Ecology Life Tables - Age Structures Age Growth in...

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Unformatted text preview: Age Structures Age Growth in populations with complex life Growth histories and long lives, such as humans and white-tailed deer, is determined by a number of processes: number 1. Age-specific survivorship 2. Age-specific fertility 3. Generation time 4. Actual numbers of individuals in each age Actual class class Age Structures Age The following example might fix in your mind The the potential importance of age distribution on population growth. Imagine a group of several thousand young Imagine people attending a concert by Daddy Yankee Daddy on an island off the coast of California. on Daddy Yankee! Daddy Suddenly a catastrophic series of earthquakes eliminates the entire local population, leaving only the concertgoers! Age Structures Age Simultaneously, California splits Simultaneously, off from the rest of North America. Assume further that a few of the Assume concertgoers are able to colonize this California island, and future population growth is now based on this group! Age Structures Age Most of the Most concertgoers would obviously have been teen-age girls (assume teen-age a few teenage boys also were dragged along). Assuming an Assuming abundance of food, this California island would rapidly be repopulated. Age Structures Age Growth would, however, be irregular, since all Growth of the girls would be the same age and reach menopause more or less simultaneously in the future. Growth would then slow until their daughters Growth began to reproduce. began Age Structures Age The Future: The Rapid, Rapid, but irregular population growth. growth. Age Structures Age Now imagine the same scenario, except that Now survivors of the disaster were individuals attending an AARP (American Association of Retired Persons) convention. Presumably, all or almost all, of the females Presumably, at the convention are over 55. AARP Convention Concert Starring: Tony Bennett Tony Age Structures Age What would the future of the California island What population be in this case? There is no future! And they don’t want to be There young again! young Age Structures Age Take home lesson: Age Age distribution can contribute to both rapid growth and extinction of a population! population! Life Tables Life Information on age-specific survivorship and Information fertility is put together into a life table. life The probability of living to different age The classes (x) is known as lx. classes The number of female offspring per female in The the population is known as mx. the Survivorship Survivorship Survivorship Type Curves: 1. Death at Senescence (Type I) 2. Equal Probability of Death at All Ages (Type II) 3. High Juvenile Mortality (Type III) Type I Type II, Constant Number Type II, Constant Probability Type III 1 Survivorship, lx 0.75 0.5 0.25 0 0 2 4 6 Age classes 8 10 Type I Type II, Constant Number Type II, Constant Probability Log survivorship, Sx 1000 Type III 100 10 1 0 2 4 6 Age classes 8 10 Arithmetic Scale 1 0.8 lx 0.6 0.4 0.2 0 0 10 20 30 40 50 Age Clases 60 70 80 90 100 Log lx 3 2.5 Log lx 2 1.5 1 0.5 0 0 20 40 60 Age Classes 80 100 Number surviving per 1000 (Sx) US Males 1985 US Males 1910 Carlise, UK 1782 Northhampton, UK 1780 1000 750 500 250 0 0 25 50 75 Age classes 100 US Males 1985 US Males 1920 Carlisle, UK 1782 Northhampton, UK 1780 3 Log Survivorship 2.5 2 1.5 1 0.5 0 0 20 40 60 Age Classes 80 100 Age Class 95 85 75 65 55 45 35 25 15 -9 -9 9 -8 9 -7 9 -6 9 -5 9 -4 9 -3 9 -2 9 -1 9 5 <1 Log of Survivorship (lx) H uman Survivorship 2010 3.5 3 2.5 2 Males 1.5 Females 1 0.5 0 Proportion alive at beginning of age interval (lx) Dall Sheep Dall 1.000 0.800 0.600 0.400 0.200 0.000 0 2 4 6 8 Age classes 10 12 14 Dall Sheep Dall 3 Log survivorship 2.5 2 1.5 1 0.5 0 0 2 4 6 8 Age classes 10 12 14 White-Crowned Sparrows White-Crowned 3 2.5 Log survivorship July 1966 2 August 1966 1.5 1 0.5 0 0 1 2 3 Age Classes 4 5 6 Gray Squirrel Gray 1 Survivorship (l x) 0.75 0.5 0.25 0 0 1 2 3 4 5 Age classes 6 7 8 Golden Lion Tamarins Golden Survivorship, lx 1 0.75 0.5 0.25 0 0 2 4 6 Age classes 8 10 12 Phlox Phlox 1 Survivorship, l x 0.75 0.5 0.25 0 1 2 3 4 5 6 Age in days 7 8 9 10 Fertility Fertility The other half of the life table is the fertility The column, mx. column Here each value represents the average Here number of female offspring produced per female of a given age. Again, gathering accurate data on fertility in Again, the field is problematic for many populations. Fertility Fertility In order to simplify calculations, we count only In the number of females. Fertility, like survivorship, can be graphed as Fertility, a function of age and the resultant fertility curve is usually triangular or rectangular in shape. Gray Squirrel Gray 2.5 Fertility mx 2 1.5 1 0.5 0 0 1 2 3 4 Age Class 5 6 7 8 Mean number of daughters per female Normal Human Fertility Curve Normal 0.6 0.5 0.4 0.3 0.2 0.1 Reproductive 0 0 10 20 30 Pre-Reproductive Age classes 40 50 60 Post-Reproductive 70 80 Population Parameters Population The Gross Reproductive Rate is the The Gross average number of female offspring per female who lives an entire lifespan. This figure is 14.96 for the gray squirrel. figure GRR = ∑m x Conditional Survivorship, px Conditional From the lx column we can develop two parallel columns, which provide information on how survivorship and mortality rates change with age. The lx column is based on the probability, at birth, of surviving to a given age class. birth The px column, by contrast, is the ageagespecific probability of surviving to the next of age class. Conditional Survivorship, px Conditional That is, p2 tells us the probability that an individual, who has survived to the age of two, will survive to be three years old. These px values are critically important when we want to project future population growth as will become clear later. The Formula for px The l x +1 px = lx Conditional Probability of Death Conditional While px is the conditional probability of surviving to another age class, its opposite, qx surviving is the conditional probability of death in the next age interval. Therefore qx is simply 1 – px Mortality Curves Mortality Just as there are typical survivorship curves, Just there are also typical mortality curves. there One such curve for mammals is the U-shaped One curve, which emphasizes that death occurs mainly in the very young and the very old age classes. classes. Human Mortality Curve for Flu of 1918 Male versus Female Mortality Male There are often some important differences There between male and female qx curves. between curves. Males have a “bump” up in the sub-adult to Males early adult age classes. early Why? Male versus Female Mortality Male In many species, young males are forced In from their homes. from If they wish to breed they must find and join a If new group (male dispersal). new Once they find a new group, in order to breed Once they must often engage in real or ritualized combat with established males. Male versus Female Mortality Male Both dispersal and combat with other males leads to Both an increase in mortality in the sub-adult and early adult age classes. adult Mortality Curves for Impalas Mortality Mortality rate (1000x) q 1000 Females Males 800 600 400 200 0 0 1 3 6 8 12 Age in years 17 23 30 qx for Humans for Mortality rate (1000qx) 3 2.5 2 1.5 Males 1 Females 0.5 0 0 10 25 40 Age classes 55 70 85 Mortality versus age 1.2 Age specific Mortality, qx 1 0.8 0.6 0.4 0.2 0 0 20 40 60 Age Classes 80 100 H uman Mortality 2010 1.200 1.000 Males 0.600 Females 0.400 0.200 Age Class 99 95 - 89 85 - 79 75 65 - 69 59 55 - 49 45 - 39 35 25 15 - 19 9 5 - 1 29 0.000 < Mortal i ty 0.800 Doubling Time Doubling Derivation: Given: Nt = N0ert We have: Nt/N0 = ert For the population to double we have: 2 = Nt/N0 = ert Take the natural log of 2 and ert. Doubling Time Doubling We have: 0.693 = rt Solve for t, which is now doubling time, and Solve we have: we Doubling time = 0.693/r Doubling Time Doubling Doubling Time = 0.693/r Or if r is negative we have the time for the Or population to reach half its present size population = -0.693/-r Population Parameters Population The Net Reproductive Rate equals the The Net mean number of female offspring produced per female per generation. This is 1.19 for the squirrel population. the R0 = ∑ l x m x Life Tables Life As above, the finite rate of increase is the As ratio of populations at time t and time t+1. ratio N t +1 λ= Nt Life Tables Life Over the long term, the survivorship and Over fertility values determine the generation time, G, the net reproductive rate and the time, the intrinsic rate of increase, r. We can also show that: λ = er Life Tables Life Given: N t +1 λ= Nt and for time zero to time 1 λ = N1/N0 t And given: Nt = N0errt and and From t =0 to t =1, we have N1 = N0er which becomes N1/N0 = er We have λ = er We Life Tables Life However, over the short term, λ is determined However, by the present number of individuals in the age classes. age Thus λ fluctuates over time until a population Thus reaches a stable age distribution. At that stable At point it equals er. point Life Tables Life A stable age distribution is reached if the stable mortality and fertility values in the life table remain constant for several generations. remain In a stable age distribution, the proportion In proportion (not the numbers) of the population belonging to the different age classes becomes constant. constant. Life Tables Life That is, for example 20% of the population That belongs to one-year old individuals. This proportion remains a constant, though This the actual numbers can change if the population is growing. population Review: Relationships among Growth Parameters Parameters Ignoring age distributions: 1. Nt+1/Nt = λ 2. Nt+1/Nt = er 3. λ = er lln λ = r n 4. Finding the Intrinsic Rate of Increase (r) from a Life Table (r) To find r from life history data we use the To Euler equation Euler 1 = ∑lxmxe-rx The problem is that r is an exponent. With more than two age classes, this is With difficult to solve explicitly. difficult Continuous Growth Models Continuous Recall that in the continuous growth model Recall we use a differential equation: we dN/dt = rN Where r = the intrinsic rate of increase and Where the r=b-d Solved form: Solved Nt = N0 ert Finding the Intrinsic Rate of Increase (r) from a Life Table (r) Where G is generation time we have: G NG = N0errG This is growth per generation. Thus: NG/N0 = erG Recall that R0 is also growth per generation. is NG/N0 = erG = R0 . Take natural logs: Take lnR0 = rG lnR and r = lnRo/G and Finding the Intrinsic Rate of Increase (r) from a Life Table (r) There is a simple estimation for generation There time, from which r can be estimated. time, There is also an optimization function in Excel There that allows a direct solution for r. that Age Structure Age The age distribution is based on the The proportion belonging to each age class. proportion If we have data on the number of individuals If per age class, nx, we can calculate the per we proportions, symbolized as cx proportions, Cx = nx/N where N = ∑nx Age Structure Age Age Class nx cx 0 120 0.48 1 60 0.24 2 40 0.16 3 30 0.12 4 0 0 ∑ 250 1.00 Stable Age Distribution Stable Whenever survivorship and fertility are Whenever survivorship fertility constant for a sufficient period of time, the population moves to a unique stable age distribution, based on the particular distribution based combination of survivorship and fertility values of the population. Stable Age Distribution Stable No matter what the original age distribution, No the population moves to its unique stable age distribution unless the survivorship or fertility distribution schedules change. schedules Stable Age Distribution Stable This stable age distribution is predictable, based on This specific equations with r and lx the independent specific variables. This is an age structure diagram. This Age Structure Diagrams Stable Age Distribution Stable Again: When a population is in the stable age When distribution, the entire population and all of the age classes grow or decline at the same rate : rate λ = er And: r = lnλ Population Projections Population In this exercise we will In learn how to project a population with an age structure and also demonstrate how quickly a population moves to its stable age distribution. distribution. Population Projections Age, x Age, lx mx px qx lxmx 0 1.00 0 0.50 0.50 0 1 0.50 2.0 0.40 0.60 1.0 2 0.20 1.0 0.50 0.50 0.2 3 4 0.10 0 1.0 0 0 -- 1.00 -- 0.1 -- Sums GRR = 4.0 R0 = 1.3 1.3 Population Projections Age, x Age, nx t=0 Cx t=0 nx t=1 Cx t=1 0 200 1.00 200 200/300 = 200/300 0.67 0.67 1 0 0 100 100/300 = 0.33 2 0 0 0 0 3 4 0 0 0 0 0 0 0 0 Sums 200 1.00 300 1.00 Population Projections Population λ = N1/N0 = 300/200 = 1.50 Population Projections Age, x Age, nx t=1 Cx t=1 nx t=2 Cx t=2 0 200 0.67 240 0.63 1 100 0.33 100 0.26 2 0 0 40 0.11 3 4 0 0 0 -- 0 0 0 -- Sums 300 1.00 380 1.00 Population Projections Population N0 = (100*2) +(40*1) = 240 (100*2) λ = N1/N0 = 300/200 = 1.50 λ = N2/N1 = 380/300 = 1.27 Population Projections Age, x Age, nx t=2 Cx t=2 nx t=3 Cx t=3 0 240 0.63 300 0.625 1 100 0.26 120 0.25 2 40 0.11 40 0.083 3 4 0 0 0 0 20 0 0.042 0 Sums 380 1.00 480 1.00 Population Projections Population N0 = (120*2) +(40*1) + (20*1) = 300 (120*2) λ = N1/N0 = 300/200 = 1.50 λ = N2/N1 = 380/300 = 1.27 λ = N3/N2 = 480/380 = 1.26 480/380 λ = N4/N3 = 586/480 = 1.22 586/480 λ = 1.23 and r = 0.206 at stable age distribution 0.206 Population Projections Age, x Age, nx t=3 Cx t=3 Cx At Stable At Age Distribution 0 300 0.625 0.625 1 120 0.250 0.258 2 3 40 20 0.083 0.042 0.082 0.035 4 Sums 0 0 1.00 0 1.00 480 Sample Problem Sample Age, x lx mx px qx nx t=0 0 1.000 0 1000 1 0.400 1.0 300 2 0.200 3.0 100 3 0.150 1.0 60 4 0 0 0 nx t=1 Sample Problem Sample Find the gross reproductive rate, the net Find gross the reproductive rate, px and qx. Would r be reproductive predicted to be positive, negative or zero? predicted Given the nx column at age zero, project the population for one year. population Sample Problem Sample Age, x lx mx px qx nx t=0 0 1.000 0 0.40 0.60 1000 1 0.400 1.0 0.50 0.50 300 2 0.200 3.0 0.75 0.25 100 3 0.150 1.0 0 1.00 60 4 0 0 -- -- 0 nx t=1 Sample Problem Age, Age, x lx mx px qx nx t=0 nx t=1 lxmx 0 1.000 0 0.40 0.60 1000 925 0 1 0.400 1.0 0.50 0.50 300 400 0.40 2 0.200 3.0 0.75 0.25 100 150 0.60 3 0.150 1.0 0 1.00 60 75 0.15 4 0 0 -- -- 0 0 0 Sums GRR = 5.0 1460 1550 R0 = 1.15 Got it?? ...
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This note was uploaded on 01/23/2012 for the course BIOL/EVPP 307 taught by Professor Crerar during the Summer '11 term at George Mason.

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