Exponential Population Growth

7 Key excerpts on "Exponential Population Growth"

• 

  eBook - PDF

  Modelling and Quantitative Methods in Fisheries

  • Malcolm Haddon(Author)
  • 2011(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  The initial, accelerat-ing growth exhibited by such populations is described as exponential growth. If birth and death rates are constant at all population sizes, it implies a constant proportional increase in population size each time period. The rate at which a population’s size changes can be described by dN dt b d N rN = -( ) ≡ (2.1) In this differential equation (Lotka, 1925, p. 101), dN / dt translates as the rate of change of the population size N relative to time t , b is the birth rate, d is the death rate, and the symbol ≡ denotes “is equivalent to.” In ecology textbooks, where r = ( b – d ), r is often termed the intrinsic rate of increase, 22 Modelling and Quantitative Methods in Fisheries © 2011 by Taylor & Francis Group, LLC the “instantaneous rate of population growth,” or even the “per-capita rate of population growth” (Krebs, 1985, p. 212). The point to note is that the per capita growth rate ( b – d ), or r , is a constant and is assumed independent of population size N . Not surprisingly, this type of growth is termed density independent. Good summaries of population growth can be found in many ecological texts (e.g., Begon and Mortimer, 1986; Caughley, 1977; Caughley and Sinclair, 1994; Christiansen and Fenchel, 1977; Krebs, 1985; Pianka, 1974). There is a possible source of confusion over the term rate . A constant birth rate at all population sizes does not mean that the same absolute number of offspring will be produced at all population sizes, but rather that the popula-tion increases by the same proportion at all sizes. Thus, a growth rate of 0.1 implies a population of 100 will increase by 10, while a population of 1,000 will increase by 100. As population size increases, the absolute number by which the population grows in each time interval will also increase, but the proportional increase will stay constant. Equation 2.1 is a differential equation relating the rate of change of popula-tion size to time.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Biology Today

  An Issues Approach

  • Eli Minkoff, Pamela Baker(Authors)
  • 2003(Publication Date)
  • Garland Science

    (Publisher)

  Because food and other resources increase more slowly than population, any pop-ulation growing exponentially will outstrip its food supply and other resources, including the available space in its habitat. Clearly, no popula-tion can continue growing exponentially. After growing exponentially for a while, a population usually follows a pattern called logistic growth . Logistic growth. Logistic growth has been demonstrated in all types of experimental populations, and we know of no biological reason why this would not also apply to human populations. Logistic growth can be modeled mathematically by the equation dN / dT = r N ( K – N )/ K In this equation, K is a new quantity called the carrying capacity of the environment. Like N , K is a number, not a rate. K refers to the maximum size of the population that can be sustained indefinitely by the environ-ment. As population size approaches this carrying capacity, the popula-tion growth (symbolized by dN / dT ) slows down, and when N = K the pop-ulation growth is zero ( Figures 9.3 and 9.4). When the population CONNECTIONS CHAPTER 11 Figure 9.4 A variety of logistic growth curves. population (billions) population (billions) time time time (A) Shape of a logistic growth curve K population (billions) r = 0.4 r = 0.2 r = 0.1 r = 0.05 r = 0.025 r = 0.01 time (D) Different rates of increase ( r ) (C) Different values of K (B) Different initial population sizes ( N 0 ) 8 N 0 = 5 billion N 0 = 1 billion N 0 = 100 million N 0 = 10 million N 0 = 1 million 10 8 6 4 2 0 K = 10 billion K = 8 billion K = 6 billion K = 4 billion population size ( N ) K Chapter 9: The Population Explosion reaches whatever carrying capacity is imposed by the environment (see Figure 9.4C), the birth and death rates are balanced. If the population size overshoots K , then a population crash follows in which deaths out-number births and the population size declines to the carrying capacity.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Complex Population Dynamics

  A Theoretical/Empirical Synthesis

  • Peter Turchin(Author)
  • 2013(Publication Date)
  • Princeton University Press

    (Publisher)

  Environ-ment here refers to all environmental influences affecting vital rates of individuals: abiotic factors, the degree of interspecific crowding, and densities of all other species in the community that could interact with the focal species. The derivation of the exponential law for the case when all individ-uals in the population are absolutely identical (in particular, there is no age, sex, size, or genetic structure) and reproduce continuously is very simple. We start by expressing population change as the balance between births and deaths (postulate 1), and then change to per capita rates using postulate 2: dN dt = B − D = bN − dN = b − dN = r 0 N (2.1) 2.2 EXPONENTIAL GROWTH 21 where B and D are the total birth and death rates, b and d are the per capita rates, N is the total number of individuals in the popula-tion, and r 0 is the per capita rate of population change. There are no immigration/emigration terms, because I assumed that the population is closed. I have also glossed over the fact that individuals typically come in discrete packages, while N is a continuous variable. This is not a serious problem, since we can think of N as the expected number of individuals per unit of area in the population at time t . (Another approach is to think of N as biomass.) This elementary derivation readily generalizes to more realistic settings: 1. For semelparous organisms (such as annual grasses or insects) we obtain the discrete form of the exponential law: N t + 1 = N t . 2. Adding age or stage structure is also relatively straightforward. However, we now have to wait awhile for the population to achieve a stable age distribution, after which all age classes (as well as total number of individuals) begin to grow according to the exponential law. 3. The general pattern of growth is still exponential when we consider finite populations and add demographic stochasticity.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Population Ecology

  • Larry L. Rockwood(Author)
  • 2015(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  In both geometric and exponential models, the growth rate is determined by a fixed parameter (R, , or r), which is not modified by competition for resources. Population growth is often curtailed by the environment even if the population is undergoing density independent growth. Major disturbances or catastrophes such as fire, wind storms, landslides, and floods signifi- cantly reduce certain populations and may even cause local extinctions. In Chapter 2 we will examine models of density dependent growth. In these models, it is assumed that the population encounters a limiting resource (food, water, nest sites, available nitrogen, space, etc.), which limits its growth. In these models the growth parameter is modified and the net growth rate eventually approaches zero at an environmental limit termed a carrying capacity. The realized growth rate is said to depend on the density of the population, hence the term, density dependent growth. 1.5 Discrete or “geometric” growth in populations with non-overlapping generations The use of an appropriate model depends first on the life history of the organism. Therefore you first need basic information on the life cycle of the species. In this first model of density independent growth, the population has a life history with discrete, non-overlapping generations. That is, there are no adult survivors from one genera- tion to the next. Examples include: annual plants, annual insects, salmon, periodical cicadas, century plants, and certain species of bamboo. In most of these cases the organ- ism passes through a dormant period as a spore, a seed, an egg, or as a juvenile stage such as a larva or pupa. Once the adults reproduce, they perish, and the future of the population is based on the dormant or juvenile stage of the organism. As noted above, when modeling such populations we usually collapse fertility and mortality into one constant, R, the net replacement rate, or net growth rate, per generation.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Stochastic Models in Biology

  • Narendra S. Goel, Nira Richter-Dyn(Authors)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  4 Population Growth and Extinction The study of the growth of human population is several centuries old, and is perhaps the oldest branch of biology which has been studied quantitatively. The first law for population growth was given by Thomas Robert Malthus in 1798 when he observed that population growth follows a geometrical progres-sion in contrast to its means of subsistence which tends to grow in an arithmetical progression. Thus he envisioned a situation in which the world's population would soon outstrip its means of support. Under conditions of unlimited sources, Malthus' geometrical growth can be expressed in the form of the equation where n(t) is the instantaneous population of the species under consideration and r is some proportionality constant. Equation (1) is known as Malthus' law, according to which the population will grow exponentially, i.e., dn/dt = rn (1) n(t) = «(0 )exp(r/) (2) 73 4 Population Growth and Extinction Malthus did not take into account the fact that in a given environment, the growth may stop due either to the tendency of one organism to destroy others or to the limit on the density of population which the environment can hold (limited food). Verhulst took into account the limitation on the population due to the latter effect, and postulated that the rate of population growth was proportional to the product of the existing population and the difference between the total available resources and the resources used by the present population. If K* is the maximum population that a given amount of food can support, then according to Verhulst, dn/dt = rn(-n/K*) (3) When η <^ K* the growth is according to Malthus' law, and as η approaches K*, the rate of population growth decreases rapidly. Equation (3) can be integrated to give AO = / ( 0 ) { / ( 0 ) + ( l -/ ( 0 ) ) e - } -1 (4) where / ( / ) = #!(/)/** (5) Equation (4) is the equation of the so-called logistic curve which has a charac-teristic S-like shape.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Essential Demographic Methods

  • Kenneth W. Wachter(Author)
  • 2014(Publication Date)
  • Harvard University Press

    (Publisher)

  Of course no one knows the population down to the nearest person, or even down to the nearest 50 million or so. Population totals for countries are thought to be accurate only within 2% or 3% at best. It is sometimes useful, however, to quote totals down to the last unit, to make it easier to pinpoint the numbers’ source. As for growth rates, demographers usually quote growth rates in parts per thousand per year. A demographer who says that the world’s growth rate peaked at “22” means it peaked at “22 per thousand per year”.

  1.3 The Exponential Curve

  We return now from our discussion of growth rates to our formula for the population over time in the case when the ratios of births and deaths to population remain constant. We rewrite this formula in the form

  Here Rt is the “exponent” of the number “e”. We are raising the number “e” to the power Rt , and another notation for this quantity is exp(Rt ), the “exponential function” of Rt . The exponential function is the inverse function for natural logarithms. That is to say, for any x or y ,

  We know from our linear equation that log(A ) is R , so we have

  These facts about logarithms show us that our equation with A and our equation with e are the same equation. The graph of exp(Rt ) as a function of t is called the “exponential curve”, the continuous-time version of the curve for geometric growth. It is important to become familiar with its shape. The shape depends on whether R is greater than, equal to, or less than zero, as shown in Figure 1.3 . If the ratios B/K and D/K remain unchanged, the population either grows faster and faster beyond all bounds, stays the same, or heads toward zero steeply at first and more gradually later. The most common case is the case in the top panel when births exceed deaths and R >0. This is the case of exponential growth that leads many people to feel alarm at the future prospects for humanity on the planet.

  Our formula for exponential growth treats the growth rate R as a constant. To apply it in practice, we break up any long period of time into shorter intervals within which the growth rate does stay more or less constant and proceed step by step. The graph of the logarithm of population size becomes a series of straight-line pieces. In each step we “run” forward in time by some amount n , and our graph has a corresponding “rise” equal to n times R . The value of R at the step is the growth rate at that step, the current slope of the logarithm of K (t

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Mathematical Modeling in the Social and Life Sciences

  • Michael Olinick(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Important factors such as wars, immigration, variations in the birth and death rates, and changes in the age structure of the population are missing from the model. Nevertheless, simple exponential growth is an accurate way of portraying the change in population in the United States during the early and middle nineteenth century. We may also conclude that if the pattern of growth established during that period continued until the present day, the population of the United States today would be well over 2.5 billion! Scientists have obtained similar conclusions about exponential growth models for other living organisms. The simple model dP dt = aP is often very accurate when the environmental conditions are close to ideal: no natural enemies of the species are present, resources are unlimited, and there is sufficient space for the organisms to develop without interfering with each other. It’s not unusual for an exponential growth model to predict fairly accurately the population of a large nation over a several decade period. Looking again at U.S. census figures, if we take the years 1940 and 1950 as our initial data values, then the value of a for the exponential model is .013539. Table 3.2 displays a comparison between the predicted and observed population figures for the half-century after 1950. Replacing the derivative with the difference in successive time periods transforms the exponential growth model dP dt = aP into the difference equation, 68 CHAPTER 3 Ecological Models: Single Species P i 1 − P i aP i 14 which we may rewrite as P i 1 1 a P i whose solution we found in Chapter 1 to be P k = 1 + a k P 0 16 Let’s use the discrete model to investigate how small changes in the initial population affect long-term predictions for the population. Small errors or “perturbations” in a constant Table 3.1 A comparison between actual population based on census data in the United States and that predicted by exponential growth with rate 0.029643.

  Sign up to read

  Learn more about book