustained by available resources over a given period of time Is dynamic as environmental conditions are always changing Fecundity: The potential for a species to produce large numbers of offspring in one lifetime.

Life is a gamble...

Factors That Affect Population Growth Birth (natality), death (mortality), immigration, emigration Population growth of any given population is calculated mathematically

More Terms! Open population – Population in which change in number and density determined by births, deaths, immigration, emigration Closed population – Change in size determined by natality and mortality alone Biotic Potential – Maximum reproductive rate (r) under ideal conditions (intrinsic rate of natural increase) eg. E. Coli...if doubled, unchecked for 24hrs they would cover the earth 1m deep!!

Population Growth Models

Geometric Growth Geometric growth () – pattern of population growth where organisms reproduce at fixed intervals at a constant rate. Eg. Animals with a specific breeding season.  = N(t +1) N (t)  = fixed growth N = Population in year (t + 1) t = year  

Geometric Growth Example 2000 seals give birth to 950 pups in May. During the next 12 months, 150 pups die. Assuming geometric growth, what will the harp seal population be in two years? Eight years? First Calculate Growth rate: N(0) = 2000 N(1) = 2000 + 950 -150 = 2800 After 2 years: After 8 years: N(t + 1) = N(t)  N(8) = N(0)  8 N(2) = 2800 x 1.4 = 2000 x (1.4)8 = 3920 = 29520 OR N (2) = N (0)  2 = 3920

Exponential growth A pattern of population growth where organisms reproduce continuously at a constant rate Ecologists are able to determine instantaneous growth rate of the population expressed in terms of the intrinsic (per capita) growth rate (r). difference between per capita birth rate, b, and per capita death rates, d, where r = (b – d) population growth rate given by the expression... dN/dt = instantaneous growth rate of population r = growth rate per capita N = population size

Exponential growth For populations growing exponentially, the time needed for population to double in size is a constant...

Exponential Growth Example A population of 2500 yeast cells in culture is growing exponentially with an intrinsic growth rate r is 0.0575 per hour. 1. What is the initial instantaneous growth rate of the population?   Given: r = 0.0575, N = 2500    dN/dt = (0.0575)(2500) = 144 per hour 2. What time will it take for the population to double in size? td = 0.69 0.0575 = 12 hours

Exponential Growth Example 3. What will the size of the population be after each of four doubling periods? Doubling Times Time in hours Population size 2500 1 12 5000 2 24 10000 3 36 20000 4 48 40000

The Cane Toad!!

Bacteria are Prolific!

Modelling Logistic Growth Food, water, light, and space within an ecosystem are factors that limit population growth as resources are consumed as the population nears the ecosystem’s carrying capacity The growth rate drops below rmax­ in this case  Stable equilibrium (births=deaths) is often reached  Population number at carrying capacity is represented by K.

Modelling Logistic Growth Logistic growth is most common growth pattern seen in nature as it represents the effect of carrying capacity on the population’s growth Logistic growth equation is as follows

Logistic Growth Example A population is growing continuously. The carrying capacity of the environment is 1000 individuals and its r max (max growth rate) is 0.50. Determine pop growth rates based on pop sizes of 100 , 500, 900, 1000

r Max Pop Size N (K-N) N Pop Growth Rate 0.50 100 900/1000 45 500 500/1000 125 900 100/1000 1000 0/1000 Question : What is the relationship between population size and growth rate? Answer: When the pop is small the growth rate is slow. It increases as the pop increases, then as it approaches carrying capacity, the growth rate declines and eventually stops!!

S-shaped (Sigmoidal) Growth

Three Distinct Phases Lag phase occurs when population is small and increasing slowly Log phase occurs when population undergoes rapid growth As available resources become limited, population experiences environmental resistance and stationary phase occurs in which the population is at dynamic equilibrium (b=d)

Practice Questions A population of spotted butterflies exhibits logistic growth. The carrying capacity of the population is 500 butterflies, and the maximum growth rate (rmax)is 0.1. [I] a) determine the population growth rates based on a population size of 50, 100, 200, 250, 480, 500 and 525 butterflies. b) describe the relationship between population size and the growth rate. The human population is currently doubling ever 40 years. Based on what you have learned about population growth rates and carrying capacity, what question should we be asking of our society? [A] It has been a wet warm summer in Northern Ontario, and a small population of mosquitoes is exhibiting exponential growth. The initial population size is 650, and the intrinsic growth rate is 0.450 per day. a) calculate the initial instantaneous growth rate for the mosquito population. b) calculate the doubling time for the population. c) how many doubling times would have to occur for the population to exceed 1 000 000? How many days does this represent?

Practice Questions A white-tailed deer population in a provincial park was estimated to be approximately 4000, with a carrying capacity of 30 000. Other than natural predators, the deer were left alone. Hunting was prohibited. After a winter in which the predators decimated the deer population, a deer population management plan was put in place. It was decided to remove the predators through hunting and trapping. Initially, the deer population flourished. Eventually, however, the deer population started to decrease. [I][C] a) Graph the population changes over the 35-year period using the data below. (more questions on the next slide) Year Deer Population 1 3 000 5 8 000 10 25 000 15 55 000 20 70 000 25 24 000 30 10 000 35 2 000

Previous question cont’d b) using a different colour, draw a horizontal line across the graph at the carrying capacity. c) why do you think the deer population decreased after year 20, despite the fact that the predators had been removed? d) Did the management plan initially seem to be effective? Why? e) Without human interference and the deer management plan, what do you think would have happened to the deer population?