BIOL 4120: Principles of EcologyBIOL 4120: Principles of Ecology

Lecture 10: Temporal And Lecture 10: Temporal And Spatial Dynamics of Spatial Dynamics of

PopulationsPopulations

Dafeng HuiDafeng Hui

Office: Harned Hall 320Office: Harned Hall 320

Phone: 963-5777Phone: 963-5777

Email: dhui@tnstate.eduEmail: dhui@tnstate.edu

Temporal and Spatial Dynamics Temporal and Spatial Dynamics of populationsof populations

Numbers of gyrfalcons exported from Iceland to Denmark during 1731 and 1770 reflected population cycles.

Topics (Chapter 12)Topics (Chapter 12)

10. 1 Fluctuation is the rule for natural 10. 1 Fluctuation is the rule for natural populationspopulations

10.2 Temporal variation affects the age structure 10.2 Temporal variation affects the age structure of populationsof populations

10.3 Population cycles result from time delays in 10.3 Population cycles result from time delays in the response of populations to their own the response of populations to their own densitydensity

10.4 Metapopulations are discrete 10.4 Metapopulations are discrete subpopulations linked by movements of subpopulations linked by movements of individualsindividuals

10.5 Chance event may cause small population 10.5 Chance event may cause small population to go extinctto go extinct

10.1 Fluctuation is the rule for 10.1 Fluctuation is the rule for natural populationsnatural populations

Domestic sheep on the island of Tasmania of Australia. Introduced in early 1800s and reached the K in about 30 years and varied slightly

What are the causes?What are the causes?

Sensitivity to environmental change and response time Sensitivity to environmental change and response time of the populationof the population

Sheep: large, greater capacity for homeostasis, and Sheep: large, greater capacity for homeostasis, and better resist physical changesbetter resist physical changes

long life, generation overlap, even out the long life, generation overlap, even out the short-term fluctuation in birth rateshort-term fluctuation in birth rate

Algae and diatom: short life span (a few days), rapid Algae and diatom: short life span (a few days), rapid turn over, high mortality, population size depends on turn over, high mortality, population size depends on continued reproduction, which is sensitivity to food continued reproduction, which is sensitivity to food availability, predation, and physical conditions. availability, predation, and physical conditions. Phytoplankton populations are intrinsically unstable.Phytoplankton populations are intrinsically unstable.

Periodic cycles of some species

Population cycles of grouse in Finland are synchronized across species and areas (Three species in two areas). (6-7 year cycles)

Populations of four moth species in the same habitat fluctuate independently

10.2 Temporal variation affects the age 10.2 Temporal variation affects the age structure of populationsstructure of populations

Variation in population size Variation in population size over time often leaves its over time often leaves its mark on age structuremark on age structure

Age structure influences the Age structure influences the rate of population growthrate of population growth

Commercial whitefish in Commercial whitefish in Lake ErieLake Erie

1944 resulted a large 1944 resulted a large population population

Age distributions of forest trees show the effects of disturbances on seedling establishment (tree ring count)

Survey in Pennsylvania, 1928 over the past 400 years

10.3 Population cycles result from time 10.3 Population cycles result from time delays in the response of populations to delays in the response of populations to

their own densitytheir own density

Cycling of populations has been observedCycling of populations has been observed E.g: hare cycles, 11-yr cycle in sunspotE.g: hare cycles, 11-yr cycle in sunspot

Oscillation and time delaysOscillation and time delays Oscillation may reflect intrinsic dynamic qualities Oscillation may reflect intrinsic dynamic qualities

of biological systems (some oscillate even with of biological systems (some oscillate even with small environmental fluctuation)small environmental fluctuation)

Time delays in responses: high birth rate -> Time delays in responses: high birth rate -> overshoot population -> high death rate -> low overshoot population -> high death rate -> low population. population.

Time delays and oscillations in discrete-time models

ΔN(t)=N(t+1)-N(t) =RN(t)R is proportional increase or decrease in N per unit of time

Let’s make R density-dependent,

ΔN(t)=RN(t) (1-N(t)/K)

=RN(t)/K (K-N(t))

K-N(t) is the difference between the size of the population and its carrying capacity at time t

Damped oscillation, limit cycle or chaos

Time delays and oscillations in continuous-time models

Logistic model:

dN/dt=r N(1-N/K) or

dN(t)/dt=rN(t)(1-N(t)/K)

(1-N(t)/K) is a component that show density dependent influence by population size

If not by current, but by a population size at time (t-τ)

dN(t)/dt=rN(t)(1-N(t-τ)/K)

Oscillation depends on (rτ):r τ=0, no oscillationr τ=1, damped oscillation, r τ=2, limit cycles

Cycles in laboratory populations

Water flea experiment

25oC, large oscillation

period of cycle: 60 days time delays: 12-15 days (average age give birth: 12-15 days at 25oC)

18oC, no oscillation reproduction fell off quickly with increasing density, and life span was longer than at 25oC. Deaths were more evenly distributed over all ages and some individuals gave birth even at high population densities.Generation overlap broadly, No time delay.

Introduction of time delays results in regular population cycles

Blowfly experimentAJ Nicholson, Australian

Control population by provide limited food supplies to larvae and unlimited food to adults.

Adults population cycles from 0 to 4000.

Period 30-40 days

Cause: a time delay in the responses of fecundity and mortality to the density of adults.

A time delayed logistic model (rt=2.1) provides a good fit for the blowfly data.

Limited food supplies to adults limited the time delay and results in the elimination of population cycles

What happens if you do not provide enough food for adults?

10.4 Metapopulations are discrete 10.4 Metapopulations are discrete subpopulation linked by movements of subpopulation linked by movements of

individualsindividuals

Habitat Patches, subpopulations and Habitat Patches, subpopulations and metapopulationmetapopulation

Processes contribute to dynamics of Processes contribute to dynamics of metapopulationsmetapopulations• Growth and regulation of subpopulations within patchesGrowth and regulation of subpopulations within patches• Colonization of empty patches by migrating individuals Colonization of empty patches by migrating individuals

to form new subpopulations to form new subpopulations • and the extinction of established subpopulations.and the extinction of established subpopulations.

Basic model of metapopulation dynamicsOne population is divided into discrete subpopulations, each subpopulation has a probability of going extinct (e).

If (p) is the fraction of suitable habitat patches occupied by subpopulations, then subpopulations go extinct at the rate (ep).

The rate of colonization of empty patches depends on the fraction of patches that are empty (1-p) and the fraction of patches sending out potential colonists (p). The rate of colonization within the metapopulation as a whole as a single rate constant (c) times the product (p(1-p)).

The rate of change in patch occupancy:

dp/dt = cp(1-p) – ep colonization extinction

Basic model of metapopulation dynamics

The rate of change in patch occupancy:

dp/dt = cp(1-p) – ep colonization extinction

A metapopulation attains equilibrium size when colonization equals extinction

cp(1-p) = ep

Thus, p^= 1 – e/c

This is the proportion of occupied patches at metapopulation equilibrium

RecapRecap

Population cycles: very commonPopulation cycles: very common

Mechanisms: result from time delays in the Mechanisms: result from time delays in the response of populations to their own response of populations to their own densitydensity

Two models: how R or rt influences the Two models: how R or rt influences the population size change.population size change.

Metapopulations and subpopulationsMetapopulations and subpopulations Basic model of metapopulation dynamicsBasic model of metapopulation dynamics

Basic model of metapopulation dynamics

dp/dt = cp(1-p) – ep colonization extinction

p^= 1 – e/c

Rate of e/c is very important

If e = 0, then p^=1, all patches occupied, none disappears

If e>=c, then p^ =0, then metapopulation heads toward extinction

0< e < c, intermediate value, results in a shifting mosaic of occupied and unoccupied patches.

Basic model of metapopulation dynamics

In this model, there are many assumptions

1.All patches are equal2.(e) and (c) for each patch are the same3.Each occupied patch contribute equally to dispersal4.Colonization and extinction in each patch occur independently of other patches5.Colonization rate is proportional to the fraction of occupied patches (p)

In reality: Patches vary in size, habitat quality, degree of isolation from other patches. Larger patch support large subpopulation, lower probability of extinction. Smaller, more isolated patches are less likely to be occupied.

Larger, less isolated patches are more likely to be occupied

Shrew on islands in two lakes in Finland

Few occupied patch area <1

Isolation?

Larger, less isolated patches are more likely to be occupied

Butterfly on patches of calcareous grassland in England (Hanski et al. 1991)

Patch size and isolation are important

Glanville fritillary butterfly study by Illka Hanski, Finland

A survey for p^: Occupied patches of dry meadows, Aland Islands, Finland

Exp. One: Total 1600 suitable patches, only 30% were occupied at any given time.

Exp. Two: Introduced populations to 10 of the 20 suitable habitat patches on the smaller, isolated island of Scottungia, in August 1991

Observed over next 10 years

Number of extinctions varies between 0 to 12 per yearNumber of colonization between 0 and 9Subpopulations: started at 10, dropped to as few as 2, and increased as high as 14, ended at 11.

None of the original 10 survived the decade, the metapopulation as a whole persisted.

The rescue effect

Immigration from large, productive subpopulations can keep declining subpopulations (small ones) from dwindling to small numbers and eventually becoming extinct. This phenomenon is known as rescue effect.

Dispersal is critical for colonizing empty patches, as well as maintaining established populations.

Model: modify the model and add rate of extinction (e) decrease as the fraction of occupied patches (p) increases (more rescuers)

dP/dt = cp(1-p) - ep(1-p) p=0, 1 if c><e.

This model predicts that p^ will either equal to 0, otherwise, it will increase to 1, as when p<1 but close to 1, 1-p is small, reduce the extinction rate. With rescue effect, all patches will be occupied.

10.5 Chance event may cause small populations to go extinction

Deterministic model and stochastic model:

Population models we described before are based on average values of birth rate and death rate, and assume no difference among individuals. Such models, whose outcomes can be predicted with certainty, are called deterministic models.

Models built in with chance factors (randomness), such as birth and death rate vary from each individual to another and from one time to another (with mean over certain time or of all individuals is fixed). The result of each model run varies and can’t be predicted, stochastic model.

Three types of randomness

1. Unpredictable catastrophe, such as appearance of a predator, disease, fire etc (birth and death)

2. Environmental variation (some rules, small variation not predictable). Physical and environmental factors (influence birth and death)

3. Stochastic processes such as death of an individual, number of offspring produced by an individual. Even under constant environment, these values could change for an individual.

(Overall, there is a probability distribution) (coin tossing is an example of a stochastic process)

Stochastic population processes produce a probability distribution of population size

N(0)=10,lambda=1.5

No stochastic process involved, what’s the population size at t=1?

Chart on left: N=10, pure birth, b=0.5 and stochastic process involved,All give birth, N1=10+10=20.

Chance events exert their influence more forcefully in small population than in large ones

Coin tossing: a set of 5 pennies, 5 heads in a trial is 1/32; 10 pennies, is 1/1024. If each individual in a population is a coin, and heads mean death, a population of 5 has a higher probability of extinction, just by chance.

Stochastic extinction of small populations

Random walk: A population subject to stochastic birth and death process is said to take a random walk, meaning that its numbers may increase or decrease strictly by chance.

When the size of such a population does not respond to changes in density, its ultimate fate is extinction, regardless of how its size might increase in the meantime.

Mathematicians have calculated the probability of extinction. For simplicity, given same birth rate and death rate, the probability of a population will die out within a time interval t is

p(t)=[bt/(1+bt)]^N.

Stochastic extinction of small populations

The probability of stochastic extinction increase over time, but decrease as a function of initial population size N.

b=0.5, p0(t)=[bt/(1+bt)]^N

N=10, t=10, p=0.16t=1,000, p=0.98

Stochastic extinction with density dependenceStochastic extinction models usually do not include density-dependent changes in birth and death rates.

If density-dependent birth and death rate includes, it rarely goes extinction (unless the population size is very small), as a population drops below K, the birth rate will increase and death rate will decrease.

Whether the density-independent stochastic models are relevant to natural populations?

They are. 1)Fragmentation by human beings creates many small subpopulation, often so isolated that eventual demise can’t prevented by immigration from other populations2)Changing environmental conditions reduce fecundity3)Endangered species can’t compete with other species4)Small populations sometimes exhibit positive density dependence (Allee effect), their number may decline more rapidly.

Size and extinction of natural populations

Small size populations become more susceptible to extinction, particularly on small islands

An Example: species lists for birds in 1917 and 1968 on two islands.

Over 51 years, 10 species disappeared from Santa Barbara Island (3 km^2), only 6 of 36 disappeared from the large Santa Cruz Island (249 km^2)

Extinction rate: 1.7% and 0.1% per year.

The EndThe End

10.9 Population extinction10.9 Population extinction

If r becomes negative If r becomes negative (birth rate < death rate), (birth rate < death rate), population declines and population declines and will go extinction. will go extinction.

Factors: Extreme Factors: Extreme environmental events environmental events (droughts, floods, cold or (droughts, floods, cold or heat etc), heat etc), loss of loss of habitat habitat (human). (human).

Small populations are Small populations are susceptible to extinctionsusceptible to extinction Allee effect, genetic drift,

inbreeding (mating between relatives)

Overgraze, only 8 in 1950

Small population size may result in Small population size may result in the breakdown of social structures the breakdown of social structures that are integral to successful that are integral to successful cooperative behaviors (mating, cooperative behaviors (mating, foraging, defense)foraging, defense)

The The Allee effectAllee effect is the decline in is the decline in reproduction or survival under reproduction or survival under conditions of low population densityconditions of low population density

There is less genetic variation in a There is less genetic variation in a small population and this may affect small population and this may affect the population’s ability to adapt to the population’s ability to adapt to environmental changeenvironmental change

Hackney and McGraw (West Virginia Hackney and McGraw (West Virginia University) examined the reproductive University) examined the reproductive limitations by small population size on limitations by small population size on American ginseng (American ginseng (Panax Panax quinquefoliusquinquefolius))• Fruit production per plant declined with Fruit production per plant declined with

decreasing population size due to decreasing population size due to reduced visitation by pollinationreduced visitation by pollination

RecapRecap

Population cycles: very commonPopulation cycles: very common

Mechanisms: result from time delays in the Mechanisms: result from time delays in the response of populations to their own response of populations to their own densitydensity

Two models: how R or rt influences the Two models: how R or rt influences the population size change.population size change.

Metapopulations and subpopulationsMetapopulations and subpopulations Basic model of metapopulation dynamicsBasic model of metapopulation dynamics