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Biology 2e

45.3 Environmental Limits to Population Growth

Biology 2e45.3 Environmental Limits to Population Growth
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  1. Preface
  2. The Chemistry of Life
    1. 1 The Study of Life
      1. Introduction
      2. 1.1 The Science of Biology
      3. 1.2 Themes and Concepts of Biology
      4. Key Terms
      5. Chapter Summary
      6. Visual Connection Questions
      7. Review Questions
      8. Critical Thinking Questions
    2. 2 The Chemical Foundation of Life
      1. Introduction
      2. 2.1 Atoms, Isotopes, Ions, and Molecules: The Building Blocks
      3. 2.2 Water
      4. 2.3 Carbon
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
    3. 3 Biological Macromolecules
      1. Introduction
      2. 3.1 Synthesis of Biological Macromolecules
      3. 3.2 Carbohydrates
      4. 3.3 Lipids
      5. 3.4 Proteins
      6. 3.5 Nucleic Acids
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
  3. The Cell
    1. 4 Cell Structure
      1. Introduction
      2. 4.1 Studying Cells
      3. 4.2 Prokaryotic Cells
      4. 4.3 Eukaryotic Cells
      5. 4.4 The Endomembrane System and Proteins
      6. 4.5 The Cytoskeleton
      7. 4.6 Connections between Cells and Cellular Activities
      8. Key Terms
      9. Chapter Summary
      10. Visual Connection Questions
      11. Review Questions
      12. Critical Thinking Questions
    2. 5 Structure and Function of Plasma Membranes
      1. Introduction
      2. 5.1 Components and Structure
      3. 5.2 Passive Transport
      4. 5.3 Active Transport
      5. 5.4 Bulk Transport
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    3. 6 Metabolism
      1. Introduction
      2. 6.1 Energy and Metabolism
      3. 6.2 Potential, Kinetic, Free, and Activation Energy
      4. 6.3 The Laws of Thermodynamics
      5. 6.4 ATP: Adenosine Triphosphate
      6. 6.5 Enzymes
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    4. 7 Cellular Respiration
      1. Introduction
      2. 7.1 Energy in Living Systems
      3. 7.2 Glycolysis
      4. 7.3 Oxidation of Pyruvate and the Citric Acid Cycle
      5. 7.4 Oxidative Phosphorylation
      6. 7.5 Metabolism without Oxygen
      7. 7.6 Connections of Carbohydrate, Protein, and Lipid Metabolic Pathways
      8. 7.7 Regulation of Cellular Respiration
      9. Key Terms
      10. Chapter Summary
      11. Visual Connection Questions
      12. Review Questions
      13. Critical Thinking Questions
    5. 8 Photosynthesis
      1. Introduction
      2. 8.1 Overview of Photosynthesis
      3. 8.2 The Light-Dependent Reactions of Photosynthesis
      4. 8.3 Using Light Energy to Make Organic Molecules
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
    6. 9 Cell Communication
      1. Introduction
      2. 9.1 Signaling Molecules and Cellular Receptors
      3. 9.2 Propagation of the Signal
      4. 9.3 Response to the Signal
      5. 9.4 Signaling in Single-Celled Organisms
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    7. 10 Cell Reproduction
      1. Introduction
      2. 10.1 Cell Division
      3. 10.2 The Cell Cycle
      4. 10.3 Control of the Cell Cycle
      5. 10.4 Cancer and the Cell Cycle
      6. 10.5 Prokaryotic Cell Division
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
  4. Genetics
    1. 11 Meiosis and Sexual Reproduction
      1. Introduction
      2. 11.1 The Process of Meiosis
      3. 11.2 Sexual Reproduction
      4. Key Terms
      5. Chapter Summary
      6. Visual Connection Questions
      7. Review Questions
      8. Critical Thinking Questions
    2. 12 Mendel's Experiments and Heredity
      1. Introduction
      2. 12.1 Mendel’s Experiments and the Laws of Probability
      3. 12.2 Characteristics and Traits
      4. 12.3 Laws of Inheritance
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
    3. 13 Modern Understandings of Inheritance
      1. Introduction
      2. 13.1 Chromosomal Theory and Genetic Linkage
      3. 13.2 Chromosomal Basis of Inherited Disorders
      4. Key Terms
      5. Chapter Summary
      6. Visual Connection Questions
      7. Review Questions
      8. Critical Thinking Questions
    4. 14 DNA Structure and Function
      1. Introduction
      2. 14.1 Historical Basis of Modern Understanding
      3. 14.2 DNA Structure and Sequencing
      4. 14.3 Basics of DNA Replication
      5. 14.4 DNA Replication in Prokaryotes
      6. 14.5 DNA Replication in Eukaryotes
      7. 14.6 DNA Repair
      8. Key Terms
      9. Chapter Summary
      10. Visual Connection Questions
      11. Review Questions
      12. Critical Thinking Questions
    5. 15 Genes and Proteins
      1. Introduction
      2. 15.1 The Genetic Code
      3. 15.2 Prokaryotic Transcription
      4. 15.3 Eukaryotic Transcription
      5. 15.4 RNA Processing in Eukaryotes
      6. 15.5 Ribosomes and Protein Synthesis
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    6. 16 Gene Expression
      1. Introduction
      2. 16.1 Regulation of Gene Expression
      3. 16.2 Prokaryotic Gene Regulation
      4. 16.3 Eukaryotic Epigenetic Gene Regulation
      5. 16.4 Eukaryotic Transcription Gene Regulation
      6. 16.5 Eukaryotic Post-transcriptional Gene Regulation
      7. 16.6 Eukaryotic Translational and Post-translational Gene Regulation
      8. 16.7 Cancer and Gene Regulation
      9. Key Terms
      10. Chapter Summary
      11. Visual Connection Questions
      12. Review Questions
      13. Critical Thinking Questions
    7. 17 Biotechnology and Genomics
      1. Introduction
      2. 17.1 Biotechnology
      3. 17.2 Mapping Genomes
      4. 17.3 Whole-Genome Sequencing
      5. 17.4 Applying Genomics
      6. 17.5 Genomics and Proteomics
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
  5. Evolutionary Processes
    1. 18 Evolution and the Origin of Species
      1. Introduction
      2. 18.1 Understanding Evolution
      3. 18.2 Formation of New Species
      4. 18.3 Reconnection and Speciation Rates
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
    2. 19 The Evolution of Populations
      1. Introduction
      2. 19.1 Population Evolution
      3. 19.2 Population Genetics
      4. 19.3 Adaptive Evolution
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
    3. 20 Phylogenies and the History of Life
      1. Introduction
      2. 20.1 Organizing Life on Earth
      3. 20.2 Determining Evolutionary Relationships
      4. 20.3 Perspectives on the Phylogenetic Tree
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
  6. Biological Diversity
    1. 21 Viruses
      1. Introduction
      2. 21.1 Viral Evolution, Morphology, and Classification
      3. 21.2 Virus Infections and Hosts
      4. 21.3 Prevention and Treatment of Viral Infections
      5. 21.4 Other Acellular Entities: Prions and Viroids
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    2. 22 Prokaryotes: Bacteria and Archaea
      1. Introduction
      2. 22.1 Prokaryotic Diversity
      3. 22.2 Structure of Prokaryotes: Bacteria and Archaea
      4. 22.3 Prokaryotic Metabolism
      5. 22.4 Bacterial Diseases in Humans
      6. 22.5 Beneficial Prokaryotes
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    3. 23 Protists
      1. Introduction
      2. 23.1 Eukaryotic Origins
      3. 23.2 Characteristics of Protists
      4. 23.3 Groups of Protists
      5. 23.4 Ecology of Protists
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    4. 24 Fungi
      1. Introduction
      2. 24.1 Characteristics of Fungi
      3. 24.2 Classifications of Fungi
      4. 24.3 Ecology of Fungi
      5. 24.4 Fungal Parasites and Pathogens
      6. 24.5 Importance of Fungi in Human Life
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    5. 25 Seedless Plants
      1. Introduction
      2. 25.1 Early Plant Life
      3. 25.2 Green Algae: Precursors of Land Plants
      4. 25.3 Bryophytes
      5. 25.4 Seedless Vascular Plants
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    6. 26 Seed Plants
      1. Introduction
      2. 26.1 Evolution of Seed Plants
      3. 26.2 Gymnosperms
      4. 26.3 Angiosperms
      5. 26.4 The Role of Seed Plants
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    7. 27 Introduction to Animal Diversity
      1. Introduction
      2. 27.1 Features of the Animal Kingdom
      3. 27.2 Features Used to Classify Animals
      4. 27.3 Animal Phylogeny
      5. 27.4 The Evolutionary History of the Animal Kingdom
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    8. 28 Invertebrates
      1. Introduction
      2. 28.1 Phylum Porifera
      3. 28.2 Phylum Cnidaria
      4. 28.3 Superphylum Lophotrochozoa: Flatworms, Rotifers, and Nemerteans
      5. 28.4 Superphylum Lophotrochozoa: Molluscs and Annelids
      6. 28.5 Superphylum Ecdysozoa: Nematodes and Tardigrades
      7. 28.6 Superphylum Ecdysozoa: Arthropods
      8. 28.7 Superphylum Deuterostomia
      9. Key Terms
      10. Chapter Summary
      11. Visual Connection Questions
      12. Review Questions
      13. Critical Thinking Questions
    9. 29 Vertebrates
      1. Introduction
      2. 29.1 Chordates
      3. 29.2 Fishes
      4. 29.3 Amphibians
      5. 29.4 Reptiles
      6. 29.5 Birds
      7. 29.6 Mammals
      8. 29.7 The Evolution of Primates
      9. Key Terms
      10. Chapter Summary
      11. Visual Connection Questions
      12. Review Questions
      13. Critical Thinking Questions
  7. Plant Structure and Function
    1. 30 Plant Form and Physiology
      1. Introduction
      2. 30.1 The Plant Body
      3. 30.2 Stems
      4. 30.3 Roots
      5. 30.4 Leaves
      6. 30.5 Transport of Water and Solutes in Plants
      7. 30.6 Plant Sensory Systems and Responses
      8. Key Terms
      9. Chapter Summary
      10. Visual Connection Questions
      11. Review Questions
      12. Critical Thinking Questions
    2. 31 Soil and Plant Nutrition
      1. Introduction
      2. 31.1 Nutritional Requirements of Plants
      3. 31.2 The Soil
      4. 31.3 Nutritional Adaptations of Plants
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
    3. 32 Plant Reproduction
      1. Introduction
      2. 32.1 Reproductive Development and Structure
      3. 32.2 Pollination and Fertilization
      4. 32.3 Asexual Reproduction
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
  8. Animal Structure and Function
    1. 33 The Animal Body: Basic Form and Function
      1. Introduction
      2. 33.1 Animal Form and Function
      3. 33.2 Animal Primary Tissues
      4. 33.3 Homeostasis
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
    2. 34 Animal Nutrition and the Digestive System
      1. Introduction
      2. 34.1 Digestive Systems
      3. 34.2 Nutrition and Energy Production
      4. 34.3 Digestive System Processes
      5. 34.4 Digestive System Regulation
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    3. 35 The Nervous System
      1. Introduction
      2. 35.1 Neurons and Glial Cells
      3. 35.2 How Neurons Communicate
      4. 35.3 The Central Nervous System
      5. 35.4 The Peripheral Nervous System
      6. 35.5 Nervous System Disorders
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    4. 36 Sensory Systems
      1. Introduction
      2. 36.1 Sensory Processes
      3. 36.2 Somatosensation
      4. 36.3 Taste and Smell
      5. 36.4 Hearing and Vestibular Sensation
      6. 36.5 Vision
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    5. 37 The Endocrine System
      1. Introduction
      2. 37.1 Types of Hormones
      3. 37.2 How Hormones Work
      4. 37.3 Regulation of Body Processes
      5. 37.4 Regulation of Hormone Production
      6. 37.5 Endocrine Glands
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    6. 38 The Musculoskeletal System
      1. Introduction
      2. 38.1 Types of Skeletal Systems
      3. 38.2 Bone
      4. 38.3 Joints and Skeletal Movement
      5. 38.4 Muscle Contraction and Locomotion
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    7. 39 The Respiratory System
      1. Introduction
      2. 39.1 Systems of Gas Exchange
      3. 39.2 Gas Exchange across Respiratory Surfaces
      4. 39.3 Breathing
      5. 39.4 Transport of Gases in Human Bodily Fluids
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    8. 40 The Circulatory System
      1. Introduction
      2. 40.1 Overview of the Circulatory System
      3. 40.2 Components of the Blood
      4. 40.3 Mammalian Heart and Blood Vessels
      5. 40.4 Blood Flow and Blood Pressure Regulation
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    9. 41 Osmotic Regulation and Excretion
      1. Introduction
      2. 41.1 Osmoregulation and Osmotic Balance
      3. 41.2 The Kidneys and Osmoregulatory Organs
      4. 41.3 Excretion Systems
      5. 41.4 Nitrogenous Wastes
      6. 41.5 Hormonal Control of Osmoregulatory Functions
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    10. 42 The Immune System
      1. Introduction
      2. 42.1 Innate Immune Response
      3. 42.2 Adaptive Immune Response
      4. 42.3 Antibodies
      5. 42.4 Disruptions in the Immune System
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
    11. 43 Animal Reproduction and Development
      1. Introduction
      2. 43.1 Reproduction Methods
      3. 43.2 Fertilization
      4. 43.3 Human Reproductive Anatomy and Gametogenesis
      5. 43.4 Hormonal Control of Human Reproduction
      6. 43.5 Human Pregnancy and Birth
      7. 43.6 Fertilization and Early Embryonic Development
      8. 43.7 Organogenesis and Vertebrate Formation
      9. Key Terms
      10. Chapter Summary
      11. Visual Connection Questions
      12. Review Questions
      13. Critical Thinking Questions
  9. Ecology
    1. 44 Ecology and the Biosphere
      1. Introduction
      2. 44.1 The Scope of Ecology
      3. 44.2 Biogeography
      4. 44.3 Terrestrial Biomes
      5. 44.4 Aquatic Biomes
      6. 44.5 Climate and the Effects of Global Climate Change
      7. Key Terms
      8. Chapter Summary
      9. Visual Connection Questions
      10. Review Questions
      11. Critical Thinking Questions
    2. 45 Population and Community Ecology
      1. Introduction
      2. 45.1 Population Demography
      3. 45.2 Life Histories and Natural Selection
      4. 45.3 Environmental Limits to Population Growth
      5. 45.4 Population Dynamics and Regulation
      6. 45.5 Human Population Growth
      7. 45.6 Community Ecology
      8. 45.7 Behavioral Biology: Proximate and Ultimate Causes of Behavior
      9. Key Terms
      10. Chapter Summary
      11. Visual Connection Questions
      12. Review Questions
      13. Critical Thinking Questions
    3. 46 Ecosystems
      1. Introduction
      2. 46.1 Ecology of Ecosystems
      3. 46.2 Energy Flow through Ecosystems
      4. 46.3 Biogeochemical Cycles
      5. Key Terms
      6. Chapter Summary
      7. Visual Connection Questions
      8. Review Questions
      9. Critical Thinking Questions
    4. 47 Conservation Biology and Biodiversity
      1. Introduction
      2. 47.1 The Biodiversity Crisis
      3. 47.2 The Importance of Biodiversity to Human Life
      4. 47.3 Threats to Biodiversity
      5. 47.4 Preserving Biodiversity
      6. Key Terms
      7. Chapter Summary
      8. Visual Connection Questions
      9. Review Questions
      10. Critical Thinking Questions
  10. A | The Periodic Table of Elements
  11. B | Geological Time
  12. C | Measurements and the Metric System
  13. Index
By the end of this section, you will be able to do the following:
  • Explain the characteristics of and differences between exponential and logistic growth patterns
  • Give examples of exponential and logistic growth in natural populations
  • Describe how natural selection and environmental adaptation led to the evolution of particular life history patterns

Although life histories describe the way many characteristics of a population (such as their age structure) change over time in a general way, population ecologists make use of a variety of methods to model population dynamics mathematically. These more precise models can then be used to accurately describe changes occurring in a population and better predict future changes. Certain long-accepted models are now being modified or even abandoned due to their lack of predictive ability, and scholars strive to create effective new models.

Exponential Growth

Charles Darwin, in his theory of natural selection, was greatly influenced by the English clergyman Thomas Malthus. Malthus published a book in 1798 stating that populations with unlimited natural resources grow very rapidly, and then population growth decreases as resources become depleted. This accelerating pattern of increasing population size is called exponential growth.

The best example of exponential growth is seen in bacteria. Bacteria reproduce by prokaryotic fission. This division takes about an hour for many bacterial species. If 1000 bacteria are placed in a large flask with an unlimited supply of nutrients (so the nutrients will not become depleted), after an hour, there is one round of division and each organism divides, resulting in 2000 organisms—an increase of 1000. In another hour, each of the 2000 organisms will double, producing 4000, an increase of 2000 organisms. After the third hour, there should be 8000 bacteria in the flask, an increase of 4000 organisms. The important concept of exponential growth is the accelerating population growth rate—the number of organisms added in each reproductive generation—that is, it is increasing at a greater and greater rate. After 1 day and 24 of these cycles, the population would have increased from 1000 to more than 16 billion. When the population size, N, is plotted over time, a J-shaped growth curve is produced (Figure 45.9).

The bacteria example is not representative of the real world where resources are limited. Furthermore, some bacteria will die during the experiment and thus not reproduce, lowering the growth rate. Therefore, when calculating the growth rate of a population, the death rate (D) (number organisms that die during a particular time interval) is subtracted from the birth rate (B) (number organisms that are born during that interval). This is shown in the following formula:

ΔN (change in number) ΔT (change in time)  = B (birth rate) - D (death rate) ΔN (change in number) ΔT (change in time)  = B (birth rate) - D (death rate)

The birth rate is usually expressed on a per capita (for each individual) basis. Thus, B (birth rate) = bN (the per capita birth rate “b” multiplied by the number of individuals “N”) and D (death rate) = dN (the per capita death rate “d” multiplied by the number of individuals “N”). Additionally, ecologists are interested in the population at a particular point in time, an infinitely small time interval. For this reason, the terminology of differential calculus is used to obtain the “instantaneous” growth rate, replacing the change in number and time with an instant-specific measurement of number and time.

dN dT  = bN  dN = (b - d)N dN dT  = bN  dN = (b - d)N

Notice that the “d” associated with the first term refers to the derivative (as the term is used in calculus) and is different from the death rate, also called “d.” The difference between birth and death rates is further simplified by substituting the term “r” (intrinsic rate of increase) for the relationship between birth and death rates:

dN dT  = rN dN dT  = rN

The value “r” can be positive, meaning the population is increasing in size; or negative, meaning the population is decreasing in size; or zero, where the population’s size is unchanging, a condition known as zero population growth. A further refinement of the formula recognizes that different species have inherent differences in their intrinsic rate of increase (often thought of as the potential for reproduction), even under ideal conditions. Obviously, a bacterium can reproduce more rapidly and have a higher intrinsic rate of growth than a human. The maximal growth rate for a species is its biotic potential, or rmax, thus changing the equation to:

dN dT = r max N dN dT = r max N
Both graphs (a) and (b) plot population size versus time. In graph (a), exponential growth results in a curve that gets increasingly steep, resulting in a J-shape. In graph (b), logistic growth results in a curve that gets increasingly steep, then levels off when the carrying capacity is reached, resulting in an S-shape
Figure 45.9 When resources are unlimited, populations exhibit exponential growth, resulting in a J-shaped curve. When resources are limited, populations exhibit logistic growth. In logistic growth, population expansion decreases as resources become scarce, and it levels off when the carrying capacity of the environment is reached, resulting in an S-shaped curve.

Logistic Growth

Exponential growth is possible only when infinite natural resources are available; this is not the case in the real world. Charles Darwin recognized this fact in his description of the “struggle for existence,” which states that individuals will compete (with members of their own or other species) for limited resources. The successful ones will survive to pass on their own characteristics and traits (which we know now are transferred by genes) to the next generation at a greater rate (natural selection). To model the reality of limited resources, population ecologists developed the logistic growth model.

Carrying Capacity and the Logistic Model

In the real world, with its limited resources, exponential growth cannot continue indefinitely. Exponential growth may occur in environments where there are few individuals and plentiful resources, but when the number of individuals gets large enough, resources will be depleted, slowing the growth rate. Eventually, the growth rate will plateau or level off (Figure 45.9). This population size, which represents the maximum population size that a particular environment can support, is called the carrying capacity, or K.

The formula we use to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. The expression “KN” indicates how many individuals may be added to a population at a given stage, and “KN” divided by “K” is the fraction of the carrying capacity available for further growth. Thus, the exponential growth model is restricted by this factor to generate the logistic growth equation:

dN dT = r max dN dT = r max N (K - N) K dN dT = r max dN dT = r max N (K - N) K

Notice that when N is very small, (K-N)/K becomes close to K/K or 1, and the right side of the equation reduces to rmaxN, which means the population is growing exponentially and is not influenced by carrying capacity. On the other hand, when N is large, (K-N)/K comes close to zero, which means that population growth will be slowed greatly or even stopped. Thus, population growth is greatly slowed in large populations by the carrying capacity K. This model also allows for the population of a negative population growth, or a population decline. This occurs when the number of individuals in the population exceeds the carrying capacity (because the value of (K-N)/K is negative).

A graph of this equation yields an S-shaped curve (Figure 45.9), and it is a more realistic model of population growth than exponential growth. There are three different sections to an S-shaped curve. Initially, growth is exponential because there are few individuals and ample resources available. Then, as resources begin to become limited, the growth rate decreases. Finally, growth levels off at the carrying capacity of the environment, with little change in population size over time.

Role of Intraspecific Competition

The logistic model assumes that every individual within a population will have equal access to resources and, thus, an equal chance for survival. For plants, the amount of water, sunlight, nutrients, and the space to grow are the important resources, whereas in animals, important resources include food, water, shelter, nesting space, and mates.

In the real world, phenotypic variation among individuals within a population means that some individuals will be better adapted to their environment than others. The resulting competition between population members of the same species for resources is termed intraspecific competition (intra- = “within”; -specific = “species”). Intraspecific competition for resources may not affect populations that are well below their carrying capacity—resources are plentiful and all individuals can obtain what they need. However, as population size increases, this competition intensifies. In addition, the accumulation of waste products can reduce an environment’s carrying capacity.

Examples of Logistic Growth

Yeast, a microscopic fungus used to make bread and alcoholic beverages, exhibits the classical S-shaped curve when grown in a test tube (Figure 45.10a). Its growth levels off as the population depletes the nutrients. In the real world, however, there are variations to this idealized curve. Examples in wild populations include sheep and harbor seals (Figure 45.10b). In both examples, the population size exceeds the carrying capacity for short periods of time and then falls below the carrying capacity afterwards. This fluctuation in population size continues to occur as the population oscillates around its carrying capacity. Still, even with this oscillation, the logistic model is confirmed.

Visual Connection

Graph (a) plots amount of yeast versus time of growth in hours. The curve rises steeply, and then plateaus at the carrying capacity. Data points tightly follow the curve. Graph (b) plots the number of harbor seals versus time in years. Again, the curve rises steeply then plateaus at the carrying capacity, but this time there is much more scatter in the data. A micrograph of yeast cells, which are oval in shape, and a photo of a harbor seal are shown.
Figure 45.10 (a) Yeast grown in ideal conditions in a test tube show a classical S-shaped logistic growth curve, whereas (b) a natural population of seals shows real-world fluctuation.

If the major food source of the seals declines due to pollution or overfishing, which of the following would likely occur?

  1. The carrying capacity of seals would decrease, as would the seal population.
  2. The carrying capacity of seals would decrease, but the seal population would remain the same.
  3. The number of seal deaths would increase but the number of births would also increase, so the population size would remain the same.
  4. The carrying capacity of seals would remain the same, but the population of seals would decrease.
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