 Population Growth

1. Population Growth Exponential and Logistic IB Environmental Studies 2004 – Darrel Holnes

2. Population Growth • What is a population? Very simply, a population is a group of organisms of the same species that live in a particular area. The number of organisms in a population changes over time because of the following: births, deaths, immigration, and emigration. • The increase in the number of organisms in a population is referred to as population growth. There are factors that can help populations grow and others than can slow down and even prevent populations from growing. Factors that limit population growth are called limiting factors. However, before we go into the limiting factors, let's talk about the biotic potential of a population.

3. Breaking It Down to Exponential • If things were perfect for a population and all the individuals in the population survived and reproduced at the maximum rate, that growth rate is called the biotic potential. The biotic potential is used as a reference when looking at growth rates of populations. Are the growth rates close to the biotic potential or far from it and how far? That type of analysis helps population ecologists understand if the conditions for the population are adequate. • It is certainly not common for a population to grow at its biotic potential for a considerable period of time; however, there are situations where this can happen. For example, when fish are introduced into a lake where there is plenty of food and space and there are no predators, the fish can reproduce at their biotic potential, but not for a long time. Another example is when a scientist grows E. coli2 on a petri dish with ideal nutrients. The bacteria will reproduce and grow at its biotic potential, which for E. coli means that the population doubles in size every 20 minutes!. The graph of a population growing at its biotic potential, which is called exponential, can be very steep.

4. For all populations, there are factors that will limit their growth. Some of these factors depend on the population density. The most common density-dependent factors that limit population growth are: • Food and water supplies - A given supply of food and water might be enough for a small population density. However, that same supply might not be enough for a high-density population, and competition among the individuals of the population would develop. • Light - Light is a very common resource needed by plants. A plant density increases those plants that don't get enough light will not grow strong enough and might even die. • Space - This is an obvious limiting factor, especially if you think of fish in a lake, bacteria in a petri dish, or plants in a forest. • Predators - Higher densities of a prey population attract more predators and as the number of prey increases, so does the number of predators. On the other hand, if the number of prey decreases, so does the number of predators. • Diseases - Diseases can certainly have an impact on birth rate and thus affect growth rate. Since many diseases are contagious, they are therefore dependent on density. • Parasitism - Parasitism is a relationship where one organism (the parasite) feeds on the tissues or body fluids of another organism (the host). In this type of relationship the parasite benefits and the host is harmed, sometimes to the point of killing the host. Like diseases, since parasites spread easier in a high-density host, their impact depends on the density.

5. Breaking It Down to Logistic • There are also limiting factors that don't depend on the population density. These density-independent factors are abiotic factors such as weather storms, fires, earthquakes, or floods. Any of these factors can have a severe impact on population sizes regardless of density. • To conclude this section, we will describe the carrying capacity of an ecosystem. The area occupied by a population does not have unlimited resources such as food, water, and supplies to build and keep a dwelling. These factors limit the population growth and many times bring about death rates that equal the birth rates. When this happens, the population size reaches a stable balance. So one could say that there is a certain number of individuals of the population that can be supported by the environmental resources in a given ecosystem. That is called the carrying capacity of that ecosystem. The graph of a population that grows until it reaches a stable size based on the carrying capacity of the ecosystem is called an S-shaped curve. - Logistic

6. The logistic curve, sometimes referred to as an "S-shaped" curve, initially follows a similar pattern as the exponential growth curve; that is, population growth is slow initially, then enters into a point where growth is rising rapidly. The logistic model is useful in describing populations which exhibit exponential growth at small populations but who live in environments which enforce an upper limit on population size. The logistic population is usually written: Exponential Formula 1 Formula 2 It is divided into two groups

7. Formula Key • Formula 1 • dN/dt refers to the population growth rate, • rm refers to the maximum instantaneous per capita rate of population growth, • r refers to the instantaneous per capita rate of population growth, • N refers to the population size, • K refers to the carrying capacity, • t refers to a specific time, • tau refers to the length of a time • lag, and theta can be used to create a model in which the response of per capita population growth decreases non-linearly with population size. • Formula 2 • Parameter r in the exponential model can be interpreted as a difference between the birth (reproduction) rate and the death rate: • where b is the birth rate and m is the death rate. Birth rate is the number of offspring organisms produced per one existing organism in the population per unit time. Death rate is the probability of dying per one organism. The rate of population growth (r) is equal to birth rate (b) minus death rate (m).

8. Exponential model • Applications of the exponential model • microbiology (growth of bacteria), • conservation biology (restoration of disturbed populations), • insect rearing (prediction of yield), • plant or insect quarantine (population growth of introduced species), • fishery (prediction of fish dynamics • Assumptions of Exponential Model: • Continuous reproduction (e.g., no seasonality) • All organisms are identical (e.g., no age structure) • Environment is constant in space and time (e.g., resources are unlimited • Graphical unchecked or nonregulated growth is commonly represented by the exponential growth curve

9. Exponential Model • Exponential modelis associated with the name of Thomas Robert Malthus (1766-1834) who first realized that any species can potentially increase in numbers according to a geometric series. For example, if a species has non-overlapping populations (e.g., annual plants), and each organism produces R offspring, then, population numbers N in generations t=0,1,2,... is equal to: • When t is large, then this equation can be approximated by an exponential function: • There are 3 possible model outcomes: • Population exponentially declines (r < 0) • Population exponentially increases (r > 0) • Population does not change (r = 0) • Parameter r is called: • Malthusian parameter • Intrinsic rate of increase • Instantaneous rate of natural increase • Population growth rate

10. Logistic Model • Logistic model was developed by Belgian mathematician Pierre Verhulst (1838) who suggested that the rate of population increase may be limited, i.e., it may depend on population density: • At low densities (N < < K), the population growth rate is maximal and equals to ro. Parameter ro can be interpreted as population growth rate in the absence of intra-specific competition.

11. Logistic Formula • Population growth rate declines with population numbers, N, and reaches 0 when N = K. Parameter K is the upper limit of population growth and it is called carrying capacity. It is usually interpreted as the amount of resources expressed in the number of organisms that can be supported by these resources. If population numbers exceed K, then population growth rate becomes negative and population numbers decline. The dynamics of the population is described by the differential equation:

12. After Thoughts • Eventually the population stops increasing and reaches its maximum level or "carrying capacity." The maximum population size that can be reached is based on the availability of light, in the case of plants, or food, shelter, etc. Most populations never approach the "carrying capacity" but instead remain at lower levels because of the regulating effects of both abiotic and biotic factors. • Note that populations do not typically remain at a steady state continually but instead tend to fluctuate or oscillate around some characteristic density.