BIOL 4120: Principles of Ecology Lecture 10: Population Growth

1. BIOL 4120: Principles of Ecology Lecture 10: Population Growth Dafeng Hui Office: Harned Hall 320 Phone: 963-5777 Email: dhui@tnstate.edu

2. World population

4. Population growth • Definition: how the number of individuals in a population increases or decreases with time • Growth is controlled by rates of birth, immigration and death and emigration. • Open or closed population: no immigration and emigration, or immig.=emig. • In closed population, growth is determined by birth rate and death rate.

5. 10.1 Population growth reflects the difference between rates of birth and death Model development A population of freshwater hydra growing in an aquarium in the laboratory. Population size N(t) when time is t. This is a closed population. Population size change is related to birth rate (b) and death rate (d) dN/dt=(b-d)N=rN The difference between birth rate and death rate is the intrinsic growth rate (r) (instantaneous per capita rate of growth). r=b-d

6. In a closed population, population size change is related to birth rate (b) and death rate (d) • The difference between birth rate and death rate is the intrinsic growth rate (r) (instantaneous per capita rate of growth). r=b-d • Population growth is related to this intrinsic growth rate (r).

7. Exponential population growth • Equations: • 1) dN/dt=rN (differential equation form) • 2) N(t)=N(0) exp(rt) (exponential growth model) • Conditions: • Initial population is small • No food or resource limitation

8. An example Started in 1910 with only 4 males and 22 females In 1940, there were nearly 2000 Reindeer, St. Paul, Alaska.

9. Whooping crane, an endangered speciesrecovered from near extinction in 1941 How to calculate r? Software, Excel (trendline) Aransas National Wildlife Refuge

10. Properties of exponential growth

11. Properties of exponential growth Widely used in biology r determines the shape of the growth. r=0, no change in population size r<0, decrease in population size; r>0, increase in population size.

12. Prediction of population growth N(t)=N(0)Exp(rt) Give a time t, we can predict the population size. An Example: Deer population: N(0)=300, r=0.5, after 5 years, what’s the population size? N(5)=N(0)Exp(rt)=300*exp(0.5*5)=3655 (495, 815, 1344, 2216, 3655) t=10, ? 44524

13. 10.2 Life table • Life table is an age-specific account of mortality. • Purpose of life table: to provide a clear and systematic picture of mortality and survival within a population

14. How to construct a life table? • 1. start with a cohort: a group of individuals born in the same period of time; • 2. Add a column of lx as the probability at birth of surviving to any given age;

15. How to construct a life table (cont.)? • 3. calculate dx, a measure of age-specific mortality • 4. Calculate age-specific mortality rate, qx

17. 10.3 Different types of life table • Two types • Cohort or dynamic life table • as the above gray squirrel • Time-specific life table Elf opine

18. 10.4 Life tables provide data for mortality and survivorship curves • Table is better than words, but a graph is worth one thousand words. • Mortality curve and survivorship curve.

19. Mortality curves

20. Survivorship curves Log scale for Y axis

21. Three basic types of survivorship Type I (convex) Humans and other mammals and some plants (k-selection) Type II (survival rates do not vary with age) Adult birds, rodent, and reptiles, perennial plants Type III. Concave Mortality rate high in the beginning (r-selection) Oysters, Fish, many plant species (most trees)

22. 10.5 Birthrate is age-specific Crude birthrate (demographers): # of birth over a period of time divided by population size at the beginning of the period*1000 Age-specific birthrates, bx Mean # of females birth to a female in each age group. (Only females give birth; birth rates vary with ages) Gross reproduction rate: sum of the bx values across all age classes, provides an estimate of average offspring born to a female over her lifetime.

23. 10.6 Birth rate and survivorship determine net reproductive rate • Fecundity table: take survivorship column, lx, from life table and add age-specific birthrate, bx. • Net reproduction rate, R0: number of female offspring a female at birth can produce (or average # of females that will be produced (left) during a lifetime by a newborn females.) • R0: depends on survivorship and fecundity • R0=1, >1 or <1

24. 10.7 Project population growth • Given a population with age structure and some other information (age-specific mortality rates and birthrates), we can project future changes of the population size. • For example, a population of squirrel with 10 adults (1-yr) and 20 juveniles females, what would happen in the next 10 years?

25. What do we need to project future population size change? Calculate age-specific survivor rate: sx=1-qx bx is age-specific birthrate

26. How to construct a population projection table?

27. How to construct a population projection table? Population size (N) increases every year. Lambda (finite multiplication rate): =N(t+1)/N(t).

28. Age distribution Stable age distribution: by year 7, the proportion of each age group remain the same year after year. Population is still growing.

29. Geometric growth vs exponential growth • N(t)=N(0) t • N(t)=N(0)exp(rt) • =exp(r) or r=ln() • These models are used to describe dynamics of populations. Geometric growth is used for population generations not overlap (discrete time interval), exponential growth model is for continuous population.

30. Fig. 11.3

31. 10.8 Stochastic processes can influence population dynamics • What’s stochastic process? • Deterministic process: Given a set of initial conditions (N(0), r), the exponential growth will predict only one exact outcome. • But the age-specific mortality rates, birth rates represent probability and averages derived from the cohort or population under study (bx=2? 0,1,2,3).

32. Stochasticity • Demographic stochasticity: stochastic (or random) variations in birth and death rates that occur in populations from year to year. (Cause change in r). • Environmental stochasticity: Random variation in the environment, such as annul variation in climate and natural disasters can have a direct influence on average birth and death rates within the population.

33. 10.9 Population extinction • If r becomes negative (birth rate < death rate), population declines and will go extinction. • Factors: Extreme environmental events (droughts, floods, cold or heat etc), loss of habitat (human). • Small populations are susceptible to extinction Overgraze, only 8 in 1950 Allee effect, genetic drift, inbreeding (mating between relatives)

34. Hackney and McGraw (West Virginia University) examined the reproductive limitations by small population size on American ginseng (Panax quinquefolius) • Fruit production per plant declined with decreasing population size due to reduced visitation by pollination

36. Small population size may result in the breakdown of social structures that are integral to successful cooperative behaviors (mating, foraging, defense) • The Allee effect is the decline in reproduction or survival under conditions of low population density • There is less genetic variation in a small population and this may affect the population’s ability to adapt to environmental change

37. END

38. Geometric Growth • When generations do not overlap, growth can be modeled geometrically. Nt = Not • Nt = Number of individuals at time t. • No = Initial number of individuals. •  = Geometric rate of increase. • t = Number of time intervals or generations.

39. Exponential Growth • Continuous population growth in an unlimited environment can be modeled exponentially. • Appropriate for populations with overlapping generations. • As population size increases, rate of population increase gets larger.