Population Growth: Age-Structured Models and Density-Independent Growth, Lab Reports of Ecology and Environment

Download Population Growth: Age-Structured Models and Density-Independent Growth and more Lab Reports Ecology and Environment in PDF only on Docsity! Lab 4 – Population Growth: p. 1 GENERAL ECOLOGY Lab 4 Population Growth: Age-structure, Density-independence, Density-dependence, and Demographic Stochasticity [NOTE: Labs this week will meet in 611 Carr (The Zoology Computer Teaching Facility). You should go there for lab and not 120 Carr).] [Credits: portions of this exercise were borrowed from EEMB 120 taught at the UC Santa Barbara. I am grateful to Dr. S.J. Holbrook for providing this material.] INTRODUCTION Much of ecology is concerned with changes in the abundance of a population over time. A considerable amount of time in lecture will be spent discussing models of population growth, and the models discussed below provide the traditional foundation for many of the models used in current studies of population dynamics. This week’s lab is designed to gives students a hands-on opportunity to see how these models behave and to begin to develop an intuition about why they behave the way they do. We have four specific objectives: 1) To explore the dynamics of age-structured populations in the absence of intraspecific competition. 2) To use the age-structured models as a vehicle to evaluate simpler models of geometric growth (discrete time) and exponential growth (continuous time), which ignore age-structure. 3) To incorporate density-dependence (e.g., intraspecific competition) into continuous time models and to compare the behavior of the exponential and logistic models. We will also explore the influence of time lags. 4) To evaluate the effects of demographic stochasticity on population growth, variance in population size, and the probability of extinction. To accomplish these goals, we primarily will rely upon a program (distributed for free) called Populus (which was developed by Don Alstad at the University of Minnesota), which we have loaded onto computers in the Zoology Computer Teaching Facility. The following exercises are written specifically for Populus, although it is also possible to do many of the problems with nothing more than pen and paper (and perhaps a calculator or a spreadsheet program). We encourage you to download Populus and use it on your own computers. Notes about downloading Populus are Lab 4 – Population Growth: p. 2 available on the course web page. Populus is a simple (albeit a bit antiquated) program that can help you develop a good intuition about population dynamics and the behavior of these common ecological models. REMINDERS ABOUT POPULUS Populus is a DOS program that can be run under Windows for Workgroups, Windows 95 (and presumably 98) and Windows NT. There are some simple instructions and help files that come with the program. Keep in mind a few useful keystrokes: F1 = help (hitting F1 twice will take you to a main help menu: once takes you to a context dependent help page). F2 = Introductory material. If you are in one of the modules, F2 will provide you with some background on the subject matter (this is instructive, but keep in mind that some of the notation may differ somewhat from what is used in lecture or your textbooks). F4 = turns on/off the “plot last data” feature. This can be useful to help you compare the results of one model run with the results of a subsequent run (i.e., both results will be superimposed on a single figure). Alt-G = turns on/off gridlines. Alt-O = options for running Populus (e.g., printer settings). Getting started: 1) Run Populus by double-clicking on the short-cut (there should be a short-cut to Populus on your computer). If it’s not on the “desktop”, your TA will provide instruction. 2) Hit return to move through the screens until you get to the “Main menu”: select “Population Growth” and hit enter. 3) Here you will see four options. We will use all four of these modules, which we discuss below. Before we get started, please note that these four models differ in their degree of complexity. Despite the differences, they all have one essential feature in common: they all function by keeping track of two counteracting processes; inputs to the population in the form of births or immigration and outputs from the population in the form of deaths or emigration. They each also make assumptions -- you should bear these assumptions in mind as you work through the exercises. What are the assumptions? One assumption that we'll make for all of the exercises is that immigration and emigration rates are very small and therefore have little impact on population dynamics. As a result, we will only consider effects of births and deaths. Lab 4 – Population Growth: p. 5 look at the examples below should make the distinction between these two types of density-independent models a little more understandable. Let's start with the discrete case and examine an annual plant that germinates from a seed, reproduces and dies in a single year. If we use years as our discrete time intervals, you can see that for all of our individuals, births and deaths occur in one discrete time interval, i.e. a year. For the sake of this example, let's assume that each plant produces two seeds. If we start our population off with 1 plant in year 0, we should have 2 plants by year 1 (1x2). By the end of year 2, each of these plants will have given rise to two offspring so we should have 4 plants (2x2). By the end of year 3, we will have 8 plants (4x2) and so on. Remember, for simplicity we are assuming that the adults in each generation die after producing their two offspring so there is no overlap in generations. We can describe this pattern mathematically using the following equation: Nt+1 = R0 Nt where: Nt = the number of plants at time t Nt+1 = the number of plants in the next time period R0 = Net Reproductive Rate Let's look at a few graphical examples. Using Populus, we can plot the size of our population through time based on values of R0 and N0 that we specify. Please note that in our plant example and in Populus, R0 = λ because each is defined on the same time scale (i.e., a generation in this case lasts one time step). This need not be the case, and in most situations the two will not be equal because they are defined with respect to different time steps (e.g., generations vs. years). Many species don't conform to the assumptions that we made in using the discrete model. These species have overlapping generations, i.e. the parents don't die immediately after giving birth, and reproduction and death occur continuously over time rather than in discrete time intervals. Although we can use the discrete time model to examine populations with overlapping generations, it is often more appropriate to use a different model. In order to work in continuous time, we have to switch from using the number of births and deaths that occur in a block of time to a system that uses instantaneous birth and death rates. Instead of describing how our population is changing from one discrete time interval to the next, now we need to describe how it is changing at any one specific instant in time (This requires some simple notation borrowed from calculus. What we are referring to is a derivative). We can model the rate of change using the following equation: dN/dt = rN where: dN/dt = the instantaneous change in population size Lab 4 – Population Growth: p. 6 N the number of individuals in the population r = b - d = per capita population growth rate b = instantaneous birth rate d = instantaneous death rate Note that r, like λ is also density independent. Also note that r=ln(λ) and that they can each be discussed in terms of per capita growth. I.e., r=.1 can be interpreted as a 10% growth rate, and λ=1.1 can also be thought of as a 10% growth rate (but see below: they do not yield the same projected population size). One important thing to keep in mind when working with continuous time models is that dN/dt represents the amount of population increase or decrease, not the actual population size. In this model, if r < 0 then dN/dt is a negative number and our population will shrink (Nt+1 < Nt), if r > 0 then dN/dt is positive and our population will grow (Nt+1 > Nt) and if r = 0 then dN/dt also equals 0 and our population will remain the same size (Nt+1 = Nt). What does a value of r = 0 mean in terms of b and d? Here are your tasks with Populus: 1) Return to the Population Growth menu and select Density-Independent Growth (hit enter). 2) Set the parameter values: N0=10, λ=1.1, t=10. 3) Run it. What happens? Does this look like population growth from the age structured model? Why? 4) Hit F4 and enter to return to the parameter menu. 5) Now, select "Continuous" and set N0=10, r=0.1, t=10; Run it. 6) Both this run and the previous one used growth rates corresponding to 10% growth. Do they give the same results? Which one grew faster? Why? 7) Rerun the discrete version (to put the run back in "memory") 8) Select "Continuous" and set N0=10, r=0.95 (i.e., less than 10% growth), t=10; Run it. 9) Now, how do the discrete and continuous versions compare? How much do these populations increase every time step? 10) Let's change r and see what happens. Set r=.1 and run it. Then set r=.2 and run it. 11) Compare these two runs. Toggle through the screens until you see the display with four figures. 12) Why is ln(N) vs. time a straight line? Compare the two slopes? Which is greater? Why? What is the slope? 13) Go back to the parameter menu and change the initial population size from 10 to 20. How will this affect the plots. E.g., will it affect the plot of log(N) vs. time? How? Run it and see for yourself. Lab 4 – Population Growth: p. 7 LOGISTIC (DENSITY-DEPENDENT) POPULATION GROWTH Density-independent models make the often unreasonable assumption that population density does not affect per capita growth rate. As a result, populations that are increasing do so forever and without bound. In reality, as the number of individuals in our population increases, the environment becomes more crowded and resources begin to be depleted. Intuitively, it makes sense that at some point all populations must stop growing and level off at some maximum number of individuals. It also seems to make sense that this maximum number should somehow be tied into the amount of resources in the environment. Making more resources available should allow the population to grow, while removing critical resources should cause the population to shrink. The logistic model is a density-dependent model of population growth which assumes that an increase in population size negatively affects the per capita rate of population growth, dN/Ndt. This decrease in dN/Ndt with increases in N may be due to a decrease in the birth rate, b, an increase in the death rate, d, or both. The end result, however, is that the value of dN/Ndt at any one instant in time depends on the population density at that same instant. We can describe this model of population growth using the following equation: dN/dt = rN(K-N)/K where: dN/dt = the instantaneous change in population size N the number of individuals in the population r = (b - d) = per capita rate of population growth K environmental carrying capacity We can think of K as a measure of the maximum population size the environment can susustain. If we subtract N from K, we get a measure of the "unused resources". By dividing this value by K, we can calculate the proportion of the environment that is still available. So, if N is almost equal to 0, then (K-N)/K is almost equal to I and the model simplifies to N/dt = rN, which is the continuous form of the density-independent model. If N is almost equal to K, then (K-N)IK is almost equal to 0 and dN/dt is almost equal to 0. When N exactly equals K, (K-N)/K equals 0 and the size of our population no longer changes; dN/dt = 0. We can use Populus to examine the shape of the logistic curve. If you haven't already done so, hit the escape key twice to return to the Population Growth menu. Use the arrow keys to select Logistic Population Growth and hit enter. Hit enter again to skip straight to the data entry window (Remember, if you are interested, you can return to the introductory text at any time from within the