Introduction to course: BSCI 462 of BIOL 708 R

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1 Introduction to course: BSCI 462 of BIOL 708 R Population Ecology: Fundamental concepts in plant and animal systems Spring 2013

2 Introduction The biology of a population = Population Ecology Issue of scale, focus, and tradition Populations evolve over time

3 Chapter 1 Density Independent Growth 1.1 Introduction What is a population? Closed population? Open population?

4 Chapter Introduction Population Biology = Popln Ecology + Popln Populations evolve thus need to think about evolutionary ecology of populations 1) Phenotypic variation 2) Some variation heritable 3) Selection acts differently on a range of phenotypes Genetics

5 Chapter Introduction Turchin (2001, 2003) Properties of Populations~ foundational ecological principles 1) Poplns tend to grow exponentially 2) Poplns exhibit self limitation (or bounded fluctuations) 3) Consumer resource interactions often oscillatory

6 Chapter Introduction What 4 factors influence population growth? 1) 2) 3) 4)

7 Chapter Introduction 1) Births: Fecundity vs. Fertility Fecundity = potential reproductive output Fertility = realized or actual reproductive output Both expressed as Rates = per individual or per 1000 individuals per unit time Often referred to as vital rates

8 1.2 Fundamentals of Population Growth 2) Mortality - Also expressed as a rate, per indiv., 1000 indiv., per unit time, per unit area. Stable age distribution (SAD) = same proportions constant in each age class over time Stable stage distributions (SSD) = same proportions constant in each stage class over time A measure of population growth (r) ~ growth rate per indiv. per unit of time = intrinsic rate of increase For a popln: r (intrinsic rate of increase) = b-d (1.1a) r (intrinsic rate of increase) = b-d (1.1b) 1000

9 1.2 Fundamentals of Population Growth 3) Immigration rate (i) = number of individuals into an area per unit time interval. I = number if individuals arrive 4) Emigration rate (e) = number of individuals leave an area per unit time interval. E = number of individuals left Referring to gene flow between populations Environment = abiotic and biotic together

10 1.3 Types of models Difference Equations most often used to model discrete popln (vs. continuous) growth: N t+1 = N t + (B-D) + (I - E) N t+1 = N t + (B+ I) - (D + E) (1.2a) (1.2b) When utilizing rates: N t+1 = N t (b-d) = N t R where R = net reproductive rate; R = b-d per generation λ = finite rate of increase = b-d per period (time or generation) R = λ when generation time = 1 year

11 1.3 Types of models Differential Equations assume continuous popln growth: dn = rn : measures the instantaneous growth of popln, N dt dn = change in N per time interval (t)~ change in N over time r measures the probability of birth minus the probability of death occurring within a popln during a particular time interval.

12 1.4 Density Independent vs. Density Dependent Growth Density independent growth = Density dependent growth = Examples? Recall: Focus of Chapter 1 is Density Independent Growth

13 1.5 Discrete ~ geometric growth in pops with non-overlapping generations Reproductive Strategies: Reproduce once and die semelparous, annual Reproduce many times iteroparous, perennial, many animals R 1 = N 1 /N 0 across generations leads to N t = N 0 R t and N t = N 0 λ t R = λ when generation time = 1 year (generation time = time period) R used when assess growth per generation and λ estimates growth per a set time period R λ when generation time 1 year; Generation time Time period

14 1.5 Discrete ~ geometric growth in pops with non-overlapping generations Recall: R = λ when generation time = 1 year R λ when generation time 1 year Therefore: R or λ > 1, pop increasing R or λ = 1, pop stable R or λ < 1, pop decreasing Fig. 1.1 (discrete growth ) Fig. 1.2 (ln growth) lnn t = lnn 0 + ln(r)t lnn t = lnn 0 + ln(λ)t with N 0 = y intercept & lnr or lnλ = slope of Linear relationship and t independent

Gypsy moths example 1.1 utility of population growth parameters http://upload.

jpg Spread of non-native Gypsy Moth: http://en.wikipedia.

15 Gypsy moths example 1.1 utility of population growth parameters antria_dispar_dispar_%28asian%29_male.jpg Spread of non-native Gypsy Moth: /commons/6/64/gypsy_moth_larva.jpg

Periodical cicadas - example 1.2 http://insects.ummz.lsa.umich.

16 Periodical cicadas - example 1.2

17 1.6 Exponential growth with overlapping pops dn = rn = measures the instantaneous growth of popln, (eq. 1.3) dt Take integral of both sides of equation and it becomes: N (t) lnn(0) = rt-r0 = rt Exponentiation of both sides of equation becomes: N (t)/n(0) = e rt with e = base of natural log

18 1.6 Exponential growth with overlapping pops Nt = N 0 e rt (eq 1.8) Figs. 1.3 expo growth & 1.4 ln growth r > 0 r = 0 r < 0 Popln growing Popln stationary Popln decreasing Made linear by taking natural log (ln) : Ln N t = ln N 0 + rt Graphing => with slope = r and y-intercept = ln N 0

1.7 Doubling time & exponential growth with invasive species Doubling time = time it takes for population to double in size Ln 2 = 0.693 Doubling time = ln 2 / r Doubling time = 0.

19 1.7 Doubling time & exponential growth with invasive species Doubling time = time it takes for population to double in size Ln 2 = Doubling time = ln 2 / r Doubling time = 0.693/ r (If r value negative, then absolute value is the time it takes a popln to decline by ½ ) Mute Swan Assignment: Prove problem with Fig. 1.5

20 1.8 Applications to human populations: How? Graphing : Slope Y-intercept Age distribution/stage distribution: Pre-reproductive Reproductive Post-reproductive Fig. 1.6 (pg.21)

21 1.9 Finite rate of increase (λ) & intrinsic rate of increase (r) ~ common currency to compare popln growth Intrinsic (~ instantaneous) rate of increase (r) Finite rate of increase (λ) over a time period & based on N t+1 = N t If popln lacks age distr. or has a stable age distribution (SAD) then the finite rate of increase is a constant. N t+1/ N t = If t = 1, then N t+1/ N t = e rt = e r λ = e r (eq. 1.12) and r = ln λ (eq. 1.13)

22 1.10 Stochastic Models & Popln Viability Analysis Up to now only considered deterministic models Now consider stochasticity in models Stochastic models produce: frequency distribution of probabilities Environmental stochasticity Demographic stochasticity Genetic stochasticity

23 1.10 Stochastic Models & Popln Viability Analysis Now also consider Demographic & Envir. stochasticity in models with Genetic stochasticity Genetic stochasticity related to small populations: Genetic drift Inbreeding depression Effective number of reproductive individuals (Ne) See Table 1.6 Prob. of repro w/r to age (p i ) x Litter size (B i ) = exp net repro. expected net repro = p i B i = λ

24 1.10 Table 1.6 Prob. of repro w/r to age (p i ) x Litter size (B i ) = exp net repro. expected net repro = p i B i = λ Prob p i of having Litter size, B i Litter size = # of female offspring/year = B i Expected net repro. = p i B i λ = = p i B i Prob. that a popln of N females will go extinct = N where N = # females Prob. that a popln of N females will double is N where N = # females Assume females die after reproducing Fig. 1.7 (pg. 28) Stochastic growth of popln with 3 females based on values Table 1.6

25 Probability of extinction (Pielou, 1977) Probability of extinction = P 0,t P 0,t = prob. of extinction at time t P 0,t = (d/b) No d = per capita death rate b = per capita birth rate

26 1.10 Arithmetic vs. Geometric Mean Relationship to level of variation - Geometric mean is always less than arithmetic mean - With less variation, then the closer the two estimates become P i = probability of a given λ when computing the geometric mean Population projections Table 1.7 (pg. 30) Arithmetic Geometric* Prob. p i λ p i λ i λ i pi p i λ i = λ i pi = Popln growing Popln declining *Geometric mean = Multiply numbers together and then take the exponent (1/number of comparisons).

27 1.10 Stochastic vs. Deterministic result Figure 1.8 Deterministic vs. Stochastic Growth With high & low variance. Initial pop size = 50 Deterministic growth with arithmetic mean Pop size Stochastic growth with geometric mean Time Table 1.8 Twenty simulations of popln growth comparing deterministic and stochastic growth with low and high variability ***Most likely probability is that a population will persist :Nt+1 =Ntλ using the geometric mean for λ

28 Chap. 1 Highlights Fundamentals of population growth r λ Difference vs. Differential equations Density dependent vs. Density independent growth Role of stochasticity Discrete (Geometric) vs. Continuous growth Exponential growth Overlapping vs. non-overlapping generations Arithmetic vs. Geometric means Deterministic vs. Geometric growth

29 Completion of Chapter 1 Up to now we have only dealt with density independent growth. In Chapter 2 we examine density dependent growth. Questions?

Chapter 2 Lecture. Density dependent growth and intraspecific competition ~ The Good, The Bad and The Ugly. Spring 2013

Chapter 2 Lecture. Density dependent growth and intraspecific competition ~ The Good, The Bad and The Ugly. Spring 2013 Chapter 2 Lecture Density dependent growth and intraspecific competition ~ The Good, The Bad and The Ugly Spring 2013 2.1 Density dependence, logistic equation and carrying capacity dn = rn K-N Dt K Where