Evolution & Ecology Lab
Evolution & Ecology Lab BIOL 3113
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Date Created: 10/12/15
Population Ecology Population Ecology 7 There are three broad fields within ecology behavioral ecology population ecology and community ecology Population ecology is that eld within ecology that deals with the biotic and abiotic factors affecting the growth of biological populations Growth of a population is determined by four factors births and immigration contributing to population growth and deaths and emigration contributing to population decreases births immigration deaths emigration PopGrowthRate 39 PopSize The intrinsic rate of increase refers to the rate of growth of a population under ideal conditions However a population s intrinsic rate of increase is seldom realized because the environment is rarely if ever providing ideal T 1 conditions for population growth Instead populations are limited by both biotic and abiotic factors in the environment These limits mean that particular environments limit population growth to some maximum sustainable size where the number of births in the population is equal to the number of deaths This maximum sustainable size is called the carrying capacity Actual population growth will tend to overshoot the carrying capacity and once the population size exceeds what the environment may support the population size will fall When the population falls below the carrying capacity births will exceed deaths until the population once again shoots over the carrying capacity to then cycle back below the carrying capacity again The population will therefore cycle around the carrying capacity see Figure 1 Various factors determine the carrying capacity for a given environment These factors prevent the population from achieving its intrinsic rate of increase and may be abiotic or biotic see Table 1 Limiting abiotic factors may be the temperature range in a particular environment or water salinity or pH Idealized owering Figure 1 Populat m growth times in plants for example may be compromised in I 7 n r those environments with a short growing season Carrying capaclty I I Optimal population growth in an aquatic microorganism may be limited by salinity or pH in a particular pond 39 39 Many of the factors that 11m1t population growth I however are due to the actions and frequency of other 1 living organisms in the environment Examples of these I biotic factors limiting population growth include Mean POPUIatlon predation availability of prey and competition either among species or between members of the same species The classic example of population regulation feedback between predator and prey is the lynx Lynx canadensis and the snowshoe hare Lepus americanus Shortly following increases in the size of the prey snowshoe hare population the size of the predator lynx population Temperature or water pH Population quotquot39 Actual population Time increases Increasing lynx populations will eventually reduce the size of the hare population and then the lynx population will in turn decrease and the cycle will begin again Understanding those factors that limit population size is also critically important for understanding evolution as well Evolution by natural selection relies on the capacity for populations to produce more offspring than may be supported by the environment This excess reproduction means that individuals will compete among one another for limited resources Any traits that allow one type to compete for resources better compared to other types in the population will result in those characteristics that aided in this particular type s competitive ability spreading in the population This spread of traits that contribute to survival and reproduction is evolution by natural selection Statistics Statistics Statistics are an interrelated class of mathematical techniques related to the collection description and analysis of data when data are taken by sampling a population Data consist of particular values of asome variable in the population Variables can be any feature that differs from one individual to another in the population and are measured in some quantitative way Knowledge of statistics is essential in many elds of science especially ecology and evolutionary biology It is virtually impossible to draw any scienti c conclusions about ecological and evolutionary processes without some statistical analyses The distinction between a population and a sample is critical in understanding statistics A population represents the totality of all observations about which inferences are made while the sample is a subset of the larger population Anytime when data cannot be collected from an entire population then inferences must be made about the population based on data collected from a sample of the population Sampling is the first step in any statistical analysis When making inferences about a population one must determine if the sample is representative of the population as a whole Sample size re ects the number of observations in a sample and is typically symbolized by the letter 71 The larger the sample size the more representative the sample will be of the population For example say you are looking at the differences in ower color between two groups of orchids The total population size of each group is far too large to record the color of each and every ower so you must sample from each group Sampling only a handful of individuals from each group runs the risk of collecting data that are not representative of the population because of the chance collection of data from an individual with a relatively uncommon ower color If for instance you collected only three owers from each group and for one group two of those owers were white while the third was pink you may conclude that the population has 23 white owers However white owers may actually represent only 10 of the population and you just happened to collect your data in an area where white owers are more common in the population In this case because you sampled only a few individuals from a limited part of the population your assumptions about the population are biased by your small sample size Generally there is no hardandfast Figure 1 Frequency Histogram of Tail Length rule as to how large one s sample size must be although there is an area of statistics known as power analysis 18 that allows for the determination of a particular sample size to detect a particular difference between two 147 groups Just remember that the larger one s sample size the better estimates one will have of key parameters of the population and thus the more power one will have to detect comparatively 8 small differences between groups Frequency H D All of the statistical tools we will 4 learn in this course are dependent on a very particular frequency distribution The frequency 0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Lengthcm distribution of data re ects the way that particular data points are spread among members of the population We will mainly concern ourselves with those data whose distribution follows a normal or parametric distribution Normally distributed data have the most frequently observed values in the middle range and the frequency of low and high values decreases in roughly equal measure relative to the middle values When the frequency of various measure are plotted out the in a frequency histogram normally distributed data take on a curved shape with a hump in the middle and two tails This characteristic shape of a normally distributed data set is known as a bell shaped curve see Figure l Distributions of data where the most frequently observed data are those values that are either large or small coupled with a long either decreasing or increasing tail end of the distribution are said to be skewed distributions Descriptive statistics are statistical metrics used to describe the central tendency and dispersion of a population The central tendency of some variable in the population refers to those values around which the population tends to cluster The population mean u is the sum of all the values in the population for some variable divided by the population size Dispersion represents the degree to which the values for some variable are spread around the central tendency The principle measure of dispersion is the population variance 02 Measures of central tendency and dispersion are estimated for the population from a sample The median of a sample is that value that divides the variables in half with half of the variables being larger than the median and half being smaller than the median The median is essentially that value expressed by the typical sample The sample mean or average 2 represents the sum of the all the values in a sample divided by the sample size 2 i1 n An important measure of dispersion in a sample is the sample variance s2 The sample variance describes just how much spread there is in the data around a sample mean Sample variance is the sum of the squared differences between the values and the sample mean divided by the sample size minus one The sample size minus one is the degrees of freedom df the data possess Differences between individual values and the mean are squared simply because we are interested in the absolute difference and not the sign of the difference positive or negative Another measure of dispersion is the standard deviation s which is simply the square root of the sample variance Analytical statistics go beyond simply describing the population and are used to test hypotheses Key in analytical statistics is probability Analytical statistics basically seeks to measure data against the probability of drawing those data from the population at random If for example differences in two groups are highly unlikely to be the result of random draw from the population then we then have a statistical basis to invoke other explanations for the observed differences This form of analytical statistics is also known as classical statistics In classical statistics hypotheses are tested against a null hypothesis An example of a null hypothesis is the idea that two population means are equal Sample means can be compared and the probability of drawing those sample means from the same population determined One very important statistical test is the ANOVA Analysis 2f E ance ANOVA tests whether 2 or more sample means could be drawn from populations with the same mean by looking at how variation is distributed between and within groups Started by the evolutionary theorist and statistician R A Fisher ANOVA like variance relies on a measure known as the sum of squares which adds the deviations of each data point from the mean for the population or group For an experimental design with multiple treatments the total sum of squares is equal to the sum of squares within treatments also known as the error sum of squares plus the sum of squares among treatments SSToml SSWithin SSAmong The within group sum of squares is the sum of the degrees of freedom times the variance across all groups where K is the number of groups SSWthin 1 39 S xi if j1 j1 i1 The sum of squares among groups is the square of the difference of the group mean minus the grand mean times the sample size summed across all groups K 7 7 2 SSAnong EnX xGerd 1 The grand mean is the mean of all samples across all groups The total degrees of freedom Nl is the sum of the among groups degrees of freedom Kl and the within groups degrees of freedom NK where N is the sample size across all groups dfToml N 1K 1N K Once the sum of squares between and within groups has been determined one needs to calculate the F statistic to determine the degree to which the variance s distributed among versus within groups The Fstatistic is computed as the mean square error MSE divided by the mean square between groups MSB Mean squares are simply the sum of squares divided by the degree of freedom SSWV SSW MSB demg K 1 MSE SSWmy SSW df Within N K The greater the numerator ie MSB compared to the denominator ie MSE the greater the Fstatistic will be The Fstatistic can be converted into a Pvalue by comparing the results to those on a table The table provides a critical value of F where P 005 The Pvalue represents the probability that these particular data could be drawn from a population with the same mean Pvalues are compared against those for a critical value of F where the probability of rejecting the null when really the samples are truly from populations with the same mean is less than 5 This probability is known as type I error or a In ecology and evolutionary biology CC is arbitrarily set at 005 however in other fields that arguably require one to me more certain they are not rejecting the null hypothesis inappropriately it may be lower A statistically significant difference is determined when the Pvalue is less than an or of 005 A higher Fstatistic means that a greater proportion of the variance in the data is distributed between rather than within groups and this will allow one to reject a null hypothesis that the two groups are sampled from a population with the same mean Rejecting the null hypothesis means that we can tentatively accept our alternative hypothesis and conclude there is likely some biologically meaningful difference in the two groups Another commonly used statistical analysis is regression Figure 1 0 0 tW variab39es Regression seeks to establish a functional relationship 20 between two variables In regression analysis data are 18 plotted against one another such that each value of the 16 i variable on the xaxis corresponds to a value for the other j 39 variable on the yaxis see gure 2 The relationship gt 10 l between two variables is de ned by the following equation 8 l 6 x y a bx 4 x 2 7 if The linear relationship between variables y and X are 0 0 5 10 15 20 25 de ned by the point where the line intersects the yaxis the yintercept a and the slope of the line b In statistical analyses the slope of the regression line b is also called the regression coef cient For any value of X the corresponding value for y can be determined with knowledge of the yintercept and slope of the relationship between the two variables Regression analysis is essentially about def1ning this linear relationship between two variables The two variables in a regression analysis are the independent variable the variable on the xaxis and the dependent variable the variable on the yaxis Real data seldom if ever end up producing a perfectly straight plot like that presented in gure 1 Regression analysis determines a line of regression through the data points that minimizes deviations from that line The line of regression created by minimizing the deviations of the data is called the least squares linear regression line The deviations of the data points from a line of regression are called residuals An actual regression analysis looks much more like gure 2 where there is some residual variation some deviation of the data from a Figure 2 Sample regression ofvariam V on Name x straight line of regression Regression analysis seeks to 30 explain the variation in the dependent variable y in terms of variation in the independent variable X The key measure in regression analysis is the coefficient of determination or R2 R2 will be a value between 0 and l and it tells you how much of the variation in the dependent variable y is explained by variation in the independent variable X For example an R2 of 091 means that 91 of the variation in the dependent variable y is explained by variation in the independent variable If R2 were to equal 0 one then that would mean that you could perfectly predict x without error the values of the dependent variables given the values for the independent variables The remainder of the variation is that variation not explained by the regression line It is the residual variation If R2 equals 091 then that means that 91 of the variation in the dependent variable is explained by the variation in the independent variable x but there is 9 of the variation in the dependent variable that is not explained by the independent variable Residual variation can be thought of as noise in the relationship It represents all those variables other than the independent variable that can affect the dependent variable A greater spread of data points around a line of regression means that the residual variation is also greater and thus R2 is lower Regression is used in instances where the researcher is looking for some indication of cause and effect It is therefore generally useful in controlled experiments where the treatment variable is the independent variable and treatments are randomized In this instance one can obtain evidence of a cause and effect relationship However a similar statistical metric called correlation is used in those cases where all the researcher is after is a measure of how two variables vary together with one another without any assumption about cause and effect Correlation simply examines how one variable changes relative to change is another and as such there are no dependent and independent variables per se Correlation is very useful in ecology and evolutionary studies when one wants to know how various characteristics of organisms are interrelated and unlike regression it does not require controlled experimental designs to implement Evolution and Ecology BIOL3 l 13 January 8 2007 Lecture Notes Instructor Herman L Mays Jr PhD Ecology and evolution de nitions Ecology 7 Ecology is the study of interactions between biological organisms and their environment Biotic environment 7 Living environment or the environment consisting of both conspecific and heterospecific biological species includes predators prey parasites mates rivals etc Abiotic environment 7 Nonliving 39 or the 39 con 139 tino of the physical non living world Abiotic interactions include how organisms respond to factors such as temperature rainfall salinity soil pH ambient light etc Evolution 7 Biological evolution is ultimately a genetic theory based on the concept of common descent with modi cation Evolution therefore is simply change in biological populations over time with time measured in generations Examples of change over time in biology that are not examples of evolutionary change Growth of the organism from juvenile to adult 7 This is not evolutionary change because individual organisms are undergoing change not populations and change is occurring not across generations but rather within the lifetime of an individual Removal of some trait in the population such as the dehorning of cattle 7 This is also not evolutionary change Horns can be removed from a population of cattle and this does represent change in the population over time the population was with homs at time t and without horns at time tl but dehomed cattle do not pass this trait on from parents to offspring therefore the change does not span generations Ecology and evolutionary biology are two scientific disciplines that are intimately linked together as each in uence the course of the other Changes in the 39 39 39 Jquot a r r 39 quot affects their evolutionary history and a population s prior evolutionary history can dictate how they respond to their biotic and abiotic environment Introduction to scienti c investigation Hypothesis testing Scientific investigation begins with observations of the natural world A potential explanation for a particular phenomenon in nature is a hypothesis The word hypothesis comes from the ancient Greek hypotithenai meaning quotto put underquot or quotto supposequot and can be based on some general theory basic physical laws previous observations or maybe even an educated guess One thing however all scientific hypotheses must have in common are predictions In order for one s hypothesis to explain some aspect of nature it must predict that nature should behave or appear in a very particular way Only if the predictions of a hypothesis are actually how nature behaves will one say the hypothesis is supported If nature behaves in a way that contradicts the predictions made by ones hypothesis then the hypothesis is rejected see figure 1 Notice that scienti c hypotheses are either supported or rejected based on whether or not the predictions of a hypothesis match the available observations in nature In science hypotheses are not proven or disproved in an absolute sense but rather they are only tentatively accepted or rejected on the basis of the available material evidence Hypotheses in science are specific towards very particular questions while theories are broad overarching concepts consisting of many interrelated hypotheses Figure 1 Scientific method Experiment versus observation Observation from nature Hypotheses may be tested through i Birds sing Gimp39ex s ngs either experimental or observational I methods In an experiment the l investigator controls variables whereas in observational studies the investigator I simply tests hypotheses based on ot s Birds sing to attract a mate Predictions Obsewatlons 0f eglsnng Varlatlon m Birds should sing more during the breeding season nature The experimental approach can Birds who sing more should attract more mates Once mated birds should sing less be a powerful tool in establishing links between cause and effect however it is I not feasible in every instance Much of Tes nature can IIOt be manipulated by the Measure predictions against what occurs in nature experimenter and therefore hypotheses Record song behavior at different times of year and for mated versus bachelor males in these cases must be tested by ecord the number of mates between birds who sing more versus less observational study Support or Reject Hypothesis lBased on whether the available data match one39s predictions the hypothesis is supported or rejected Variati on Understanding biology especially biology from the standpoint of ecology and evolutionary history is ultimately about explaining variation Why does one population do things differently than another Why does one person contract an infectious disease while another does not Why is one species found one place and not another Why do some individuals in the population behave one way while others behave differently Dealing with variation requires one to use statistics Statistics is especially designed to describe analyze and explain variation in data Variation in a population can have all sorts of causes and not all of those causes are necessarily due to biology In humans males tend to be large than females but there are some who are shorter than the average woman and some women taller than the average man Because populations vary and variation can be caused by any number of factors sampling of the population is vital in scientific investigation This means scientific experiments must have replication that is the same experimental procedure must be repeated independently across different individuals Taking only a single individual from say each of two groups like male and female and comparing them may not be adequate in detecting differences
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