Week 2: Population Growth Simulations

Transcription

1 Week 2: Population Growth Simulations Before Lab: Remember to read, print out and bring this to your lab. You will be having your first Pre-Lab Quiz on this material. What to Bring to Lab: Please bring your Gotelli book, A Primer of Ecology, with you to lab and a calculator. Meet in your assigned Lab room this week and then we ll head over to the computer lab from there. Background: Introduction: Ecologists are interested in understanding the dynamics of populations and in predicting how populations might respond to changing environmental conditions. In biology, a population is defined as a group of individuals from a single species that interact. Ecologists have developed simple equations that will predict the state of a population at some point in the future as a first step toward understanding growth and fluctuations of populations. Although these models are conceptually simple, it will become clear shortly that even these simple population models can lead to complex behavior. Although there are many population growth models used in ecology, they can be classified into two general categories; density-independent models and density-dependent models. In density-independent models, species do not limit their own growth as is the case in density-dependent models. This limit on growth is the result of finite resources that deplete as a population grows. In these densitydependent models, time lags may also be present, meaning that population growth is based on the population at some point in the past, rather than the current population. This can lead to very interesting population dynamics. We will explore the behavior of different population models using a program called Populus. (Details on how to use Populus are included later in the lab.) We will specifically examine the dynamics of density-independent and density-dependent population growth for continuous and discrete models. 1

2 We will also incorporate time lags into the density-dependent models and study how this affects population dynamics. Notation. The initial size of a population at time t is indicated by N 0. The size of the population in the future is represented by N t. Many factors go into predicting the size of a population in the future such as the number of individuals that were born and the number that died as well as the number the moved into the population from another population (immigration) and the number that left (emigration). For most examples we do, we will be assuming that immigration (I) and emigration (E) cancel each other out and therefore will be ignored. N t = N 0 + B D + I E Continuous vs. Discrete Population Models As we ve heard in lecture, there are 2 types of models that are used to predict population size in the future, continuous models and discrete models. Continuous models are used to predict populations of organisms that reproduce all the time like bacteria or humans. These organisms do not undergo large pulses in reproduction like many organisms do because their environment is relatively constant in relation to their lifespan. In contrast, organisms like mosquitoes will have pulses of reproduction based on rainfall and wildebeest will all reproduce in the same season to limit predation on the young. For these types of organisms, discrete models may be better at capturing changes within a population. Density independent vs. dependent growth In addition to knowing whether a population of interest is growing continuously or discretely, you also need to know whether the growth is density-dependent or density- independent. Density-independent models assume that the number of individuals in a population has no bearing on the rate of the population growth. This can only be true in very specific circumstances and is rarely, if ever, seen in nature for extended periods of time. Populations assumed to be growing in a densityindependent fashion will have what is called exponential growth and can theoretically grow forever. This requires that the population is not limited by any resource; they have more than everything they need to survive or there is a constant input of resources that exceeds population requirements. This means that the growth rate of the population can remain constant. This type of growth has been observed for short periods of time when, for instance, a species invades into a new range without competitors or back into one from which they have previously been excluded. 2

3 Remember: Density-independent growth can happen in populations that reproduce both continuously or discretely The equations to represent this phenomenon are as follows: Continuous Growth: N t = N 0 e rt Using this equation we can determine the size of a population at some time in the future (N t ) by knowing the initial population (N 0 ), the intrinsic growth rate (r) and the time that we want to project over (t). Discrete Growth: N t = λ t N 0 You ll notice that in this equation we have the same two symbols for initial and future population size as well as for time however, we ve added λ. This represents the finite growth rate which measures the proportional change in population size between 2 time periods. Therefore: λ= N t / N 0 Additionally, λ can be related to r by using the following equation: λ = e r The other more realistic scenario that population growth models use is one of finite resources. This is called density-dependent growth or logistic growth. It assumes that individuals within a population will compete for resources once they become scarce. In this case, growth rate (r) can change as time passes and the population grows. Think of some examples of finite resources that could limit population growth. Since ecosystems in the real world have finite resources, there will always be a maximum number of individuals that they can sustain. This number is called the carrying capacity (K). It is this carrying capacity that leads to density dependence in population growth. In today s lab we will examine how populations act as they approach this K value and the affect that different breeding strategies (continuous vs. seasonal) have on the behavior of the system. Just as in exponential growth scenarios, populations experiencing logistic growth can reproduce continuously or discretely. In a continuously growing population we measure instantaneous changes in population or growth rate using the following formula: 3

4 As previously defined, r is the intrinsic growth rate, N is the number of individuals and K is the carrying capacity. In discretely reproducing populations this equation is used: All of the notation should be familiar although the subscript on the N has changed. This is simply using the population at time t (N t ) to predict the population size at 1 time step past t (N t+1 ). They could very easily be replaced with N 0 and N t. Now let s see how well you understand the concepts described above. For each of the following populations described, identify whether the population is growing continuously or discretely and whether it is a densitydependent or density-independent system. 1) Earthworms were introduced into North America by the Jamestown settlers in 1607 for use in gardens and are able to reproduce yearround in un-frozen soil. How would you describe the population growth of earthworms living in Virginia in 1608? 2) Polar bears are being affected by global warming because of a lengthening of the ice-free period in the Arctic. This limits the time they are able to hunt on the ice (this is how they catch most of their prey) and the quality of the hunting habitat. They need this time to build up fat stores for the winter when they hibernate and give birth. How would you describe the current population growth of polar bears in the Arctic? 3) Dandelions are an annual plant found all over the world. They reproduce only once per growing season by releasing hundreds of airborne seeds. A single dandelion was experimentally planted in an abandoned agricultural field in New Hampshire from which all other plants and animals will be excluded for the length of the study. How would you describe the population growth of dandelions in that field over the next few years? 4

5 4) The diet of a population of sea otters living off the Oregon coast consists primarily of sea urchins. Since otters have a notoriously high metabolism, which comes from living in very cold water year round, they require a LOT of these urchins to survive. In terms of their reproduction, the climate of the Pacific Coast is fairly consistent year round, which causes there to be no distinct breeding season for sea otters. How would you describe the population growth of sea otters living off the Oregon Coast? Time lags In models with density dependence, time lags could arise between the size of a population and the resultant effects on growth rates. Imagine a population of butterflies where the current population size is near Carrying Capacity (K) when they lay eggs. The effect of that reproductive event will not be felt by the butterflies until the caterpillars complete their metamorphosis at which time they will be competing for the same resources as their parents. The time it takes for that effect to be felt by the adult population from the time the eggs were laid is the time lag (τ) in this situation. In this equation, τ represents the lag that exists between when population size is measured and when the population actually increases. In contrast to continuous models, discrete population models have a time lag of 1 built into them. Why? How to Use Populus: Log into the computers using the default username and password on the log-in screen. To open Populus, look for the icon on the desktop. It may look different from machine to machine. If not installed go to and save the Windows version of Populus 5.4 with JVM into the C:\temp folder on the hard drive. 5

6 Once open, there are a couple of options. Pull down the Model menu and you will see all the different model types available in Populus. Today we will only be using the Single-Species set of models. Begin by opening the Density Independent model. You will notice that it has options for continuous or discrete models, as well as a number of plot types. It also allows you to change r or λ depending on what type of model you are working in. You can also set initial population size and length of run. If you forget what a particular variable represents, the help file can explain each. You can access these help files by clicking on the light bulb at the top of each menu. You may also display several plots on the same graph by checking the box under the A, B, C and D tabs. In order to more accurately make predictions from the plots, you can add gridlines. This can be done in the Options menu of the plot window. You can also increase or decrease the resolution of those gridlines. Exercises: Complete these before leaving lab and hand into your TA 1) A colony of Daphnia (a small aquatic crustacean) is being raised in ideal conditions for use in the BCOR 12 laboratory. There are 150 students enrolled in the course and each student needs 4 Daphnia in order to perform an experiment. The scientific literature reports the intrinsic growth rate (r) of the particular species of Daphnia used in that 6

7 lab to be 0.25 day -1. Also, under ideal conditions, Daphnia reproduce continuously. With an initial population of 10, how long will it take to reach the necessary number of Daphnia? (Assume the time scale is in days) What does an r value of 0.25 represent in words? 2) In 1992, a pride of 5 lions was moved from South Africa to Europe in an attempt to re-colonize their ancient range. The new territory in Europe had virtually unlimited resources for the lions. The lions adapted to reproduce in the spring time and each lioness typically had one cub per year. After the first year when a group of researchers went to check on the pride they saw that two cubs had been born and survived their first year*. Describe the type of growth this population of lions is experiencing. What is the finite growth rate for this population? If this current growth rate remains constant, in what year will the lions reach a sustainable population size of 30 *? What if only one cub survived that first year? How long would it take to achieve a sustainable population? (Try plotting them both on the same graph.) What would the intrinsic growth rate be for the original population if they were to start breeding year-round? * Although this is based on fossil record, obviously no such re-introduction program was ever carried out. But wouldn t that be crazy!? 7

8 3) The critically endangered Sunburst Orchid (Solarplexia caliente) has recently been located in Williston, VT in an abandoned field scheduled to be the site of a new Flowers R Us Mega-Store. The population there currently consists of 50 individuals with a 1.1 proportional rate of increase. Under the current plans, only 10 individuals would remain after construction. However, Flowers R Us, because of the persistence of the UVM Horticulture Club, has agreed to take one of the following measures to conserve the orchid: 1) Reduce the Mega-Store to a Super-Store, leaving 45 of the orchids to rebuild the population. 2) Keep the Mega-Store but fertilize the remaining orchids with Selenium which has been shown to increase the finite growth rate to 1.2. Model the original population, and the populations resulting from each of the two conservation options over a 10 year period on the same graph. Being a member of the Horticulture Club, which action would you recommend Flowers R Us take? Why? Now look at the population projections for 30 years. Would your recommendations to Flowers R Us change? Now let s try using the Density Dependent Model 8

9 Now you can see that there are three models to choose from in this menu; continuous logistic, lagged logistic and discrete logistic. What are the differences between them? (Thought Question) You can also set initial population (N o ), carrying capacity (K), intrinsic growth rate (r) and lag time (T) in addition to the length of time that will be graphed. Exercises: 1) Examine how carrying capacity (K) affects initial growth rate of a population. Does it affect populations with continuous growth differently from those with seasonal reproductive cycles? 2) With the following initial conditions, examine the differences between the Discrete and Continuous plots when the r value is increased from 0 to 4. Be sure to view the N vs. t and ln(n) vs. t plots. N o = 5 K = 400 Run Time = 50 a) Describe the population dynamics you observe as you increase r in the continuous model. At low values of r (r < 1) At medium values of r (1 < r < 2.5) At high values of r (r > 3) 9

10 b) Describe the population dynamics you observe as you increase r in the discrete model? At low values of r (r < 1) At medium values of r (1 < r < 2.5) At high values of r (r > 3) Why is there little change in the continuously growing population as r increases, but the in the discretely growing population, there are lots of changes in growth pattern? 3) The purple-toed tree frog (Arborea-tadpolium viola) although threatened, has a very large geographic range which spans from Northern New England to Central America. Through extensive research, different populations of this frog have been identified as having very different reproductive characteristics. The timing of reproduction is entirely dependent on tree cover, meaning that they will only reproduce when the leaves are on the trees and shade the forest floor. Population 1 is found in where, although broad-leafed, the majority of the forest canopy is evergreen. In contrast, population 2 is located in Southern where the majority of the tree species are deciduous. Although the frogs habitats are very different, it was discovered that each would support exactly 500 frogs. Amazing! 10

11 In a fifty year study which began in 1955, both populations were found to have the same intrinsic population growth rate of And in another shocking coincidence, both populations at the beginning of the study contained only 12 individuals. a) Determine the size of each population in 1995 (40 years). b) Determine the size of each population in c) Suppose that in 1955, methods for counting frogs were inexact and in actuality there were 13 frogs in each population. How would this affect the population sizes in 1995? d) What if there were 11 frogs? e) Measuring carrying capacity is very difficult. What if the scientists were off by 10 and the actual K is 510? f) Or 520? 11

12 g) What does this tell you about the ability of scientists to accurately predict populations with discontinuous reproduction? Finally, suppose that two additional populations (populations 3 and 4) of Purple-toed Tree frogs were studied which were identical to populations 1 and 2 except that their intrinsic growth rate was Answer questions (a) through ( f ) from above under these new conditions. a) b) c) d) e) f) 12

13 g) How did altering the intrinsic growth rate alter the predictability of the different frog populations? See you next week! 13