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In graphs of survivorship curves, I'm seeing that the Type II curves are straight lines, and the supplementary text says that the mortality rate is constant (i.e. the slope of the line is constant). However, it's also clearly stated that the y-axis is a logarithmic scale, which means that the original Type II curve is exponential: $$ln y=-rx+b$$ $$y=Ae^{-rx}$$

This implies that the real mortality rate is not constant, but changes as $$y'=-Ar(e^{-rx})$$ How are we defining the word *rate*, anyway?

$r$ is the individual mortality rate per time step. Survivorship curves (plotted on a log scale) show the proportion of individuals surviving with time, and with a Type II curve a constant proportion is dying at each time step (constant mortality with age, $r$). When the model is expressed as $y$ you are looking at the number of individuals surviving to a time step, which is an negative exponential for Type II, since a smaller and smaller proportion of the population remains at each time step. Therefore $y'$ is the change (slope) in the number of individuals surviving over time. This can be seen as a rate, but it is not the mortality rate of individuals, and it is not accurate to describe it as the "… *real mortality rate… *".

For an introduction to survivorship curves you can also look at "Survivorship Curves" from Nature Education

**For Discrete Time**

There are two quantities you should be careful not to mix up.

- One is the
*number*of individuals who will die during a given interval: $d_x = N_x - N_{x+1}$. - One is the
*fraction*, out of those alive at the beginning of a given interval, who will die during the interval: $q_x = frac{d_x}{N_x}$

($N_x$ being the number of survivors at age $x$.)

On a *regular* plot, if $d_x$ is constant then survivorship will decrease linearly; if $q_x$ is constant then survivorship will decrease geometrically. On a *semilog* plot, when $q_x$ is constant then this geometric decrease will look like a straight line. This is, as far as I know, the main reason to even use semilog plots in the first place; it makes geometric decrease easy to recognize. $d_x$ is sometimes called the *death rate* and $q_x$ is sometimes called the *mortality rate*.

(In my opinion this usage is careless; they aren't rates, but just numbers. Compare: there is a distinction between travelling 10 miles in an hour, and travelling *at* 10 miles *per* hour.)

$q_x$ can also be thought of as estimating the probability that someone who is age $x$ will die before age $(x+1)$. It is therefore sometimes also called the *"Age-Specific Probability of Death"*. Notice that it is bounded between 0 and 1.

(Probabilities can't be lower than 0 or higher than 1; and the fraction dying has to be somewhere in between "none of them" and "all of them"!)

Here's an example survivorship curve, on a regular plot, I made for constant $q_x = 0.1$.

This much should be enough to answer your question. But, for completeness…

**For Continuous Time**

I complained about calling $d_x$ the death rate. That's because rates have a unit of $time^{-1}$ ("… *per second*", "… *per hour*" etc.) Of course, dividing $d_x$ by the *length* of the interval, does give you the average death rate over the interval. And,

- The
*slope*of the survivorship curve (multiplied by -1; we want a positive rate of decrease, instead of a negative rate of increase) gives you the*instantaneous*death rate at age $x$: $-N^{'}_{x}$.

(For instantaneous rates to be useful, we have to "smooth-out" the survivorship curve instead of letting it be a stepped curve. If it was stepped, then the slope would just always be horizontal or vertical.)

This brings us to another quantity: the "*Force of Mortality*":

- The instantaneous, age-specific,
*per-capita death rate*; or, the*death rate*at age $x$, divided by the*cohort size*at age $x$: $mu_x = -frac{N^{'}_{x}}{N_x}$

To illustrate, if the death rate at age $x$ is "*60 deaths per minute*", and if at that instant there are 60 people, then the per-capita death rate is "*1 death per person per minute*". Because choice of time unit is arbitrary, this is the same as saying "*60 deaths per person per hour*" or "*525600 deaths per person per year*".

The Force of Mortality is unintuitive ("per-capita" death rate? - but everybody only dies once!) but the formula is coherent. The point of having a "per-capita rate" is to enable comparison between cohorts of different sizes, or the *same* cohort at different ages when it was bigger vs. smaller, and so on; in general the idea is that death rate has something to do with the number of members, but also something to do with the condition of each member, and $mu_x$ is trying to get at the latter. It can be visualized as the *slope* of the survivorship curve (multiplied by -1 to make it positive), divided by the *height* of the survivorship curve.

There is a similarity between these two quantities, and the two which were defined for discrete time. If $-N^{'}_{x}$ is constant, then so is $d_x$; and if $mu_x$ is constant, then so is $q_x$. And, again: on a regular plot, if $-N^{'}_{x}$ is constant then survivorship will decrease linearly; and if $mu_x$ is constant then survivorship will decrease exponentially.

But they are distinct. It is the $time^{-1}$ dimension, and the fact that time units are interchangeable, which makes the difference.

If there are only 60 people, then it's obviously not possible for more than 60 people to die in the next year $(0 le d_x le 60)$; and it's not possible for the fraction dying to be higher than "all of them" $(0 le q_x le 1)$.

But suppose the *rate* of death at this instant is "*60 deaths per hour*". That's the same thing as saying it's "*1 death per minute*" or "*525600 deaths per year*". There's no implication that that many people will in fact die, because there's no implication that the rate will be *kept up* for the whole year. If this particular death rate were held constant as long as possible, then the cohort would all be dead in exactly an hour and then the rate would then hit 0. The instantaneous death rate can be as high as you like; it just can't stay high *forever*.

(If a death rate of "*60 deaths per hour*" is measured at an instant when there are 60 people, then the per-capita death rate is "*1 death per person per hour*" - or, equivalently, "*$frac{1}{60}^{th}$ of a death per person per hour*", or "*8760 deaths per person per year*".)

Although the absolute death rate can't stay high forever, the per-capita death rate ($mu_x$) *can*. That's what happens during constant $mu_x$ (exponentially decreasing $N_x$), for example. The reason why this is not paradoxical is that the slope ($-N^{'}_{x}$; the numerator) is constantly decreasing; it's just that so is the height ($N_x$; the denominator).

The Force of Mortality $mu_x$ has *also* sometimes been referred to as the "*mortality rate*". The equivocal use of language is unfortunate. It is also, especially in reliability engineering, known as the "hazard rate".

**By the way**

$q_x$ and $mu_x$ have often been conflated, at least in gerontology (with which I am most familiar). But they are not the same. The first estimates a probability, and is bounded between 0 and 1. The second is *not* a probability, and has no upper bound.

One way they have been conflated is with the "**Gompertz equation**".

A Type II Survivorship Curve is one which is decreasing exponentially. When this is the case, both $q_x$ and $mu_x$ are constant. We would call such a species "non-aging" or "non-senescing": your age makes no difference to your vulnerability.

But many species do age.

Benjamin Gompertz proposed that for many species including humans $mu_x$ grows exponentially throughout adulthood:

- Gompertz' Law of Mortality: $mu_x = mu_0 cdot e^{Gx}$
- Or, in its logarithmic form: $ln(mu_x) = ln(mu_0) + Gx$

(Where $mu_0$ is the "initial mortality" and $G$ is the exponential "Gompertz parameter". In the logarithmic form, $ln(mu_0)$ is the intercept, and $G$ is the slope.)

Whether or not $mu_x$ follows such a law *indefinitely* is an empirical matter. (It's actually debated whether it decelerates in late life, and if so why.) But notice that $q_x$ couldn't *possibly* grow exponentially forever, since $q_x$ couldn't *possibly* ever exceed 1. And yet, due to the conflation, in recent times it has often been in terms of $q_x$ that the Gompertz equation has been presented and discussed.

**Further Reading**

Unfortunately, as far as I know the literature (both professional and pedagogical) is sorely lacking in treatments which are both intuitive and accurate!

Peter Medawar's "The Definition and Measurement of Senescence" (Chapter 1 in *The CIBA Foundation Colloquia on Ageing, Vol. 1*) is probably still the best introduction to the rationale behind why we care about survivorship curves and mortality, from the perspective of biogerontology. Medawar gives the correct definition and formula for $mu_x$ (though he doesn't stress its distinction from $q_x$; indeed, he doesn't mention $q_x$ at all).

The distinction between $q_x$ and $mu_x$ is discussed in Gavrilov and Gavrilova's *The Biology of Life Span: A Quantitative Approach*, as well as in some of their papers. They present the correct forumula for $mu_x$, but do not really try to explain it.

This is a statistical property of the curve - in time to event analysis (which is what a survivorship curve is), a constant hazard (the instantaneous probability of an event occurring in time t given it has not occurred already) will yield an exponentially distributed survival function. When graphed on a log axis, this function looks like a straight line.

## What animals have a Type 2 survivorship curve?

A **type II survivorship curve** shows a roughly constant mortality rate for the species through its entire life. This means that the individual's chance of dying is independent of their age. **Type II survivorship curves** are plotted as a diagonal line going downward on a graph.

Also, what animal has a Type 3 survivorship curve? The Type III curve, characteristic of small mammals, **fishes**, and invertebrates, is the opposite: it describes organisms with a high death rate (or low survivorship rate) immediately following birth. In contrast, the Type II curve considers birds, mice, and other organisms characterized by a relatively constant&hellip

Keeping this in view, what animals have a Type 1 survivorship curve?

**Type** I. Humans and most primates **have a Type** I **survivorship curve**. In a **Type** I **curve**, organisms tend not to die when they are young or middle-aged but, instead, die when they become elderly.

What type of survivorship curve Do turtles have?

The hypothesis that the local population of **turtles** in Cougar Lake would follow a **Type** II **survivorship curve** was completely supported by the data (Figure 1). This **type of survivorship curve** indicates that **turtles have** about the same probability of surviving to the next year throughout their entire lives.

## Survivorship curve

Our editors will review what you’ve submitted and determine whether to revise the article.

**Survivorship curve**, graphic representation of the number of individuals in a population that can be expected to survive to any specific age.

There are three general types of curves. The Type I curve, illustrated by the large mammals, tracks organisms that tend to live long lives (low death rate and high survivorship rate) toward the end of their life expectancies, however, there is a dramatic increase in the death rate. The Type III curve, characteristic of small mammals, fishes, and invertebrates, is the opposite: it describes organisms with a high death rate (or low survivorship rate) immediately following birth. In contrast, the Type II curve considers birds, mice, and other organisms characterized by a relatively constant mortality or survivorship rate throughout their life expectancies.

This article was most recently revised and updated by John P. Rafferty, Editor.

## Which type of organism would be most likely to have a type II survivorship curve?

In contrast, the **Type II curve** considers birds, mice, and other **organisms** characterized by a relatively constant mortality or **survivorship** rate throughout their life expectancies. Certain lizards, perching birds, and rodents exhibit this **type** of **survivorship curve**.

Secondly, which type of survivorship curve applies to humans? **Type** I or convex **curves** are characterized by high age-specific survival probability in early and middle life, followed by a rapid decline in survival in later life. They are typical of species that produce few offspring but care for them well, including **humans** and many other large mammals.

Herein, which organism is most likely to have a type III survivorship curve?

Trees and marine invertebrates exhibit a **type III survivorship curve** because very few of these **organisms** survive their younger years, but those that do **make** it to an old age are more **likely** to survive for a relatively long period of time.

What type of survivorship curve do elephants have?

**Elephants have** a **Type** I **survivorship curve** (mortality increases with age), and fecundity decreases with age.

## Species Distribution

In addition to measuring size and density, further information about a population can be obtained by looking at the distribution of the individuals throughout their range. A species distribution pattern is the distribution of individuals within a habitat at a particular point in time—broad categories of patterns are used to describe them.

Individuals within a population can be distributed at random, in groups, or equally spaced apart (more or less). These are known as **random, clumped, and uniform distribution patterns**, respectively (Figure 2). Different distributions reflect important aspects of the biology of the species. They also affect the mathematical methods required to estimate population sizes. An example of random distribution occurs with dandelion and other plants that have wind-dispersed seeds that germinate wherever they happen to fall in favorable environments. A clumped distribution, may be seen in plants that drop their seeds straight to the ground, such as oak trees it can also be seen in animals that live in social groups (schools of fish or herds of elephants). Uniform distribution is observed in plants that secrete substances inhibiting the growth of nearby individuals (such as the release of toxic chemicals by sage plants). It is also seen in territorial animal species, such as penguins that maintain a defined territory for nesting. The territorial defensive behaviors of each individual create a regular pattern of distribution of similar-sized territories and individuals within those territories. Thus, the distribution of the individuals within a population provides more information about how they interact with each other than does a simple density measurement. Just as lower density species might have more difficulty finding a mate, solitary species with a random distribution might have a similar difficulty when compared to social species clumped together in groups.

Figure 2. Species may have a random, clumped, or uniform distribution. Plants such as (a) dandelions with wind-dispersed seeds tend to be randomly distributed. Animals such as (b) elephants that travel in groups exhibit a clumped distribution. Territorial birds such as (c) penguins tend to have a uniform distribution. (credit a: modification of work by Rosendahl credit b: modification of work by Rebecca Wood credit c: modification of work by Ben Tubby)

**Life tables** provide important information about the life history of an organism and the life expectancy of individuals at each age. They are modeled after actuarial tables used by the insurance industry for estimating human life expectancy. Life tables may include the probability of each age group dying before their next birthday, the percentage of surviving individuals dying at a particular age interval, their mortality rate, and their life expectancy at each interval. An example of a life table is shown in Table 1 from a study of Dall mountain sheep, a species native to northwestern North America. Notice that the population is divided into age intervals (column A).

As can be seen from the mortality rate data (column D), a high death rate occurred when the sheep were between six months and a year old, and then increased even more from 8 to 12 years old, after which there were few survivors. The data indicate that if a sheep in this population were to survive to age one, it could be expected to live another 7.7 years on average, as shown by the life-expectancy numbers in column E.

Table 1. This life table of

*Ovis dalli*shows the number of deaths, number of survivors, mortality rate, and life expectancy at each age interval for Dall mountain sheep.

Life Table of Dall Mountain Sheep 1 | ||||
---|---|---|---|---|

Age interval (years) | Number dying in age interval out of 1000 born | Number surviving at beginning of age interval out of 1000 born | Mortality rate per 1000 alive at beginning of age interval | Life expectancy or mean lifetime remaining to those attaining age interval |

0–0.5 | 54 | 1000 | 54.0 | 7.06 |

0.5–1 | 145 | 946 | 153.3 | — |

1–2 | 12 | 801 | 15.0 | 7.7 |

2–3 | 13 | 789 | 16.5 | 6.8 |

3–4 | 12 | 776 | 15.5 | 5.9 |

4–5 | 30 | 764 | 39.3 | 5.0 |

5–6 | 46 | 734 | 62.7 | 4.2 |

6–7 | 48 | 688 | 69.8 | 3.4 |

7–8 | 69 | 640 | 107.8 | 2.6 |

8–9 | 132 | 571 | 231.2 | 1.9 |

9–10 | 187 | 439 | 426.0 | 1.3 |

10–11 | 156 | 252 | 619.0 | 0.9 |

11–12 | 90 | 96 | 937.5 | 0.6 |

12–13 | 3 | 6 | 500.0 | 1.2 |

13–14 | 3 | 3 | 1000 | 0.7 |

Another tool used by population ecologists is a **survivorship curve**, which is a graph of the number of individuals surviving at each age interval versus time. These curves allow us to compare the life histories of different populations (Figure 3). There are three types of survivorship curves. In a type I curve, mortality is low in the early and middle years and occurs mostly in older individuals. Organisms exhibiting a type I survivorship typically produce few offspring and provide good care to the offspring increasing the likelihood of their survival. Humans and most mammals exhibit a type I survivorship curve. In type II curves, mortality is relatively constant throughout the entire life span, and mortality is equally likely to occur at any point in the life span. Many bird populations provide examples of an intermediate or type II survivorship curve. In type III survivorship curves, early ages experience the highest mortality with much lower mortality rates for organisms that make it to advanced years. Type III organisms typically produce large numbers of offspring, but provide very little or no care for them. Trees and marine invertebrates exhibit a type III survivorship curve because very few of these organisms survive their younger years, but those that do make it to an old age are more likely to survive for a relatively long period of time.

**Attribution**

Population Demography by OpenStax is licensed under CC BY 4.0. Modified from the original by Matthew R. Fisher.

## Biology 2e (0th Edition) Edit edition

Survivorship curve is a graphical representation of surviving individuals of same age at a particular time (cohort). There are three types of survivorship curves- type I, type II and type III.

Type I survivorship curves consist of those species of population which shows high survival capabilities during its young and middle age followed by decline in their population due to mortality in later life. Such organisms reproduce few offspring and invest huge amount of energy and time in caring for them.

Type II survivorship curves show a constant survival and mortality rate of its individuals regardless of age. Example: lizards

Type III survivorship curves produces offspring in large numbers but do not care for them. The offspring that survives the harsh conditions during its young stage will survive for a considerable amount of time. This type show high birth rate and very low mortality rate. Example: trees.

There is no type IV survivorship. There are only type I, type II and type III survivorship curves.

## Multiple Choice

Which of the following methods will provide information to an ecologist about both the size and density of a population?

A. mark and recapture

B. mark and release

C. quadrat

D. life table

Which of the following is best at showing the life expectancy of an individual within a population?

A. quadrat

B. mark and recapture

C. survivorship curve

D. life table

Human populations have which type of survivorship curve?

A. Type I

B. Type II

C. Type III

D. Type IV

## 2.4: Life Tables and Survivorship

Population ecologists use life tables to study species and identify the most vulnerable stages of organisms&rsquo lives to develop effective measures for maintaining viable populations. Life tables, like **Table (PageIndex<1>)**, track **survivorship**, the chance of an individual in a given population surviving to various ages. Life tables were invented by the insurance industry to predict how long, on average, a person will live. Biologists use a life table as a quick window into the lives of the individuals of a population, showing how long they are likely to live, when they&rsquoll reproduce, and how many offspring they&rsquoll produce. Life tables are used to construct **survivorship curves**, which are graphs showing the proportion of individuals of a particular age that are now alive in a population. Survivorship (chance of surviving to a particular age) is plotted on the y-axis as a function of age or time on the x-axis. However, if the percent of maximum lifespan is used on the x-axis instead of actual ages, it is possible to compare survivorship curves for different types of organisms (**Figure (PageIndex<1>)**). All survivorship curves start along the y-axis intercept with all of the individuals in the population (or 100% of the individuals surviving). As the population ages, individuals die and the curves goes down. A survivorship curve never goes up.

* Table (PageIndex<1>):* Life Table for the U.S. population in 2011 showing the number who are expected to be alive at the beginning of each age interval based on the death rates in 2011. For example, 95,816 people out of 100,000 are expected to live to age 50 (0.983 chance of survival). The chance of surviving to age 60 is 0.964 but the chance of surviving to age 90 is only 0.570.

Age (years) | Number Living at Start of Age Interval | Number Dying During Interval | Chance of Surviving Interval | Chance of Dying During Interval |

0-1 | 100000 | 606 | 0.993942 | 0.006058 |

1-5 | 99394 | 105 | 0.998946 | 0.001054 |

5-10 | 99289 | 60 | 0.999397 | 0.000603 |

10-15 | 99230 | 70 | 0.999291 | 0.000709 |

15-20 | 99159 | 242 | 0.997562 | 0.002438 |

20-25 | 98917 | 425 | 0.995704 | 0.004296 |

25-30 | 98493 | 475 | 0.995176 | 0.004824 |

30-35 | 98017 | 553 | 0.994362 | 0.005638 |

35-40 | 97465 | 681 | 0.993015 | 0.006985 |

40-45 | 96784 | 968 | 0.989994 | 0.010006 |

45-50 | 95816 | 1535 | 0.983982 | 0.016018 |

50-55 | 94281 | 2306 | 0.975541 | 0.024459 |

55-60 | 91975 | 3229 | 0.964895 | 0.035105 |

60-65 | 88746 | 4378 | 0.950668 | 0.049332 |

65-70 | 84368 | 6184 | 0.926698 | 0.073302 |

70-75 | 87184 | 8670 | 0.889101 | 0.110899 |

75-80 | 69513 | 12021 | 0.827073 | 0.172927 |

80-85 | 57493 | 15760 | 0.725879 | 0.274121 |

85-90 | 41733 | 17935 | 0.570241 | 0.429759 |

90-95 | 23798 | 14701 | 0.382258 | 0.617742 |

95-100 | 9097 | 7169 | 0.211924 | 0.788076 |

100 and over | 1928 | 1928 | 0 | 1.000000 |

SOURCE: CDC/NCHS, National Vital Statistics System.

Survivorship curves reveal a huge amount of information about a population, such as whether most offspring die shortly after birth or whether most survive to adulthood and likely to live long lives. They generally fall into one of three typical shapes, Types I, II and III (**Figure (PageIndex<1>)a**). Organisms that exhibit **Type I** survivorship curves have the highest probability of surviving every age interval until old age, then the risk of dying increases dramatically. Humans are an example of a species with a Type I survivorship curve. Others include the giant tortoise and most large mammals such as elephants. These organisms have few natural predators and are, therefore, likely to live long lives. They tend to produce only a few offspring at a time and invest significant time and effort in each offspring, which increases survival.

In the **Type III** survivorship curve most of the deaths occur in the youngest age groups. Juvenile survivorship is very low and many individuals die young but individuals lucky enough to survive the first few age intervals are likely to live a much longer time. Most plants species, insect species, frogs as well as marine species such as oysters and fishes have a Type III survivorship curve. A female frog may lay hundreds of eggs in a pond and these eggs produce hundreds of tadpoles. However, predators eat many of the young tadpoles and competition for food also means that many tadpoles don&rsquot survive. But the few tadpoles that do survive and metamorphose into adults then live for a relatively long time (for a frog). The mackerel fish, a female is capable of producing a million eggs and on average only about 2 survive to adulthood. Organisms with this type of survivorship curve tend to produce very large numbers of offspring because most will not survive. They also tend not to provide much parental care, if any.

**Type II** survivorship is intermediate between the others and suggests that such species have an even chance of dying at any age. Many birds, small mammals such as squirrels, and small reptiles, like lizards, have a Type II survivorship curve. The straight line indicates that the proportion alive in each age interval drops at a steady, regular pace. The likelihood of dying in any age interval is the same.

In reality, most species don&rsquot have survivorship curves that are definitively type I, II, or III. They may be anywhere in between. These three, though, represent the extremes and help us make predictions about reproductive rates and parental investment without extensive observations of individual behavior. For example, humans in less industrialized countries tend to have higher mortality rates in all age intervals, particularly in the earliest intervals when compared to individuals in industrialized countries. Looking at the population of the United States in 1900 (**Figure (PageIndex<1>)b**), you can see that mortality was much higher in the earliest intervals and throughout, the population seemed to exhibit a type II survivorship curve, similar to what might be seen in less industrialized countries or amongst the poorest populations.

## Population Size and Density

Populations are characterized by their population size (total number of individuals) and their population density (number of individuals per unit area). A population may have a large number of individuals that are distributed densely, or sparsely. There are also populations with small numbers of individuals that may be dense or very sparsely distributed in a local area. Population size can affect potential for adaptation because it affects the amount of genetic variation present in the population. Density can have effects on interactions within a population such as competition for food and the ability of individuals to find a mate. Smaller organisms tend to be more densely distributed than larger organisms (Figure (PageIndex<1>)).

Figure (PageIndex<1>). Australian mammals show a typical inverse relationship between population density and body size. As this graph shows, population density typically decreases with increasing body size. Why do you think this is the case?

### Estimating Population Size

The most accurate way to determine population size is to count all of the individuals within the area. However, this method is usually not logistically or economically feasible, especially when studying large areas. Thus, scientists usually study populations by sampling a representative portion of each habitat and use this sample to make inferences about the population as a whole. The methods used to sample populations to determine their size and density are typically tailored to the characteristics of the organism being studied. For immobile organisms such as plants, or for very small and slow-moving organisms, a quadrat may be used. A **quadrat** is a square structure that is randomly located on the ground and used to count the number of individuals that lie within its boundaries. To obtain an accurate count using this method, the square must be placed at random locations within the habitat enough times to produce an accurate estimate.

For smaller mobile organisms, such as mammals, a technique called **mark and recapture** is often used. This method involves marking captured animals in and releasing them back into the environment to mix with the rest of the population. Later, a new sample is captured and scientists determine how many of the marked animals are in the new sample. This method assumes that the larger the population, the lower the percentage of marked organisms that will be recaptured since they will have mixed with more unmarked individuals. For example, if 80 field mice are captured, marked, and released into the forest, then a second trapping 100 field mice are captured and 20 of them are marked, the population size (*N*) can be determined using the following equation:

Using our example, the equation would be:

These results give us an estimate of 400 total individuals in the original population. The true number usually will be a bit different from this because of chance errors and possible bias caused by the sampling methods.

### Species Distribution

In addition to measuring size and density, further information about a population can be obtained by looking at the distribution of the individuals throughout their range. A species distribution pattern is the distribution of individuals within a habitat at a particular point in time&mdashbroad categories of patterns are used to describe them.

Individuals within a population can be distributed at random, in groups, or equally spaced apart (more or less). These are known as **random, clumped, and uniform distribution patterns**, respectively (Figure (PageIndex<2>)). Different distributions reflect important aspects of the biology of the species. They also affect the mathematical methods required to estimate population sizes. An example of random distribution occurs with dandelion and other plants that have wind-dispersed seeds that germinate wherever they happen to fall in favorable environments. A clumped distribution, may be seen in plants that drop their seeds straight to the ground, such as oak trees it can also be seen in animals that live in social groups (schools of fish or herds of elephants). Uniform distribution is observed in plants that secrete substances inhibiting the growth of nearby individuals (such as the release of toxic chemicals by sage plants). It is also seen in territorial animal species, such as penguins that maintain a defined territory for nesting. The territorial defensive behaviors of each individual create a regular pattern of distribution of similar-sized territories and individuals within those territories. Thus, the distribution of the individuals within a population provides more information about how they interact with each other than does a simple density measurement. Just as lower density species might have more difficulty finding a mate, solitary species with a random distribution might have a similar difficulty when compared to social species clumped together in groups.

Figure (PageIndex<2>). Species may have a random, clumped, or uniform distribution. Plants such as (a) dandelions with wind-dispersed seeds tend to be randomly distributed. Animals such as (b) elephants that travel in groups exhibit a clumped distribution. Territorial birds such as (c) penguins tend to have a uniform distribution. (credit a: modification of work by Rosendahl credit b: modification of work by Rebecca Wood credit c: modification of work by Ben Tubby)

**Life tables** provide important information about the life history of an organism and the life expectancy of individuals at each age. They are modeled after actuarial tables used by the insurance industry for estimating human life expectancy. Life tables may include the probability of each age group dying before their next birthday, the percentage of surviving individuals dying at a particular age interval (their mortality rate, and their life expectancy at each interval. An example of a life table is shown in Table 1 from a study of Dall mountain sheep, a species native to northwestern North America. Notice that the population is divided into age intervals (column A).

As can be seen from the mortality rate data (column D), a high death rate occurred when the sheep were between six months and a year old, and then increased even more from 8 to 12 years old, after which there were few survivors. The data indicate that if a sheep in this population were to survive to age one, it could be expected to live another 7.7 years on average, as shown by the life-expectancy numbers in column E.

Table 1: Life Table of Dall Mountain Sheep

Age interval (years) | Number dying in age interval out of 1000 born | Number surviving at beginning of age interval out of 1000 born | Mortality rate per 1000 alive at beginning of age interval | Life expectancy or mean lifetime remaining to those attaining age interval |
---|---|---|---|---|

0&ndash0.5 | 54 | 1000 | 54.0 | 7.06 |

0.5&ndash1 | 145 | 946 | 153.3 | &mdash |

1&ndash2 | 12 | 801 | 15.0 | 7.7 |

2&ndash3 | 13 | 789 | 16.5 | 6.8 |

3&ndash4 | 12 | 776 | 15.5 | 5.9 |

4&ndash5 | 30 | 764 | 39.3 | 5.0 |

5&ndash6 | 46 | 734 | 62.7 | 4.2 |

6&ndash7 | 48 | 688 | 69.8 | 3.4 |

7&ndash8 | 69 | 640 | 107.8 | 2.6 |

8&ndash9 | 132 | 571 | 231.2 | 1.9 |

9&ndash10 | 187 | 439 | 426.0 | 1.3 |

10&ndash11 | 156 | 252 | 619.0 | 0.9 |

11&ndash12 | 90 | 96 | 937.5 | 0.6 |

12&ndash13 | 3 | 6 | 500.0 | 1.2 |

13&ndash14 | 3 | 3 | 1000 | 0.7 |

Another tool used by population ecologists is a **survivorship curve**, which is a graph of the number of individuals surviving at each age interval versus time. These curves allow us to compare the life histories of different populations (Figure (PageIndex<3>)). There are three types of survivorship curves. In a type I curve, mortality is low in the early and middle years and occurs mostly in older individuals. Organisms exhibiting a type I survivorship typically produce few offspring and provide good care to the offspring increasing the likelihood of their survival. Humans and most mammals exhibit a type I survivorship curve. In type II curves, mortality is relatively constant throughout the entire life span, and mortality is equally likely to occur at any point in the life span. Many bird populations provide examples of an intermediate or type II survivorship curve. In type III survivorship curves, early ages experience the highest mortality with much lower mortality rates for organisms that make it to advanced years. Type III organisms typically produce large numbers of offspring, but provide very little or no care for them. Trees and marine invertebrates exhibit a type III survivorship curve because very few of these organisms survive their younger years, but those that do make it to an old age are more likely to survive for a relatively long period of time.

Figure (PageIndex<3>). Survivorship curves show the distribution of individuals in a population according to age. Humans and most mammals have a Type I survivorship curve, because death primarily occurs in the older years. Birds have a Type II survivorship curve, as death at any age is equally probable. Trees have a Type III survivorship curve because very few survive the younger years, but after a certain age, individuals are much more likely to survive.

## Life tables are a valuable tool to examine how age structure can change a population’s growth trajectory

Population demography is the study of numbers and rates in a population and how they change over time. The basic tool of demography is the **life table**. Life tables are an analytical tool that population ecologists use to study age-specific population characteristics such as survival, fecundity, and mortality. These data can be critical in conservation efforts (such as reintroductions or pest reductions) where ecologists would like to know how well an endangered or transplanted population is doing.

Life tables determine the number of individuals that survive from one age group to the next. Cohort life tables follow one group of individuals born at the same time, called a cohort, until the death of all individuals. This technique of demographic assessment requires key assumptions:

- The population sample of each age class is proportional to its numbers in the population
- Age-specific mortality rates remain constant during the time period, meaning that subsequent cohorts will exhibit similar pattern of birth and death.

Life Table:

The first row represents the birth year of the cohort, and each subsequent row of the life table shows that same group one year older. Assuming that the unit of age (x) is years, the number alive (nx) column indicates that not all individuals survive from year to year. **Survivorship** converts that mortality into a proportion alive of the original cohort (lx = nx/n0). The average number of offspring born to individuals of each age is **age-specific fecundity**, and it cannot be calculated from other information provided in the table but instead must be estimated from data.

Here’s the best bit and the reason we bother to gather all the age-specific survivorship and fecundity information: if the assumptions (1 and 2 above) are met, then the sum of the product of survivorship and fecundity at each age gives a population growth parameter called R0 (pronounced R-nought), defined as the **net reproductive rate**. When R0 exceeds 1, the population is producing more offspring than it is losing from deaths. In other words, the population is growing.

- Is the population above growing, shrinking, or stable?
- At what age is fecundity maximized? Survivorship?

Because of life history trade-offs, patterns of age-specific survival are predictive of the general life history of a population. While a life table shows the survivorship in a numerical form, assessing pattern from columns of data is difficult. Instead, ecologists create **survivorship curves** by plotting lx versus time.

Population biologists look for three types of patterns in survivorship curves (note that the y-axis is a log scale):

Survivorship curves show the distribution of individuals in a population according to age. Humans and most mammals have a Type I survivorship curve, because death primarily occurs in the older years. Birds have a Type II survivorship curve, as death at any age is equally probable. Trees have a Type III survivorship curve because very few survive the younger years, but after a certain age, individuals are much more likely to survive. (Source: OpenStax Biology)

Type I curves are observed in populations with low mortality in young age classes but very high mortality as an individual ages. Type II curves represent populations where the mortality rate is constant, regardless of age. Type III curves occur in populations with high mortality in early age classes and very low mortality in older individuals. Populations displaying a Type III survivorship curve generally need to have high birth rates in order for the population size to remain constant. High birth rates ensure that enough offspring survive to reproduce, ensuring the population sustains itself. In contrast, populations characterized by a Type I survivorship curve often have low birth rates because most offspring survive to reproduce, and very high birth rates result in exponential population growth.

This video provides an overview of survivorship curves and some of the nuance in interpreting these plots:

As noted in the video, species with Type I survivorship curves tend to have K-selected traits, while species with Type III (or sometimes Type II) survivorship curves tend to have r-selected traits.

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