Lec #5: Population Growth Part 2

Logistic Growth

No natural populations can maintain exponential growth indefinitely

Population density typically fluctuates around a constant number of individuals (it is constant when extrapolated)

As population size increases resources become limiting for growth and reproduction

Modify the exponential growth model to incorporate changes in (r) as the population size grows toward (K)

When there is limited resources, the predictions are correct (environment is constant, no predators, no competition from other species)

K changes by birth and death, not migration

rN = when there is no competition; unlimited resources

At large values of (N), the value of (K-N / N) becomes small and the population growth is small

When population size is below (K) pop growth increases (available resources; biomass and reproductive output is high)

When population size is near (K) pop growth decreases

To check/validate this model, test in the natural environ

How well does the logistic model fit the growth of natural populations?

- Populations of very small organisms fit fairly well

- Many populations do not stabilize at (K) and deviate from sigmoid curves

So why does the logistic curve not fit well for most natural population growth patterns? (3 Assumptions)

Each individual added to the population has the same negative effect on population growth

Population approaches (K) smoothly

Populations are large, and density is important in regulation (boom and burst)

For the assumption that each individual added to the population has the same negative effect on population growth, what are some examples of exceptions?

- Chance events

- Isolated plants (savannah trees): has shade below (less evaporation and more water moisture in the soil/tree) and has a reproductive partner next to them

- Flamingos: won’t reproduce in captivity unless there are many of them (safety), so mirrors are used (easily fooled)

• They cluster because they need water

• Reproductive output is high, and they grow fast the more you add (intense density independence)

• Algae blooms in lake, which is exponential in nature (unlimited resources)

• Predators cant eat all the flamingos because there are too many (escape in numbers)

For the assumption that population approaches (K) smoothly, what are some examples of exceptions?

- Time lag (takes long) between the negative effect of population size increase and when they are realized

- Time lags cause population size to overshoot and undershoot (K)

- Populations may oscillate about (K), but is smooth when extrapolated (model prediction) (math predicts instantaneous change)

For the assumption that populations are large, and density is important in regulation (boom and burst), what are some examples of exceptions?

Insects, Microorganisms

- sensitive to environmental fluctuations

Carrying Capacity (K)

Def: The maximum population size that an environment can support

- (K) varies over space and time

- Crowding and resource limitation effects the population growth rate (r)

- Population growth slows as its density approaches (K)

Limiting factors

Any essential resource that is in short supply can limit population growth

Ex: space, light, nutrients, etc.

Density - Dependent Control

Factors that alter per capita birth or death rates in a population are dependent on population density (can stabilize population density)

More dense, more death

Ex: Predation, parasitism, competition, herbivory, and disease can be density-dependent

Graphing the % killed shows that for every 100, 1 dies

Density - Dependent Control Example: Bubonic Plague

- Blood pathogen, exists in rats, rabbits and other small mammals and transmitted by flea bites (fleas are pests with altered digestive tracts and never get full; their bites transfer the pathogen)

- 25 million deaths in European cities (25% of Europe)

- 1994 Indian Bubonic plague outbreak

- The greater the density of rats, fleas, and humans the higher the infection rate (passage of disease) and the higher the mortality

Density - Independent Control

Factors that alter per capita birth or death rates in a population are independent of population density (do not stabilize population density)

Ex: Physical/chemical factors: weather, drought, freezes, flood and fire, etc.

% killed is constant because the death rate is independent of density

Density - Independent Control Example: Monarch Butterfly

- Migrate from Canada to Mexico

- Illegal logging removed temperature buffer for storms

- 2002 freezing storm killed over 200 million Monarchs (>80 % population mortality)

- The density of Monarchs had no impact on mortality rates; just had physiological lower limit temperature tolerance (no vegetation)

Natural Population Control and Regulation

- A mix of density - dependent and density - independent factors

- Over long time scales many populations remain stable and close to (K) (Density - Dependent Control)

- Short term fluctuations in populations due to density - independent factors

Oscillating logistic curves

Stable and Unstable Populations

At equilibrium the population does not change unless disturbed

- Due to moderate rates of biotic interactions (natural forces): competition, predation, or herbivory

Stable Equilibrium (birth = death)

- If perturbed population will return to initial density

- Stabilizing forces dampen population fluctuations and return to line

- Density dependent control

- Over time, there will be a stable equilibrium

Ex: like a cone

Unstable Equilibrium

- If perturbed, population may not return to initial density

• Unstable equilibriums can stabilize but at a different density

- Destabilizing forces enhance population fluctuations (not dampen)

- Inverse density dependence: the bigger the population, the smaller the impact of regulating force/effect, like flamingoes

- D-D with long time lags: elephants have a 2 year baby development, so the resources are different

• Assumptions about birth/death rate models are that they are instantaneous (they eat and pop out a baby), but there is always a lag

Metapopulation:

Def: Population is divided into discrete sub-populations connected by immigration & emigration (Levins 1969)

Dynamics: Growth and reproduction within patches

- Migration between patches or colonization of empty patches

Big populations have more resistance to birth and decay

Little populations have a fast/high frequency and little resistance to change

Increasing Rates of Migration for Metapopulations

Differences between local pop fluctuations are dampened, there is synchrony, higher probability of extinction during the down part (disease) (no species persistence)

Shrinks differences between sub-populations and oscillate together

Decreasing Rates of Migration for Metapopulations

Differences between local pop fluctuations are enhanced, there is asynchrony, higher probability of extinction for the small population since there are more down parts (disease) (big populations can go extinct too)

If they don’t recolonize any open habitat

Intermediate Rates of Migration for Metapopulations

Shifting mosaic of occupied & unoccupied patches (species persistence), there is partial synchrony (sometimes together and apart) and repopulation

Populations slightly speed up and slow down

No extinction globally because they recolonize (maybe some extinction in the short term)

Metapopulation dynamics model

How does the fraction of occupied sites ( f ) change with time?

If f = 1 then all sites are occupied by populations

If f = 0 then sites are unoccupied, and populations are extinct

E is not emigration

1. If the immigration rate is high & the extinction rate is low the fraction of occupied sites will increase with time (I > E then f )

2. If the immigration rate is low & the extinction rate is high the fraction of occupied sites will decrease with time (I < E then f )

Can predict the increase and decrease pattern based on colonization and extinction

Metapopulation Example: Bay checker spot butterfly

- Larvae (caterpillar) feeds on specialized plants which grow on patches of serpentine soils (Metapopulation) (serpentine soil is what goldfield flowers grow on; sandstone soil surrounds it)

- Drought years most of the host plants die killing larvae (3 sub-populations became extinct)

- Largest sub-population (Morgan Hill) serves as source of new colonists

- Butterfly has recolonized formerly extinct patches of habitat

Species persistence over time with intermediate migration (small population went extinct first); E<I

If the species did no migrate during a drought, there would be extinction as the little population goes extinct first and there is no recolonization