Population Growth: Exponential vs. Logistic - Prof. Thomas W. Schoener, Study notes of Ecology and Environment

Download Population Growth: Exponential vs. Logistic - Prof. Thomas W. Schoener and more Study notes Ecology and Environment in PDF only on Docsity! EVE101 Lecture 6: Part I: Page 1 ©2008 Catherine A. Toft Lecture 6 Outline Single-species population growth Readings: Molles Chapters 10-11 Part I: I. Introduction: elements of population growth II. Types of population growth • A. Exponential (geometric) -- Density independent PG • B. Logistic -- Density dependent PG • C. Examples of exponential and logistic population growth • D. Mechanisms of density dependence III. Simple models of population growth • A. Background • B. Exponential • C. Logistic PG and density dependence • D. Allee effect (reverse density dependence) Part II: IV. Demography • A. Age structured population growth • B. Life tables and estimating population growth rates V. Reproductive strategies revised January 17, 2008 Molles 4th edition pages EVE101 Lecture 6: Part I: Page 2 ©2008 Catherine A. Toft Lecture 6 - Single species population growth - Part I Molles Chapter 11 I. Introduction: Elements of population growth We are now going to take a step "back" and stop focusing on the individual. Instead, we will focus on the population, a group of individuals, and look at the sum of all individuals' behavior, physiology, survival, and reproduction collectively. A population is a group of individuals and population "growth" is the change in numbers of individuals with time at a given place. This change in numbers of individuals with time is the result of the collective births and deaths there and immigration to and emigration from that place. This definition allows us the flexibility to talk about positive "growth" (increase in numbers of individuals with time) or negative "growth" (= decrease in numbers of individuals with time). Population growth is also referred to as population "behavior" or "dynamics". First, we consider a population to be mostly closed, with only a little immigration or emigration, so we can concentrate on the birth and death terms of population growth. In other words, the contribution of birth and death of individuals will outweigh that of individuals coming into (immigration) or leaving (emigration) the population (from an unknown, "outside" world). In more advanced courses in ecology, you will learn about the concept of metapopulation (Ch. 10.2), which is a group of local populations connected by some degree of immigration and emigration. In studying the metapopulation, we focus away from the local birth and death processes (assuming that we understand them well) and focus instead on the movement of individuals between populations (immigration and emigration). When do we need to the metapopulation scale (also called landscape scale, Molles Ch. 21)? We need to consider a larger spatial scale when there is marked spatial heterogeneity in birth and death rates. For example, within a species, a local population might be increasing at a fast rate (births>> deaths) in one area and another local population might be declining (births < deaths) somewhere else. To understand the dynamics of the entire (global) population, we have to consider all the local populations separately and consider dispersal among them. If there is not much spatial heterogeneity, we can learn what we need to know about a certain species by studying a fairly average local population--and in doing so we can focus n birth and death processes instead of immigration and emigration. We will concentrate on local processes in EVE 101. Everything we've talked about in the unit on individual ecology leads to population growth. Assimilating energy from food, getting mates, surviving predation, elements, and so on, all lead to the positive term in population growth, that is to having offspring. Alternatively, old age and death from hazards during feeding and reproduction (predation, death from elements etc.) all lead to the negative term in population growth. We can ask then: What is the rate at which new individuals are produced and what is the rate at which individuals die? EVE101 Lecture 6: Part I: Page 5 ©2008 Catherine A. Toft Examples of longer-term exponential population growth include populations that were depressed for some reason and are now increasing exponentially in population size, such as that of the Scotch pine, Pinus sylvestrus, which was depressed after the last ice age (Fig. 10.6; see also Fig. 10.17) . And then there is the infamous example of the human population, Homo sapiens: The human population The world population of humans has approximated exponential growth for much of its history, showing no decrease in per capita population growth rate with increasing density of humans world-wide (at least not so far). The human population growth over a long period of time is actually not a perfect exponential because the per capita growth rate is not a constant. Typically, the per capita population growth rate has increased with increasing density because of advances in human culture aiding use of resources and developing human knowledge, for example in areas such as medicine. Thus the human population has actually grown at a rate faster than exponential. Ecologists and demographers (in particular Raymond Pearl) thought that the human population had reached the inflection point of the logistic curve (the place where the total population growth rate is maximum, i.e., where the slope of the logisitic curve is at its maximum) in the middle of this century. Based on this assumption, they projected a steady-state size of the human population to be about 2-3 billion. Unfortunately they were wrong! The human population passed the 3 billion mark in 1960, with no suggestion whatsoever of an inflection. In Winter 2008 there are 6.64 billion humans on the planet. Fig. 11.26 p. 269 in Molles The human population still seems to be growing approximately exponentially, with a current doubling time of just over 40 years. Lately, the per capita population growth has decreased a little, as a consequence of education and improving quality of life world wide. However, improving quality of life means a higher per capita consumption of resources, so although the population's growth rate in units of numbers of humans added per unit time is less, the per capita impact on the planet's resources remains if anything greater. Links for further information include: • The U.S. Census Bureau: http://www.census.gov/ipc/www/popclockworld.html • Museum of Natural History in Paris France has an outstanding web site exhibit on the growth of the human population. If you go to one site, I recommend this one: http://www.popexpo.net/english.html We will see that most organisms DO NOT withhold their reproduction so as not to overrun the environment (including humans!). Virtually all organisms will reproduce exponentially (at slower or faster rates) until they begin to encounter some limit imposed from outside, either by EVE101 Lecture 6: Part I: Page 6 ©2008 Catherine A. Toft resources or predation & disease. As we proceed, we will learn how this regulation of populations takes place. First and foremost, be aware that self-regulating individuals in a population cannot be favored by natural selection "for the good of the species" or for the "good" of the environment. This can never happen. Why? Definition of population regulation: 1. the upper limit to population size imposed from outside this population ; or 2. any process by which a populations is limited in size; 3. (strictest of all) the mechanisms by which the per capita population growth rate is density-dependent (decreases with increasing size of the population). B. Logistic (sigmoidal) population growth Lecture handout, Molles Ch. 11.2-3 pp. 259-66. Logistic growth, in contrast, is population growth that slows down as the population size gets large. When plotted as numbers of individuals against time, logistic population growth appears as a S-shaped or "sigmoidal" curve. This type of population growth is almost exponential at small enough population sizes. Then as population density increases, the per-individual contribution to births and deaths gets smaller, and rate of population growth slows. Eventually, the population's growth slows to zero, and population size is unchanging with time, because births = deaths. We term this maximum population size the population's carrying capacity. This is density-dependent population growth. That is, the per-capita (individual) population growth rate depends on the population density. In the plot of N against time, the higher the density, the slower the rate of growth, hence the "flatter" slope to the population "curve" in the phase after early exponential-type population growth. Eventually, the growth rate of the population decreases to zero, and we observe the flat or asymptotic region of the population size plot. We will look at more plots of per capita population growth when we use the population growth model (below). The reason for the density-dependence in births and death rates is virtually always some form of resource limitation: there is not enough to go around. Thus when we use the term "density dependence" in talking about per capita rates of population growth, we mean that per capita population growth rate decreases with increasing density. Examples of logistic growth: [Lecture; Molles Figs. 11.8-11.15] We can follow new introductions (deer in Tasmania, yeast or paramecium in a laboratory culture; Figs. 11.9-10) or populations depressed by another form of regulation (Kaibab deer after predators were eliminated; African buffalo after rinderpest was eliminated, Fig. 11.12) over an entire region and see the way the population eventually levels off because resources become limiting. The per capita population growth rate is density-dependent, meaning that as EVE101 Lecture 6: Part I: Page 7 ©2008 Catherine A. Toft population density increases, the per capita population growth rate decreases. How this is accomplished varies, but in general the birth rate decreases or the death rate increases, or both. As you can imagine, density-dependent processes are not pretty. Sometimes the population overshoots the resources there and crashes. Later we will try to understand these dynamics. C. Mechanisms for density-dependent population growth First, let's make a list of the factors that can produce density-dependent effects on population growth: (The list is from The Science of Ecology, by Ehrlich and Roughgarden). • 1. Resource depletion and fecundity • a. The amount of food available determines the growth rate and body size of an individual, and the number of offspring (tissue devoted to reproduction) is related to body size. Most common in organisms with indeterminate growth: many insects and other invertebrates, fish, amphibians, reptiles, i.e., grow after they reach a reproductive size • b. The amount of food determines how much you can put into reproduction at any given time. If not enough, (or less) you produce fewer eggs, or offspring, or you wait a year. • 2. Resource depletion and survival. In these cases, individuals starve to death, or poor nutrition makes it more likely to die of predation, exposure or disease. Here the principle of allocation comes in: the more time foraging (animals) or more physiological effort invested in growth or resource uptake, the less effort an individual will devote to defense against predators. • 3. Space depletion. Animals increase territory size when they face lower food availability, and this may mean not enough room for everyone to have a territory. If you don't have a territory, you don't reproduce, a situation common in many birds. In sessile organisms, individuals simply take up space until none is left. • 4. Increased time in social interaction. Individuals spend more time defending territories as population size increases, and less time foraging and mating; this situation is common in many fish. • 5. Intraspecific predation or other harassment. Many insects, frogs, salamanders, etc., will eat the eggs of others of their own species. The more dense the eggs and the adults etc., the higher proportion will be eaten. (Cannibalism). • 6. Density-dependent interspecific predation: For example, predators may use search images. The more common a particular prey species, predators eat it disproportionately. So in that population of prey, predation is greater at higher densities. EVE101 Lecture 6: Part I: Page 10 ©2008 Catherine A. Toft "compounding" the production of new individuals in the same way that your bank account compounds interest, giving you interest on the interest you earned, etc. This is why this form of population growth (r is constant) is called logarithmic population growth. When you reduce the whole picture to rates, in particular instantaneous rates, you reduce time t to a very small number, as t approaches 0, and use the science of calculus to predict population behavior. C. Logistic growth. Ch. 11.2-3 1. Basic Model: (lecture handout p. 18) We will use a really simple minded, totally descriptive but also very well studied equation to get a logistic curve. [Pianka's book Evolutionary Ecology 5th edition has a very clear explanation on pg. 187 on how the two scientists Verhulst and Pearl originally justified and derived this model.] First we postulate some population size N = K, which will be the upper limit of population density based on resource abundance. We don't care about the type of resources, or the animal's behavior, we just pick a value K that describes the population's upper limit. This is the asymptote of the sigmoidal (logistic) curve and it is called the carrying capacity of the environment. ["Carrying capacity" comes from Paul Erlington, a wildlife biologist; in this and other fields of ecology, it was apparent that population size had some upper limit, based resources available, particularly during the worst time of the year during which the population goes through a "bottleneck" (i.e., winter). As N approaches K, then N/K approaches 1, and so 1 - N/K approaches 0, and (K-N)/K approaches 0. In other words, these are all the possible ways to view the process of the population's size approaching carrying capacity so that we can understand the effect on the population's growth rate. where ractual = rmax (1 - N/K) and ra = bo (1 - N/K) - do (1 - N/K) so that when N is high, and close to K, ractual will become closer to zero, as birth and death rates get closer to being equal. The idea is that the per capita rate of growth of the population will slow down as the population nears K, for various reasons we will talk about in a minute. Now we can see that rmax (your book uses rm, m means maximum) is the same thing as the "plain" r in the exponential growth model. This rmax is the biotic potential, because the per capita population EVE101 Lecture 6: Part I: Page 11 ©2008 Catherine A. Toft growth rate is maximum when individuals are rare and resources are as yet unlimited. What the VP model does is cause the population to grow at less than rmax, the closer it gets to the carrying capacity. In other words, “r-actual” is less than r-max the closer you get to K. Let's look now at what this density dependence looks like on a graph, and how density dependence in r related to density dependence in birth and death rates considered individually. 2. Density Dependence: a look in more detail (lecture handout pp 19) Population Density, N This is a picture of just one possibility: both births and deaths are dependent on density. Other scenarios are: births only, deaths only, and everything in between. No matter what the exact slopes of the birth and death rate lines, there is a place where they cross, where b=d and r=0--that point is the carrying capacity, K. The biotic potential, rmax, is the greatest difference between births and deaths, where bmax > dmin. as pictured on the next graph: Lecture handout, Molles Figs. 11.14-15, p. 261) The VP logistic model is entirely linear, as you can see from this graph. The actual r is equal to the maximum r, the biotic potential, only when N is very small, close to zero, where there is a a EVE101 Lecture 6: Part I: Page 12 ©2008 Catherine A. Toft virtually no density dependent negative effects, such as would occur when individuals in the population competed for limited resources. The actual r then is zero when the population size is at its maximum (above); this is pictured in the graph. This graph illustrates what is meant by an equilibrium. Try this yourself: when N is below K (N<K) then ractual is what? Will N increase or decrease? Likewise, when N>K, ractual is what? Will N then increase or decrease? We see that when N<K, ra >0 and when N>K, ra <0 so that N always returns to K. This property of returning to the same point, the same population size, no matter what, is known as an equilibrium. If we look at density dependence in the population's growth rate as a whole, instead of the per individual's population growth rate, we get this (lecture handout p. 20) and compare this to the plot of N against time in the logistic:: Unlike the graph of the per capita rate, the graph of the populations' density dependent growth shows more about what happens in the second graph on the population size and its trajectory in time. We can see that the maximum rate of growth of the population, dN/dt, occurs where the graph of the logistic curve is steepest, or at N = K/2. D. The Allee effect or reverse density dependence (lecture handout) Under the simple Verhulst-Pearl logistic equation, r of the simple exponential model is actually rmax, because the population is always at the maximum growth rate during exponential population growth. In density-dependent (logistic) population growth, ractual decreases as a linear function of population density N, as we can see from the above graphs. a a