Name Problem Set - Population Dynamics Perhaps the most successful conjunction of math and biology has been in the study of ecology and evolution. This problem set focuses on one

aspect of population biology: population size. We will try to get an understanding, via mathematics, of how populations change over time. THE EXPONENTIAL EQUATION Let's think of a population of individuals that reproduces asexually via fission. It could be yeast or bacteria, it doesn't really matter what it is, just what it does. We start with 1 individual at time (t) = 0. 1. If after every hour, each individual divides into two individuals, list how many individuals there will be at each time interval: time, t (h) 0 1 2 3 4 5 6 #indiv (N) But tables like this would get very unwieldy after even one day of growth. So, let's try to simplify the relationship with a function that tells us how many individuals there are at any interval. 2. Complete the following function by giving 2 an exponent (HINT: it will be a variable): N₁= 2- Throughout this problem set, N will refer to population size and N, is population at time t. We're not going to worry about what the units of time are, however. Now we will try to make this a more general equation that can be applied to populations with different growth rates. It is possible to represent 2 as any combination of numbers, e.g., : 2=1+1,=√ =√√2² 148 So 74 we can begin to make the function in question 2 more general by substituting a new combination of values for 2. Because it is an exponential relationship, we'll try using e and to make it equal to 2, we will raise it to the power of 0.693. For this population: N₁ = 2¹ = e0.693t We're still not at the general equation, however, because we haven't really changed the equation, just its form. It's still just 2¹. Now we said that this population doubles every hour. But not every population does. So, which part of this equation is responsible for making it double every hour? 0.693. e is a constant, so if we want the population's rate of change to be something different from doubling every hour, we should change the exponent. To make the exponent a variable, let's replace 0.693 (which applies just to this population) with r: rt N₁ = e't So now depending on the population, we can have r = whatever is appropriate: 0.693 in these yeast or 0.214 in some flies or 0.014 in elephants, for example. But there's one more thing! Not all populations begin at the same size. If you have a sexually reproducing population, you definitely need more than one individual, and populations can start with many more than one. So, where should we add the value for the population at time 0? Remember, that in this example No = 1, therefore, our final, general equation becomes: N₁ = Noert This equation demonstrates the size of a population as a function of time and is called the exponential equation. (NOTE: This is the "integrated form" of the equation. We will also look at the differential form later.) No is a constant it is a constant for any one population, but it will differ among populations. It is the variable time in whatever units are appropriate. We'll just consider it unitless. r is perhaps the most important component of the equation, and it determines the rate at which a population grows over time. ris called the intrinsic rate of natural increase, and it is a constant for each population that represents essentially how quickly a population can grow when there are no limitations. POPULATION DEMOGRAPHICS - integrated form Population size will change based on the number of (3.) minus the number of (4.) (Ignore immigration and emigration.) r is essentially the first answer minus the second answer per each individual under conditions that are optimal for population growth. 5. Give an example of a biological factor that would result in a difference in r between two populations. All else being equal, there would be a difference in r between two populations, if one population had a higher/larger than the other. Now this gives us the growth of a population over time. But what we are really interested in is taking any given point in time and predicting how quickly the population will grow at that point in time. So let's make a differential equation relative to time. No 6. Differentiate the exponential equation with respect to time. The result is below, just show how to get there. (HINT: before differentiating, start by taking the natural log of both sides. If you get to the end and it N doesn't look like the final answer, then it might help to remember that: ert 7. First, let's plot the integrated form of the exponential equation on the axes below. Plot three different functions, each with N(0) = 1 but with three different r's: 0.1, 0.2 and 0.5. You do not need to show your work, but of course you must label the functions: 1000 dN(t) dt means. It Now what does this tell us about population size and how it changes? Keep in mind what means the change in population size over a given time. Most of the time we think about dN (t) as dt⇒0. That is, at any instant, how quickly the population is growing. dt 900 800 700 dN, dt 600 €500 400 300 200 100 0 0 =rN₁ - differentiated form 10 = 20 30 40 t 50 60 70 80 8. OK, so what do these three curves tell you about r? As r increases, how do the curves change? dN₁ dt 9. Another approach to problem 6 above is to ask what happens to the differential equation that you derived in question 6.) as r increases? (Think about 10. What does the exponential equation (in its integrated/non-differential form) predict about population size after a long time? Solve for the limit as t→∞. Show your work. 11. Does your answer to 10 make biological sense? Explain. THE LOGISTIC EQUATION So let's agree that the exponential equation is unrealistic in at least one way: there are no populations that are infinitely large. Real populations can generally be grouped into two major groups: (1) those that are unstable and fluctuate wildly and (2) those that are relatively stable. The ones that fluctuate wildly tend to have booms and busts. They increase rapidly and then quickly die back. Locusts that swarm and destroy crops are a good example. The populations that are more stable are the ones we're going to discuss now. The idea is that in a given habitat, each population has a carrying capacity. This is sort of a ceiling of stable population size. That is, there is a certain amount of room and food and water and other resources in a habitat and these combine to determine the number of individuals that can live there sustainably. That population size that is the maximum sustainable population size is the carrying capacity, symbolized as K. dN₁ t = dt So, how do make the model better fit reality and incorporate K into our equation? We want the population size not to change when it is equal to K. In other words, when N = K, 0. Because we are talking about changes, we should start with the differential form of the equation. of K by adding to the differential version of the exponential equation: K-N₁ So, we can incorporate the effect dN₁ =rN₁ dt K Set N = K and see what you get for This equation is called the logistic equation and when the integrated form is graphed it produces the logistic growth curve. Note that this form here is the differentiated form. dN₁ t dt The version above is the more common way to see this equation, but if you want to see how many individuals there are in a population at any time, you would need the integrated form: Ket - integrated form N₁ Note that we use No here. It's a constant just like in the exponential equation. A) r= 0.1, B) r = 0.1, C) r = 0.2, 12. On the graph to the right, plot the integrated form of the logistic No= 1, No= 250, No=1, e + equation (using whatever means you wish) for the three given populations. Label them, of course (3 pts.): K = 500 K = 500 K = 500 K-No No - differentiated form N(t) 550 500 450 400 350 300 250 200 150 100 50 0 0 10 20 30 40 50 60 70 80 90 100 t