Population

DR. SHAHRUL ANUAR MOHD SAH

Basic terms: Ecology - the study of relationships between organisms and their environment. Population - a group of interacting individuals of the same species. Community - a group of interacting populations. Ecosystem - a functional environmental unit, consisting of a biotic community and the abiotic (nonliving) factors on which the organisms depend. Biosphere - the total of all ecosystems. In other words, all the inhabited area on earth. Characteristics of Populations ●A

population is a group of plants, animals, or other organisms, all of the same species, that live together and reproduce. ●To study populations, you need to study the individuals that make up the population. But are all individual created equal? What is a population? Webster's Third New International Dictionary -

"The total number or amount of things especially within a given area." "The organisms inhabiting a particular area or biotype." "A group of interbreeding biotypes that represents the level of organization at which speciation begins." Krebs (1972:139) A group of organisms of the same species occupying a particular space at a particular time.

1

DR. SHAHRUL ANUAR MOHD SAH

La Monte Cole (1957) A biological unit at the level of ecological integration where it is meaningful to speak of birth rate, death rate, sex ratios, and age structure in describing properties or parameters of the unit. Things to consider about individuals: ●Almost

all species pass through a number of stages in their life cycle. Insects metamorphose from eggs to larvae to adults; plants pass from seeds to seedlings to photosynthesizing adults, etc. ●In

all such cases, the different stages are likely to be influenced by different factors and to influence birth, death, immigration, and emigration of individuals, and should be treated separately. A second thing to consider about individuals: ●Even

within a stage, or where there are no separate stages, individuals commonly differ in “quality,” such as size, stored reserves, etc. And yet one more thing to consider about individuals: ●What

constitutes an individual depends on whether the organism is unitary or modular. ●In unitary organisms, form is highly determinant. For example, all dogs have four legs, all locusts have six legs. Humans are unitary organisms.

2

DR. SHAHRUL ANUAR MOHD SAH

Unitary Organisms

Life begins when a sperm fertilizes an egg to form a zygote. The fetus grows until birth, then the infant grows until it reaches sexual maturity. The reproductive phase lasts a certain number of years, followed by a phase of senescence. The succession of phases is completely predictable. ●

But not all organisms are unitary ●In

modular organisms, in contrast, the zygote develops into a unit of construction (the module) which then produces further modules like the first. The product is almost always branched and, except for a juvenile phase, immobile. ●Individuals are composed of a highly variable number of basic elements, and the program of development is unpredictable and strongly dependent on their interaction with the environment. ●Most plants are modular, and so are sponges, corals, and bryozoans. Population Biology

•Population

Ecology Changes in the number of individuals in time and/or space -Inter- and intra-specific competition -Predation -Resource availability Population Ecology - the science devoted to understanding: ●The distribution of organisms (what factors shape distributions?) ●The abundance of organisms (what factors control abundances?) •Evolutionary

Biology

Evolutionary biology - the science devoted to understanding:

3

DR. SHAHRUL ANUAR MOHD SAH

●How

populations change through time in response to modifications in their social, biological, and physical environments (adaptation) ●How new species come into being (diversity) Population genetics -microevolution – what maintains genetic variability Evolutionary ecology -design of phenotypes for reproductive success (e.g. age and size at reproduction, clutch size).

●Basic Population Growth

POPULATION MODELS

–Elements of population growth –BIDE parameters –Instantaneous birth and death rates –Instantaneous rate of increase, r

●Continuous growth equation, dN/dt = rN

Elements of Population Growth

●Let

the variable N indicate the size of the population, and the subscript t indicate the point in time we are talking about. ●Thus, Nt is the number of individuals in the population at time t. ●Nt = 4 deer Some formalities: ●By

convention, we use t = 0 to indicate the starting population. For example, say you census a population of bees and count 500 bees at the beginning of the study. You revisit the population in one year and count bees. ●N0 = 500 ●N1 = 800. Why do populations change in number?

4

DR. SHAHRUL ANUAR MOHD SAH

Populations can increase through births (B) or can decrease through deaths (D). ●Populations can increase or decrease by movement of individuals to or from other populations. The population can increase by immigration (I) or decrease by emigration(E). ●

●

Birth Immigration Death Emigration (BIDE) N0 = 500 ; N1 = 800 How did the population of bees increase by 300?

●Initial population of 500 bees might have produced 400 new bees during

the year and lost 100 adults to death, with no movement of individuals to or from the population. ●Or, there might have been 50 births and 50 deaths, with 300 individuals leaving (emigration) and 600 bees arriving from a different population. ●How would you write these mathematically? BIDE Parameters ●Birth,

Immigration, Death, and Emigration are called the BIDE parameters, and can be incorporated into a mathematical expression for population growth: Nt+1 800 bees

= Nt + B - D + I - E. = 500 bees (Nt) + 200 baby bees (B) - 300 dead bees (D) + 400 new bees (I) 0 emigrants (E)

Population Change:

5

DR. SHAHRUL ANUAR MOHD SAH

●Ecologists

are often interested in knowing what the difference between this year’s population and last year’s population was. ● ΔN indicates the change in size from one time period to the next. We get that by subtracting Nt from both sides of our expression. ●Nt+1 - Nt = Nt - Nt + B - D + I - E. ● ΔN = B - D + I – E ΔN = B - D + I – E ●To

simplify things, we will assume the population is closed; or there is no movement of individuals between sites (I and E = 0). ●ΔN = B - D ●For our population of ants that increased by 300 in a year, by how much would the population have increased after 6 months? ●ΔN = 150 ants … or (300 * 182 days /365 days = 150 ants) And now let’s look at what happens to the population when you consider even shorter time intervals: ●How many ants would there be if we censused the population after just 2 days? ●DN= (2 days / 365 days) * 300 = 1.64384 additional ants! Instantaneous Birth Rates ●If

each individual produces the same number of offspring during a short time interval, the number of births in the population will be directly proportional to population size. (As size of population gets bigger, the number of births gets bigger). ●Let

b denote the instantaneous birth rates. The units of b are the number of births per individual per unit time (births / individual * time). Over a short time interval, the number of

6

DR. SHAHRUL ANUAR MOHD SAH

births in the population (B) is the product of the instantaneous birth rate and the population size: B = bN B=bN ●What

is the instantaneous birth rate, b, for our ant population? Recall that B = 300, and N = 800 ●B=bN, therefore: ●300 = b (800); solve for b, b = 300/800; thus b = 0.375 births per individual per year. For a population in which 2 young are born per individual per year (b = 2), the total population in the next year is proportional to the initial population size. ●

Instantaneous Death Rates ●Similarly,

we can define the instantaneous death rate d, with units being number of deaths per individual per unit time (deaths / individual * time). ●Of course, an individual either lives or dies, but this represents death rates for a population in a short period of time. ●D = d N Now we’ll do a little substituting: B = b N and D = d N ●dN

/ dt = B - D

●dN

/ dt = b N - d N

●dN

/ dt = b-d (N)

substituting

dN / dt = b-d (N) 7

DR. SHAHRUL ANUAR MOHD SAH

b - d equal the constant, r, the instantaneous rate of increase. Sometimes r is called the intrinsic rate of increase, or the Malthusian parameter. ●The value of r determines whether the population increases exponentially (r >0), decreases exponentially (r < 0), or remains stable (r = 0). ●The units of r are individuals per individual per unit time. dN / dT = r N (more on r) ●r is the per capita rate of population increase over a short period of time. Why do you suppose it’s called that? ●That rate is simply the difference between b and d. ●dN / dT = r N what does it mean? dN / dT = r N ●Let

●This

is a simple model of population growth. ●It says the population growth rate (dN / dt) is proportional to r and that populations increase only when the instantaneous birth rate ( b ) is greater than the instantaneous death rate ( d ). If b = d, then the population remains stable. An example: You are studying a population of beetles of size 3000. During a one month period, you record 400 births and 150 deaths. Estimate b, d, and r. ●b = 400 / 3000 = 0.1333 births / ind * mo. ●d

= 150 / 3000 = 0.050 deaths / ind * mo.

●r

=b-d = 0.0833 ind / ind * mo. When will the population grow?

8

DR. SHAHRUL ANUAR MOHD SAH

●If

r is positive, the population is growing exponentially (unchecked) and population growth (d N / dt) is proportional to N. (The bigger the population N, the faster its rate of increase. ●If r is negative, the population is declining exponentially and population growth is proportional to N. ●Population growth is 0 when either:  r = 0 (births and deaths are perfectly balanced), OR  N = 0 (the population is extinct, and we don’t allow immigration). Exponential Population Growth dN / dt = r N ●Calculating

population size ●Calculating doubling time ●Assumptions of the exponential model ●Do populations really grow exponentially??? Calculating Population Size ●dN / dt = r N tells us population growth rate, but not how big the population is expected to be. ●If you integrate this equation (remember your Calculus!), you can predict population size: rt ●Nt = N0 e Integration of dN/dt=rN ●dN/dt=rN

●dN/N=rdt

dN/N = ∫ rdt for t = T and t=0 ●ln(N(T)) – ln(N(0)) = r(T)-r(0) ln(N(T)) -ln(N(0 ) ●e e = erT rT ●N(T)-N(0) = e rt ●Nt = N0e ●∫

9

DR. SHAHRUL ANUAR MOHD SAH

Verbally, population size at time t = initial population size multiplied by base e raised to r multiplied by time ●

Nt = N0 e rt ●N0 is the initial population size ●Nt is the population size at time t ●e is a constant, the base of the natural logarithm (e = 2.717). ●Knowing the starting population size and r, we can use this equation to forecast population size at a later period in time. An example: You are studying a population of turtles of size 3000. During a 1 month period, you record 400 births and 150 deaths in this population. Estimate b, d, and r, and project population size in 6 months. Turtles don’t reproduce like this, they would be better fit by a discrete equation”. ●Remember however that the continuous and the discrete equations are qualitatively similar. ●Let’s revisit our turtles ●λ =Nt+1/N0 ●Nt+1 = 3000+400-150=3250 ●Thus λ = 3250/3000=1.08333 6 ●And N6 = 1.08333 *N0 ●N 6 = 4849 turtles.

r ≈ ln(λ) Exponential growth of a herd of 50 cattle, with r = 0.0365 cows (cow * year): ●Year 0 50 cows ●Year 1 72 cows ●Year 2 103.8 cows 10

DR. SHAHRUL ANUAR MOHD SAH

●Year

3 149.5 cows ●Year 10 1923.7 cows 9 ●year 50 4.2 * 10 17 ●year 100 3.6 * 10 year 200 2.5 * 10 33 Organisms vary in their reproductive potential (the number of offspring that a single individual (or pair of individuals in a sexually reproducing species) can produce under optimal conditions, assuming that all progeny survive to breed, over various time intervals. ●Aphis fabae (an aphid) ●Elephant ●Housefly ●Mycophila speyeri (a fly that feeds on mushrooms) ●Staphylococcus aureus (a bacterium) ●524

billion in one year. ●19 million in 750 years 18 ●191 x 10 in 5 months ●20,000 / sq. ft in 35 days ●cells would cover the Earth 7 ft deep in 48 hrs.! Trajectories of exponential population growth:

What does this graph tell you? ●When r > 0, populations increase exponentially. ●The larger the value of r, the faster the rate of increase. ●The larger the population, the faster its rate of increase.

11

DR. SHAHRUL ANUAR MOHD SAH

Calculating Doubling Time ●One important feature of a population that is growing (increasing or decreasing) exponentially is a constant doubling time….no matter how big or small a population, it will always double (or halve) in size after a fixed period. How long it takes to double is called the doubling time. Calculating Doubling Time ●If the population has doubled in size, it is exactly twice as large as the initial population size: ●Nt double = 2 N0 rt, ●Remember that Nt = N0 e so….. rt double ●2 N 0 = N0e so…. rt double ●2 = e and now solve for t, which will you tell you how long the doubling time is. 2 = e rt double and now solve for t •A

little math review…solve y = e x for x. •Because the base is e, take the natural log, ln, of both sides x x •ln y = ln e (then, ln(e) cancels itself out, ln (e ) = x, so the solution is that: ln y = x.) So now let’s apply that trick to solving our doubling time problem. •ln (2) = r t double and now the finale……. •t double = ln (2) / r

An example: Currently, the human population is expected to double in ~ 50 years. Calculate r for the human population. If the population size in 1993 was 5.4 billion, what is the projected population size for the year 2000? •t double = ln (2) / r

12

DR. SHAHRUL ANUAR MOHD SAH

r = ln(2) / t = ln(2) / 50 years = 0.01386 individuals per individual / year. rt •Nt = N0 e N0 = 5.4 billion and t = 7 N7 = 5.4 * e (.01386) (7) = 5.95 billion humans An Example – Discrete Populations A population of wasps increases by 18% annually. What is the approximate doubling time? •Wasps are short-lived (annual reproduction and death) and are best fit by a discrete population growth model. •If the population is growing by 18% per year then we know that λ = Nt/N0 = 1.18 •We know that r ≈ ln(λ), so we can convert the discrete equation to a continuous equation •Thus r = 0.1655 individuals/individual*year •Finally, tdouble = ln(2)/r = ln(2)/0.1655 •tdouble= 4.1 years •Note, because the equation is actually for a continuous population, the answer is an approximation. •

Some properties of doubling time ●the larger the r, the faster the doubling time. ●r varies considerably among different plants and animals: smallbodied organisms grow faster and have larger rates of population increase than large-bodied organisms.

13

DR. SHAHRUL ANUAR MOHD SAH

For example, bacteria can reproduce by asexual fission (clones) every few minutes. Larger organisms tend to reproduce only after long maturation periods. ●

Estimates of r (individuals / individual * day) and doubling times for different organisms ●Virus 300.0 3.3 min ●Bacterium 58.7 17.0 min ●Protozoan 1.59 10.5 hrs ●Hydra 0.34 2 days ●Flour Beetle 0.101 6.9 days ●Brown Rat 0.0148 46.8 days ●Domestic Cow 0.001 1.9 years ●Mangrove tree 0.00055 3.5 years ●Southern Beech 0.000075 25.3 years

Assumptions of the exponential growth model (what is the underlying biology of a population that is growing exponentially?) NO Immigration or Emigration. The population is closed; changes in population size depend only on local births and deaths ●Constant b and d. If a population is going to grow with constant birth and death rates, an unlimited supply of space, food, and other resources must be available. ●No genetic structure. All individuals in the population have the same birth and death rates, so there cannot be any underlying genetic variation ●No age or size structure. There are no differences in b and d among individuals due to their age or body size. Basically, we are modeling a sexless, parthenogenic population in which individuals are immediately reproductive when they are born! ●Continuous growth with no time lags. ●

Do populations REALLY grow exponentially? 14

DR. SHAHRUL ANUAR MOHD SAH

Sometimes, but not usually for long!! Population censuses of Muskox on Nunivak Island between 1936 and 1968 (Spencer and Lensink 1970). Logistic Growth Equation ●Logistic

Growth Equation, introduced to ecology in 1838 by P. F. Verhulst. It is the simplest equation describing population growth in a resource-limited environment. ●K is more than a mathematical convenience. It is the carrying capacity of the environment. It represents the maximum population size that can be supported. ●Because K represents the maximum sustainable population size, its units are numbers of individuals. How does this equation work? ●The logistic growth equation looks like the exponential growth equation (r N) , but is multiplied by an additional term, ( 1 - N / K). ●This term represents the unused portion of the carrying capacity. An example: Suppose K = 100 and N = 7. ●The unused portion of the carrying capacity is large: [1-(7 / 100 ) ] = 0.93. ●The population is relatively uncrowded and growing at 93% of the growth rate of an exponentially increasing population: dN / dt = r N (0.93). ●

Full steam ahead!! (Well, 93% of full steam). An example: Suppose K = 100 and N = 98.

15

DR. SHAHRUL ANUAR MOHD SAH

If the population is close to K (N = 98 ), the unused carrying capacity is small: [ 1 - (98 / 100) ] = 0.02. ●Consequently, the population grows very slowly, at 2% of the exponential growth rate: dN / dt = r N (0.02). ●

●Finally,

if populations should ever exceed carrying capacity ( N > K ), the term in parentheses becomes negative, which means that the growth rate is less than zero and the population declines back to K. ●Thus, density - dependent birth and death rates provide an effective break on exponential growth. When will the population stop growing (d N / d t = 0) ? In the exponential model populations stop growing when N or r = 0. In logistic growth model, the population also reaches a stable equilibrium (N = K) at the intersection of the curves, where b` = d`. We can integrate the growth equation to express population size as a function of time: ●This equation gives the classic s - shaped logistic growth curve. ●The

population grows at its highest rate when N = K / 2. ●Regardless of what N0 is, a population will reach K, but faster growing populations (high r) will reach K more quickly. An example: Suppose a population of butterflies is growing according to the logistic equation. If the carrying capacity is 500 butterflies and r = 0.1 (ind / ind * mo), what is the maximumpossible growth rate for the population? ●d

N / d t = r N ( 1 - N / K). We know K and r, but need to determine N.

16

DR. SHAHRUL ANUAR MOHD SAH

●The

maximum possible growth rate occurs when N = K / 2, so N = 250 butterflies. ●d N / d t = 0.1 (250) [1 - ( 250 / 500) ] ●d N / d t = 12.5 ind / month Population growth rate ( d N / d t) as a function of population size.

Comparing the Integrated Exponential and Logistic Models Assumptions of the Logistic Growth Model: ●This model shares four of the five assumptions of the exponential growth model (NO Immigration or Emigration, No genetic structure, No age or size structure, Continuous growth with no time lags). ●But resources are limited, so there are two additional assumptions. ●Constant carrying capacity. ●Linear density dependence. Does this really happen in nature? ●In many instances, the logistic model is a gross oversimplification of population growth ●It appears to “work well” in predicting population growth of a laboratory animal population (such as flour beetles). ●But for many populations, the assumptions of the model are severely violated. Problems with Assumptions of the Logistic ●Carrying capacity constant –Even in what we consider very stable environments, carrying capacity will vary, sometimes considerably. ◆Fluctuations in food supply ◆Changes in the abundance of competitors 17

DR. SHAHRUL ANUAR MOHD SAH

Climate ●Linear density dependence –Density dependence may be different at different population densities. ◆Allee effects ◆Thresholds –Direct vs. delayed-density dependence ◆

18