BIOL 4120: Principles of Ecology Lecture 10: Population GrowthDafeng HuiOffice: Harned Hall 320Phone: 963-5777Email: dhui@tnstate.edu

World population

Population growthDefinition: how the number of individuals in a population increases or decreases with timeGrowth 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.

10.1 Population growth reflects the difference between rates of birth and deathModel developmentA 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

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-dPopulation growth is related to this intrinsic growth rate (r).

Exponential population growthEquations:1) dN/dt=rN (differential equation form)

2) N(t)=N(0) exp(rt) (exponential growth model)

Conditions:Initial population is smallNo food or resource limitation

An exampleReindeer, St. Paul, Alaska. Started in 1910 with only 4 males and 22 females

In 1940, there were nearly 2000

Whooping crane, an endangered species

recovered from near extinction in 1941

How to calculate r? Software, Excel (trendline)Aransas National Wildlife Refuge

Properties of exponential growth

r determines the shape of the growth.r=0, no change in population sizer0, increase in population size. Properties of exponential growthWidely used in biology

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, whats the population size?

N(5)=N(0)Exp(rt)=300*exp(0.5*5)=3655(495, 815, 1344, 2216, 3655) t=10, ?Prediction of population growth44524

10.2 Life tableLife 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

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;

How to construct a life table (cont.)?3. calculate dx, a measure of age-specific mortality

4. Calculate age-specific mortality rate, qx

10.3 Different types of life tableTwo typesCohort or dynamic life tableas the above gray squirrelTime-specific life table

Elf opine

10.4 Life tables provide data for mortality and survivorship curvesTable is better than words, but a graph is worth one thousand words.

Mortality curve and survivorship curve.

Mortality curves

Survivorship curvesLog scale for Y axis

Three basic types of survivorshipType 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. ConcaveMortality rate high in the beginning (r-selection)Oysters, Fish, many plant species (most trees)

10.5 Birthrate is age-specificCrude birthrate (demographers): # of birth over a period of time divided by population size at the beginning of the period*1000

Age-specific birthrates, bxMean # 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.

10.6 Birth rate and survivorship determine net reproductive rateFecundity 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 fecundityR0=1, >1 or

10.7 Project population growthGiven 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?

What do we need to project future population size change?

Calculate age-specific survivor rate: sx=1-qxbx is age-specific birthrate

How to construct a population projection table?

How to construct a population projection table?Population size (N) increases every year.

Lambda (finite multiplication rate): =N(t+1)/N(t).

Age distributionStable age distribution: by year 7, the proportion of each age group remain the same year after year.

Population is still growing.

Geometric growth vs exponential growthN(t)=N(0) tN(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.

Fig. 11.3

10.8 Stochastic processes can influence population dynamicsWhats 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).

StochasticityDemographic 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.

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 extinctionAllee effect, genetic drift, inbreeding (mating between relatives)Overgraze, only 8 in 1950

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

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 densityThere is less genetic variation in a small population and this may affect the populations ability to adapt to environmental change

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Geometric GrowthWhen generations do not overlap, growth can be modeled geometrically.

Nt = Not

Nt = Number of individuals at time t.No = Initial number of individuals. = Geometric rate of increase.t = Number of time intervals or generations.

Exponential GrowthContinuous 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.

Abundance, distribution & densitySpace and timeIn one place, population size varies. How to describe the population size change mathmatically (model), and to predict population size in the future.N=1000, in one month, 50 death, 100 birth, b=10/100=0.10, death rate=5/100=0.05, r=0.05, population growth rate=50 per 1000 per month.t, t+1N(t)=100,N(t+1)=105N(t+1)=N(t) +B(t)- D(t)B(t)=bN(t)*delta(t)D(t)=dN(t)*delta(t)

N0=100, in one year, 5 death, 10 birth, b=10/100=0.10, death rate=5/100=0.05, r=0.05, population growth rate=5 per year.First one: predict the rate of population size change with timeSecond one: predict the population size change with timeDerive the exponential growth model

islandR is the parameter that determining the shape of the growth form**R is the parameter that determining the shape of the growth form

Beer-larmert law, metabolic rate and body mass, growth, (decomposition)

**R is the parameter that determining the shape of the growth form**Population growth model provides a general picture of population size change without considering age structure and difference in birth and death rate at different ages. To obtain a better picture of mortality (survival) within a population, another approach is commonly used, life table.

Life table provide a summary of how survial and reproductive rates vary with the age of the organisms.

A life table is an age-specific account of mortality

It includes several columns. We will explain what these mean and how to construct a life table.

A cohort is a group of individuals born in the same period of time

x = age classesnx = the number of individuals from the original cohorts that are alive at the specified age (x)lx = the probability at birth of surviving to any given age (x)

dx = age-specific mortality = the difference between the number of individuals alive for any age class (nx) and the next older age class (nx + 1)

qx = age-specific mortality rate = the number of individuals that died in a given time interval (dx) divided by the number alive at the beginning of that interval (nx)

CohortX represent age in unit of yearnx represent the # of individuals from the original cohort that are alive at the specific age (x).Track all individuals**Pupae instars elf orpine Time specific life table??? is a snap-shot of a long-lived species. It usually starts out as a survivorship curve and they usually have a normalization to 1,000 individuals. Time-specific life tables work best for long-lived species, and they assume that there are equal birth rates among all age classes.

A cohort or dynamic life table is used to track the fate of a group of individuals born at a given timeThese individuals are followed from birth to death Modified version: A dynamic composite life table constructs a cohort from individuals born over several time periodsA time-specific life table is a distribution of age classes during a single time periodSeveral assumptions are made in this approachEach age class was sampled in proportion to its numbers in the populationAge-specific mortality rates (and birthrates) are constant over time

Many animals (e.g., insects) live only one breeding season. Because generations do not overlap, all individuals belong to the same age classnx is measured by estimating the population size several times over its annual seasonThe life table is useful for studying several areas of plant demographySeedling mortality and survivalPopulation dynamics of perennial plants marked as seedlingsLife cycles of annual plants

**Life table data are generally presented as:A mortality curve that plots the qx column against age (x)A survivorship curve that plots the lx column against age (x)Life tables and curves are based on data from one population at a specific time and under certain environmental conditionsRosette Sedum: Crassullceae**Good for comparison

Example of survivorship curves Next slide for:I. deer, sheep, human, convexII: squire and adult birds, linear, not change with ageIII. Plants, fish, young bird, concave There are three idealized types of survivorship curvesType I: typical of populations in which individuals have long life spans, survival rate is high throughout the life span with heavy mortality at the endHumans, other mammals, some plantsType II: survival rates do not vary with ageAdult birds, rodents, reptiles, perennial plantsType III: mortality rates are extremely high in early lifeFish, many invertebrates, and plants

Some show different types at different stages, like bird in the previous slide

**Life table can also be used to estimate the popultaiton size change

Next is the reproduction

The crude birthrate is expressed as births per 1000 population per unit timeOnly females give birthBirthrate of females generally varies with ageBirthrate is better expressed as the number of births per female of age xbx = mean number of females born to a female in each age groupContinuing with the gray squirrel example = gross reproductive rate = the average number of female offspring born to a female over her lifetime

Assume that the female will survive and live a full life

Total number of offsprings produced not only depends on the age-specific birthrate, but also the survivorship.Lxbx: mean number of females born in each age group adjusted for survisorshipR0=1, replace itselfA fecundity table combines the survivorship (lx) with the age-specific birthrates (bx)lxbx = mean number of females born in each age group, adjusted for survivorship

R0 = net reproductive rate = the average number of females that will be produced during a lifetime by a newborn femaleR0 = 1; on average, females will replace themselves in the populationR0 < 1; females are not replacing themselves in the populationR0 > 1; females are more than replacing themselves in the population

**For simplicity, age-specific mortality (qx) is converted to age-specific survival (sx)sx = 1 qxA population projection table can be constructed using sx and bx

Only females hereWe will use the life and fecundity table values of the gray squirrel to illustrate a hypothetical population of squirrels introduced into an unoccupied forestFemales are only used to construct the population projection table

An age distribution for each successive year can be calculated from a population projection tableAge distribution is the proportion of individuals in the various age classes for any one yearA stable age distribution is attained when the proportions of each age group in the population remain the same year after year (even though N(t) increases)An estimate of population growth can be derived from a population projection table = finite multiplication rate = N(t +1)/N(t)Once the population reaches a stable age distribution, the value of remains constant > 1.0 indicates a population that is growing < 1.0 indicates a population in decline= 1.0 indicates a stable population size through timeThe population projection table demonstrates two concepts of population growth (estimated population growth rate) is a function of sx and bxThe constant rate of population increase from year to year and the stable age distribution are results of sx and bx that are constant through timeIf does not vary (under conditions of stable age distribution), population size in the future can be projectedN(t) = N(0) tDescribes a pattern of population growth similar to the exponential growth modelGeometric population growth = N(t) = N(0) tFiniteExponential population growth = N(t) = N(0)ertContinuous = er or r = ln

Plant population size change of phlox

Population dynamics represent the combined outcome of many individual probabilitiesAge-specific survival rates (sx) represent the probability that a female of that age will survive to the next age classThis reality has led ecologists to develop probabilistic or stochastic models of population growth to account for these variationsDemographic stochasticity is the random (stochastic) variations in birth and death rates from year to yearThe variations in d and b cause populations to deviate from the predictions based on deterministic modelsEnvironmental stochasticity is the random variations in the environment or the occurrence of natural disastersThese events directly influence d and b

Individual survival and reproductionAverage birth and death rates caused by environmental factorsUnder the following conditions, a population can become so small that it declines toward extinction:When deaths exceed births, populations declineR0 becomes less than 1.0r becomes negativeUnder the following conditions, a population can become so small that it declines toward extinction:Extreme environmental eventsSevere shortage of resourcesUnder the following conditions, a population can become so small that it declines toward extinction:Introduction of a novel predator, competitor, or parasite (disease)Habitat loss (due to human activities)Small population size

Allee effect: decline of either reproduction or survive under conditions of small population. Reasons: difficult to find a mate for animal, produce less seeds for plants, breakdown social structure for animals (hard to defend predators etc).

Small populations are more susceptible to both demographic and environmental stochasticityWhen only a few individuals make up a population, the fate of each individual can be crucial to population survivalOver large territories, it can be impossible to find a mate (large cats)Chemical signals will not be intercepted (insects)Pollination is unlikely (plants)

Inbreeding cause some rare, recessive, deleterious genes become expressed. decease in fertility, slow growth, even deaths.

**Figure 10.13*