Models for Population Growth

11 Key excerpts on "Models for Population Growth"

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  Population Dynamics

  Alternative Models

  • Bertram G. Jr. Murray(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  Mathematical analysis of populations is essential for developing an understanding of population processes, and, although idealized models are simpler than the natural phenomena they intend to describe, they are useful for organizing data and thoughts and for providing insights. Idealized mathematical models are often unappreciated by field ecologists, but Bartlett (1973) correctly asked, Until we can understand the proper-ties of these simple models, how can we begin to understand the more complicated real situation? This book is a field ecologist's attempt to discover quantitative relation-ships between population parameters in order to understand the observ-able world and perhaps to serve as an antidote to the recent trend toward mathematical elegance (e.g., May, 1976b; Pielou, 1977). For the field biologist, mathematics is no more than a tool, and the minimal set of tools falls into three distinct categories: (1) the exponential and logistic equations describing population growth (Verhulst, 1838; Pearl and Reed, 1920), (2) actuarial equations relating several population parameters (Lotka, 1925), and (3) the projection matrix (Leslie, 1945, 1948). These will be considered only to the extent necessary to follow the theoretical de-velopment in later chapters. Fuller treatment of population mathematics can be found in Pielou (1969, 1977), Mertz (1970), and Poole (1974). E X P O N E N T I A L AND LOGISTIC EQUATIONS 19 E PONENTIA LAN DLOGISTI CE ATION S The dynamics of population growth has been described most frequently by exponential and logistic equations (Verhulst, 1838; Pearl and Reed, 1920). These describe the change in numbers of populations in unlimited and limited situations, respectively. In constant and unlimited conditions, the rate of change in population size is given by dN ~dT = rN > ( 2 J ) where r is the instantaneous rate of increase per head and is the population's size.

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  Modelling and Quantitative Methods in Fisheries

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

    (Publisher)

  19 © 2011 by Taylor & Francis Group, LLC 2 Simple Population Models 2.1 Introduction 2.1.1 Biological Population Dynamics A biological population is a collection of individuals that has identifiable emergent properties not possessed by individual organisms. These properties include the population’s size, its growth rate, its immigration and emigra-tion rates, its age and size structure, and its spatial distribution. The dynamic behaviour of a population relates to changes in these properties through time. One objective of population modelling is to describe and possibly explain how a population’s properties change through time. Mathematical equations used to model biological populations provide an abstract representation of their dynamics. This requires emphasis because the equations in many population models can exhibit dynamic behaviours that biological populations either do not or could not exhibit; models are, after all, only models. For example, a mathematical model might predict that under some circumstances the modelled population was made up of a negative number of organisms or produced a negative number of recruits. Such obvi-ous discrepancies between the behaviour of the mathematical equations and possible biological behaviours are of little consequence because usually they are easily discovered and avoided. Unfortunately, purely mathematical behav-iours can also arise that are less obvious in their effects. It is thus sensible to understand the dynamic behaviour of any equations used in a modelling exercise to avoid ascribing nonsensible behaviours to innocent populations. 2.1.2 The Dynamics of Mathematical Models The purpose of this chapter is to give a brief introduction to the proper-ties of models and how their dynamic behaviours are dependent upon both their particular mathematical form and the particular values given to their parameters.

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  Fundamentals of Ecological Modelling

  Applications in Environmental Management and Research

  • S.E. Jorgensen(Author)
  • 2011(Publication Date)
  • Elsevier

    (Publisher)

  5 Modelling Population Dynamics

  5.1. Introduction

  This chapter covers population dynamic models where state variables are the number or biomass of individuals or species. The growth of one population is used — see Sections 5.2 and 5.3 — to present the basic concepts. Afterward, the interactions between two or more populations are presented. The famous Lotka-Volterra model and several more realistic predator-prey and parasitism models, are shown. Age distribution is introduced and computations with matrix models are illustrated, including the relations to biological growth. Finally, the last three sections illustrate the use of fishery/harvest models, metapopulation dynamics, and infection models.

  5.2. Basic Concepts

  This chapter deals with biodemographic models, which are population models characterized by numbers of individuals or kilograms of biomass of individuals or species as typical units for state variables. As early as the 1920s, Lotka and Volterra developed the first population model, which is still widely used (Lotka, 1956 ; Volterra, 1926 ). So many population models have been developed, tested, and analyzed since that it would not be possible to give a comprehensive review of these models here. This chapter mainly focuses on models of age distribution, growth, and species interactions. Only deterministic models will be mentioned. Those interested in stochastic models can refer to Pielou (1966 , 1977 ), which gives a very comprehensive treatment of this type of population dynamic model.

  A population is defined as a collective group of organisms of the same species. Each population has several characteristic properties, such as population density (population size relative to available space), natality (birth rate), mortality (death rate), age distribution, dispersion, growth forms, and so forth.

  A population changes over time, and we are interested in its size and dynamics as it grows or shrinks. If N represents the number of organisms and t the time, then dN/dt = the rate of change in the number of organisms per unit time at a particular instant (t), and dN/(Ndt) = the rate of change in the number of organisms per unit time per individual at a particular instant (t). If the population is plotted against time, then a straight line tangential to the curve at any point represents the growth rate.

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  Population Ecology

  First Principles - Second Edition

  • John H. Vandermeer, Deborah E. Goldberg(Authors)
  • 2013(Publication Date)
  • Princeton University Press

    (Publisher)

  62 3 Applications of Simple Population Models I n the previous two chapters we showed how simple mathematical models can illustrate general principles about population dynamics. In this chapter we illustrate the application of some of these models and associated tools to address a number of different kinds of problems. We start with the basic mod-els of population dynamics from chapter 1 and apply them to the problem of the evolution of life histories and then move to using the structured models of chapter 2 to describe more complex patterns in life histories. We then turn to applications of population projection matrices both for natural resource man-agement and for conservation. The most obvious uses of population-dynamic models are to examine the range of kinds of dynamics we might observe in real populations and, more directly, to make at least short-term population projections. However, we start this chapter by focusing on a very different and important use of pop-ulation-dynamic models, the study of life history evolution. This activity has a long history in ecology, from the classic studies of Lack (e.g., 1947) and Cole (1954) to the present time. Cole (1954), for example, showed that add-ing one offspring in a previous year could have much greater effects on long-term population growth than adding several more offspring in a current year, which suggests strong selection for earlier reproduction. Or, if evolutionary pressure were exerted on individuals within a population to be more efficient in their use of resources, modeling that population with the logistic equa-tion would begin with the assumption that evolutionary pressure was caus-ing the carrying capacity, K, to increase. On the other hand, if evolutionary pressure on the individuals in a population caused each individual to require more space in the environment, we might model that situation with a logistic equation again, but this time incorporating a decreasing K.

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  Models for Ecological Data

  An Introduction

  • James S. Clark(Author)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  E L E M E N T S O F P R O C E S S M O D E L S • 31 (e.g., May 1981). Some ecologists will find it confusing to see polynomials referred to as "linear models." Recall that statisticians come from a tradition of thinking about parameters. Of course, models can be nonlinear both in param- eters and in state variables, such as f(n) = - —— and f(n) = 1 - e~ pn . I retain this confusion of terms, because it is entrenched. Throughout, I define how the terms are used in specific cases. Except when used in a statistical con- text, I use the terms linear and nonlinear with respect to the state variable, such as concentration of a nutrient, population size or density, and so forth. A second source of confusion comes from the fact that population ecolo- gists often apply the terms linear and nonlinear to the effect of density on the per capita growth rate, lln dnldt. This terminology is fine, so long as it is qual- ified by context. However, a model that is linear in per capita growth rate is not a linear differential equation. In human demography and population biology, linear models (in terms of the state variable density or abundance) are typically used to describe current population growth. Standard human population growth statistics, such as "country X is growing at a rate of Y percent per year," are examples. They are rarely used to predict long-term population trajectories (i.e., over generations), because they cannot be extrapolated with confidence. Many factors that might be ignored in the short term have long-term or cumulative effects. These reg- ulatory factors can directly or indirectly involve density or age structure. Nonlinear models, when used to describe population growth, are termed density-dependent, because the per capita rate of growth depends on the state variable density.

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  From Malthus' Stagnation to Sustained Growth

  Social, Demographic and Economic Factors

  • Bruno Chiarini, Paolo Malanima, Gustavo Piga(Authors)
  • 2012(Publication Date)
  • Palgrave Macmillan

    (Publisher)

  Today, Malthus’s original intuition is always formalised by means of the fol- lowing differential equation: (1) where: N(t) is the number of human individuals at time t; b is birthrate; d is death rate; n is population growth rate; Thus, the time path of population is an exponential function, as follows: (2) 146 Giovanni Scarano The concept of carrying capacity was originally introduced by Pierre-François Verhulst, a Belgian mathematician, in 1838, eight years after the last demographic essay by Malthus, to endogenously correct the “Malthusian” exponential model of population growth (Verhlust, 1838). The Verhlust model, also known as lo- gistic growth model, shows a positive constant K, referred to as the environmental carrying capacity, which enters into the new function that determines the values of n over time (Clark, 1990): (3) where n i is the intrinsic growth rate, that is, the balance of birthrate and death rate, as they are genetically determined, excluding death causes produced by the environment, such as predation, illness, famine, and so on. K is exactly the max- imum number of individuals that can survive at each time, given the yearly flows of natural resources, such as food, water, air and so on, provided by the ecosystem in which the population lives. Thus, the motion equation for population becomes: (4) The population time path is consequently modified as follows: (5) where c is the initial condition: (6) that is, the potential maximum rate of variation of the population. From equation (5) it follows that: (7) ( ) ( ) ( ) ( ) Population, Earth Carrying Capacity and Economic Growth 147 This means that K is a stable dynamic equilibrium for the population. But humanity is a very particular species, because it utilises production, which is in fact a specific human way to transform useless things into goods, i.e.

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  Concepts of Mathematical Modeling

  • Walter J. Meyer(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Prerequisites The first of the three sections requires only high school algebra. The second requires understanding matrix multiplication. Some exercises use a bit more linear algebra. The third section requires some probability theory: the independence rule and expected values.

  How many people will there be in the population of a certain country in 10 years? How many births? In this section we build the simplest possible model for answering these questions.

  We will use the variable t to measure time in years, with t = 0 denoting the present. Let P (t ) denote the size of the population at time t . Let B (t ) be the number of births in the 1-year interval between time t and t + 1 and let D (t ) denote the number of deaths between t and t + 1. The main assumption of our model is that certain rates stay the same. The rates we have in mind are defined as follows:

      Definition

  1. B (t )/P (t ) is called the birth rate for the time interval t to t + 1.
  2. D (t )/P (t ) is called the death rate for the time interval t to t + 1.

  Assumptions

  1. The birth rate is the same for all intervals. Likewise, the death rate is the same for all intervals. This means that there is a constant b , called the birth rate, and a constant d , called the death rate so that, for all t ≥ 0,
     (1)
  2. There is no migration into or out of the population; i.e., the only source of population change is birth and death.  
     As a result of Assumptions 1 and 2 we deduce that, for t ≥ 0,
     (2)
     Setting t = 0 in (2) gives
     (3)
     Setting t = 1 in Equation (2) and substituting Equation (3) gives
     Continuing this way yields
     (4)
     for t = 0, 1, 2, . . . . The constant 1 + b – d is often abbreviated r and called the growth rate or, in more high-flown language, the Malthusian parameter, in honor of Robert Malthus who first brought this model to popular attention. In terms of r , Equation (4) becomes
     (5)
     P (t ) is an example of an exponential function. Any function of the form cr t , where c and r are constants, is an exponential function.

  Example 1

  Suppose the current population is 250,000,000 and the rates are b = 0.02 and d

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  Introduction to Population Biology

  • Dick Neal(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  In contrast, the growth rate per capita (r) in the logistic growth model declines linearly as the density increases (Figure 5.1c). When the density is close to zero, r is approximately equal to r m because there are no effects of density. The value of r declines to zero when the density reaches the carrying capacity, and at densities above the carrying capacity, r is negative. Assuming no immigration or emigration, the population adjusts the value of r in relation to density by altering the birth and death rates, and a stable equilibrium (N = K) is reached at a density where the birth rate is equal to the death rate. The model has many unrealistic assumptions. It assumes that all individuals are identical, but in reality they may vary in size, age, sex and genotype. These factors affect birth and death rates, and the use of resources, and so we cannot expect r m and K to be constants. The model also assumes that individuals adjust their birth and death rates (i.e. r) instantaneously as the population changes in size, but in reality there will be time lags to any such response (see Section 5.3). Finally, it assumes that the environment is constant, but environments change over the course of time and this is another reason why we cannot expect r m and K to be constants. Let us relax some of the restrictive assumptions of the model to see how the form of growth may change. 5.3 TIME LAGS Most populations have time lags in the way that they adjust their birth and death rates in relation to population density. For example, many species lay eggs that hatch independently of the parent, and so the birth rate cannot be adjusted if the population density changes between the laying and hatching of the eggs. In this case, the birth rate is related to the density at the time of egg deposition, not the time of hatching, and the time lag will correspond to the length of the incubation period. Similarly, when the young are born, they are usually much smaller than adults.

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  Lectures, Problems And Solutions For Ordinary Differential Equations

  • Yuefan Deng(Author)
  • 2014(Publication Date)
  • World Scientific

    (Publisher)

  76 Chapter 2 Mathematical Models 2.1 Population Model 2.1.1 General Population Equation It is customary to track the growth or decline of a population in terms of its birth rate and death rate functions, which are defined as follows: • œ is the number of births per unit population per unit time at time  • … is the number of deaths per unit population per unit time at time  2.1 Population Model 77 Then the numbers of births and deaths that occur during the time interval ,  + Δ is approximately given by births: œhΔ and deaths: …hΔ . The change Δh in the population during the time interval ,  + Δ of the length Δ is ∆h = birt hs − deaths = œhΔ − …hΔ = $œ − …%hΔ Thus, ∆h ∆ = $œ − …%h Taking the limit ∆ → 0 , we get the DE h  =  œ − …  h (2.1) which is the General Population Equation where œ = œ and … = … where œ and … can be either a constant or functions of  or they may, indirectly, depend on the unknown function h . Example 1 Given the following information, find the population after 10 years. h0 = h B = 100 œ = 0.0005h … = 0 Solution Substituting the given information in the above formula we have h  = œ − …h h  = 0.0005h  h h  = 0.0005  h h  =  0.0005 Chapter 2 Mathematical Models 78 − 1 h = 0.0005 + 1 Substitution of h0 = 100 in the above DE given 1 = − 1 100 Therefore h = 2000 20 −  and h10 = 200 Thus, the population after 10 years is 200.

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  Modeling of Living Systems

  From Cell to Ecosystem

  • Alain Pavé(Author)
  • 2012(Publication Date)
  • Wiley-ISTE

    (Publisher)

  8 :

  – a f (

  xn

  ) represents “input” into the population (reproduction, immigration);

  – b g (

  xn

  ) represents “output” from the population (mortality, emigration).

  3.7.2. The Fibonacci model

  As we mentioned in section 1.4.1 , the first known demographic model, i.e. the first known model relating to population dynamics, may be attributed to Leonardo of Pisa, otherwise known as Fibonacci. Nowadays, the value of this model is essentially historical, although its matrix form, which we shall discuss in the following section, may be seen as a simple example of the Leslie model, the theory of which was published in 1945. This same class of models is the deterministic version of a two-state branching process which we shall present later in section 3.7.3 and Appendix 4 . Figure 3.26 shows a page from the manuscript of the Liber Abaci , available online.

  Figure 3.26

  .

  Page from the Liber abaci showing, in Latin, the growth model of a rabbit population (more precisely, of pairs of rabbits). The book contains one of the first uses of the Arabic numeral system. Copyright: public domain

  The translated text is as follows9 :

  “A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the above-written pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there are 8 pairs, of which 5 pairs bear another 5 pairs; these are added to the 8 pairs making 13 pairs in the fifth month; these 5 pairs that are born in this month do not mate in this month, but another 8 pairs are pregnant, and thus there are in the sixth month 21 pairs; [p284] to these are added the 13 pairs that are born in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eighth month; there will be 55 pairs in this month; to these are added the 34 pairs that are born in the ninth month; there will be 89 pairs in this month; to these are added again the 55 pairs that are born in the tenth month; there will be 144 pairs in this month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month. To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the above-written pair in the mentioned place at the end of the one year. You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the first to the second, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the above-written sum of rabbits, namely 377, and thus you can, in order, find it for an unending number of months”.

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  Dynamical Systems for Biological Modeling

  An Introduction

  • Fred Brauer, Christopher Kribs(Authors)
  • 2015(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  The resulting model does not decouple, however, so we shall reserve its further study for Chapter 6. 3.2.3 The spread of infectious diseases Another biological science to which mathematics has made important con-tributions is epidemiology. Mathematical models for the spread of infectious diseases typically divide populations into distinct classes based on epidemi-ological or demographic factors relevant to the transmission of the disease: 118 Dynamical Systems for Biological Modeling: An Introduction susceptible and infected individuals, high-risk and low-risk individuals, highly infectious and less infectious individuals, etc. For this reason they are often referred to as compartmental models. Most compartmental models involve keeping track of several populations, an activity which requires as many state variables (and equations) as compartments. The simplest possible epidemic model, however, can be rewritten as a single logistic equation, and so we in-clude it here. Consider a population of constant size N in which an infectious disease is introduced. We divide the population into two classes: susceptibles (not infected) and infectives (infected and contagious). Let S ( t ) denote the num-ber of susceptibles at time t and let I ( t ) denote the number of infectives, so that S ( t ) + I ( t ) = N . We assume that the disease is spread from infectives to susceptibles through contact. Suppose that an “average” infective makes po-tentially infective contacts with a constant number βN of individuals in unit time. Then, presumably βI ( t ) of these individuals are already infected, and the number of new infections caused by an “average” infective in unit time is βS ( t ). Thus the total number of new infections in unit time is βS ( t ) I ( t ). We assume also that the disease is never fatal, and that in unit time a fraction γ of the infectives recover, so that the number of recoveries in unit time is γI ( t ).

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