BIOL 4120: Principles of Ecology Lecture 10: Population Growth Dafeng Hui Office: Harned Hall 320 Phone: 963-5777 Email: dhui@tnstate.edu Abundance, distribution & density Space and time In one place, population size varies. How to describe the population size change mathmatically (model), and to predict population size in the future.

World population

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.

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 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+1 N(t)=100, N(t+1)=105 N(t+1)=N(t) +B(t)- D(t) B(t)=bN(t)*delta(t) D(t)=dN(t)*delta(t)

Population growth is related to this intrinsic growth rate (r). 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). 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.

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 First one: predict the rate of population size change with time Second one: predict the population size change with time Derive the exponential growth model

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

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 4/21/2017 Properties of exponential growth R is the parameter that determining the shape of the growth form

Properties of exponential growth 4/21/2017 Properties of exponential growth R is the parameter that determining the shape of the growth form Beer-larmert law, metabolic rate and body mass, growth, (decomposition) 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.

Prediction of population growth 4/21/2017 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, ? R is the parameter that determining the shape of the growth form 44524 12

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 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 classes nx = 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)

How to construct a life table? 4/21/2017 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; Cohort X represent age in unit of year nx represent the # of individuals from the original cohort that are alive at the specific age (x). Track all individuals

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 table 4/21/2017 10.3 Different types of life table Two types Cohort or dynamic life table as the above gray squirrel Time-specific life table 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 time These individuals are followed from birth to death Modified version: A dynamic composite life table constructs a cohort from individuals born over several time periods A time-specific life table is a distribution of age classes during a single time period Several assumptions are made in this approach Each age class was sampled in proportion to its numbers in the population Age-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 class nx is measured by estimating the population size several times over its annual season The life table is useful for studying several areas of plant demography Seedling mortality and survival Population dynamics of perennial plants marked as seedlings Life cycles of annual plants Elf opine

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. 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 conditions

4/21/2017 Mortality curves Rosette Sedum: Crassullceae

Survivorship curves Log scale for Y axis Good for comparison Example of survivorship curves Next slide for: I. deer, sheep, human, convex II: squire and adult birds, linear, not change with age III. Plants, fish, young bird, concave Log scale for Y axis

Three basic types of survivorship 4/21/2017 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) There are three idealized types of survivorship curves Type I: typical of populations in which individuals have long life spans, survival rate is high throughout the life span with heavy mortality at the end Humans, other mammals, some plants Type II: survival rates do not vary with age Adult birds, rodents, reptiles, perennial plants Type III: mortality rates are extremely high in early life Fish, many invertebrates, and plants Some show different types at different stages, like bird in the previous slide

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. 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 time Only females give birth Birthrate of females generally varies with age Birthrate is better expressed as the number of births per female of age x bx = mean number of females born to a female in each age group Continuing 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

10.6 Birth rate and survivorship determine net reproductive rate 4/21/2017 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.) 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 survisorship R0=1, replace itself A 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 female R0 = 1; on average, females will replace themselves in the population R0 < 1; females are not replacing themselves in the population R0 > 1; females are more than replacing themselves in the population R0: depends on survivorship and fecundity R0=1, >1 or <1

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?

What do we need to project future population size change? For simplicity, age-specific mortality (qx) is converted to age-specific survival (sx) sx = 1 – qx A population projection table can be constructed using sx and bx Calculate age-specific survivor rate: sx=1-qx bx is age-specific birthrate

How to construct a population projection table? Only females here We will use the life and fecundity table values of the gray squirrel to illustrate a hypothetical population of squirrels introduced into an unoccupied forest Females are only used to construct the 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 distribution An age distribution for each successive year can be calculated from a population projection table Age distribution is the proportion of individuals in the various age classes for any one year A 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) Stable 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 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. 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 time The population projection table demonstrates two concepts of population growth  (estimated population growth rate) is a function of sx and bx The constant rate of population increase from year to year and the stable age distribution are results of sx and bx that are constant through time If  does not vary (under conditions of stable age distribution), population size in the future can be projected N(t) = N(0) t Describes a pattern of population growth similar to the exponential growth model Geometric population growth = N(t) = N(0) t Finite Exponential population growth = N(t) = N(0)ert Continuous  = er or r = ln

Fig. 11.3 Plant population size change of phlox

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

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. Individual survival and reproduction Average birth and death rates caused by environmental factors Under the following conditions, a population can become so small that it declines toward extinction: When deaths exceed births, populations decline R0 becomes less than 1.0 r becomes negative Extreme environmental events Severe shortage of resources Introduction of a novel predator, competitor, or parasite (disease) Habitat loss (due to human activities) Small population size

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 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 stochasticity When only a few individuals make up a population, the fate of each individual can be crucial to population survival Over 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. Overgraze, only 8 in 1950 Allee effect, genetic drift, inbreeding (mating between relatives)

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

Figure 10.13

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

END

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.

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.