Density-independent growth: At times, populations invade new habitats that contain abundant resources. For a while at least, these populations can grow rapidly because the initial number of individuals is small and there is no competition for resources. This is called density-independent growth because the density of individuals does not have any effect on future growth. As you can imagine, this cannot continue indefinitely. The first person to mathematically describe a population's potential to reproduce was Thomas Malthus, and his writings would influence the ideas of Charles Darwin. Show Take the equation below and run through 10 generations. Start with an initial population size (Ni) of 100, and use a constant growth rate (r) of 1. (A growth rate of 0 indicates no reproduction, a value of 1 means doubling, higher values would yield more rapid population increases.) ΔN is the change in number. Nf is the final number, after reproduction has occured, and is calculated as the initial number, Ni plus the change in number, ΔN. In generation 2, Nf becomes the new Ni and we run through the equation again. This kind of growth is called "exponential" and is fairly typical of bacterial cultures in fresh medium. Bacteria divide by binary fission (one becomes two) so the value of 2 for a growth rate is realistic. Graph your results. ΔN = r Ni Nf = Ni + ΔN Density-dependent growth: In a population that is already established, resources begin to become scarce, and competition starts to play a role. We refer to the maximum number of individuals that a habitat can sustain as the carrying capacity of that population. If a population overshoots its carrying capacity by too much, nobody gets enough resources and the population can crash to zero. If the population approaches its carrying capacity more gradually, these limiting factors, such as food, nesting sites, mates, etc. tend to regulate further growth and the population stabilizes. The "logistic equation" models this kind of population growth. ΔN = r Ni ((K-Ni)/K) Nf = Ni + ΔN Compare the exponential and logistic growth equations. The rN part is the same, but the logistic equation has another term, (K-N)/K which puts the brakes on growth as N approaches or exceeds K. Take the equation above and again run through 10 generations. Start with an initial population size (Ni) of 100. Again, use a constant growth rate (r) of 2. K is the carrying capacity of the population, which we will set at 500. Graph your results. Download the Excel file  Take a look at World Population Growth among humans. Predicting population growth accurately depends on a variety of factors. Clearly nutrition and disease are two important factors that affect survival to reproductive age, but also the ratio of males to females in a population (because females are the limiting factor) and the age distribution of the population (because younger populations have higher reproductive rates) are important parameters. How do war, famine, and environmental degradation relate to population size? Consider China's one child policy to limit population growth, and the social practices that favor one gender over the other in some cultures. Further Reading: http://www.nature.com/scitable/knowledge/library/how-populations-grow-the-exponential-and-logistic-13240157 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 ModelIn 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). 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 “K – N” indicates how many individuals may be added to a population at a given stage, and “K – N” 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: dNdT= rmaxdNdT=rmaxN(K - N)KNotice 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), 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 CompetitionThe 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 GrowthYeast, a microscopic fungus used to make bread and alcoholic beverages, exhibits the classical S-shaped curve when grown in a test tube (Figurea). 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 (Figureb). 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. Art ConnectionIf the major food source of the seals declines due to pollution or overfishing, which of the following would likely occur?
What is it called when a population exceeds the carrying capacity?Overshoot. A population is in overshoot when it exceeds available carrying capacity. A population in overshoot may permanently impair the long-term productive potential of its habitat, reducing future carrying capacity.
Can population growth exceed carrying capacity?However, a real population's size typically oscillates around its carrying capacity. This means it's common even for a stable population to briefly exceed or dip below its carrying capacity, even though the average growth rate of the population is zero. Carrying capacities can change.
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