Population Ecology ES 100 8/21/07

Remember from last time: Population ecology Life Tables Cohort-based vs. Static Identifying vulnerable growth stages Age-specific birth rate Computing fitness, net reproductive rate and generation time Population projections

Today: Metapopulation Theory Population Models Immigration and Emigration Source and Sink Populations Maintaining Genetic Diversity Population Models Exponential and Logistic growth Assumptions Doubling time When should this model be used?

Is the Population Increasing, or Decreasing? Fitness is one indication….. But… Populations vary dramatically over time (boom/bust cycles) Individuals move in (immigration) and out (emigration) of populations Metapopulations (18.5 Bush) Nt+1 = Nt + (B-D) + (I-E)

Threatened Species:Western Snowy Plover Before 1970, 53 breeding locations in CA (including Santa Barbara) Now, 8 breeding sites support 78% of the CA metapopulation

Populations across the landscape Metapopulation: sum of multiple interacting sub-populations sub-population A sub-population C sub-population B sub-population D

Populations across the landscape Genetic diversity is maintained by exchange of genes between the sub-populations sub-population A sub-population C sub-population B sub-population D

Populations across the landscape Most mating occurs within a sub-population sub-population A sub-population C sub-population B sub-population D

Populations across the landscape Some habitat patches are better than others hot and dry few nesting sites most ideal many predators

Populations across the landscape Sub-populations can be source populations or sink populations sink sink hot and dry source few nesting sites most ideal many predators sink

Populations across the landscape In source population habitats: living conditions are good, so births meet or exceed deaths competition may be great, forcing some members out sink sink hot and dry source few nesting sites most ideal many predators sink

Populations across the landscape If a sub-population goes extinct, it can be revived by recruits from a source population…. But sinks are important too! sink source locally extinct source of recruits

Controls on immigration Distance to source population Lots of immigration mainland Little immigration Obstacles Mountains Waterways mountains hills

Sample Metapopulation Data Is this population assessment static or cohort based? Which sub-population(s) are sources? Sinks? Can you develop a life table for each sub-population? Can you develop a life table for the total population?

Mathematical Models Uses: synthesize information look at a system quantitatively test your understanding predict system dynamics make management decisions

Population Growth dN dt t = time N = population size (number of individuals) = rate of change in population size (ind/time) r = maximum/intrinsic growth rate (1/time) = fractional increase, per unit time, when resources are unlimited dN dt

Population Growth dN dt Lets build a simple model (to start) = r * N Constant growth rate  exponential growth Assumptions: Closed population (no immigration, emigration) Unlimited resources No genetic structure No age/size structure Continuous growth with no time lags dN dt

Projecting Population Size Nt = N0ert N0 = initial population size Nt = population size at time t e  2.7171 r = intrinsic growth rate t = time

Doubling Time

Let’s Try It! The brown rat (Rattus norvegicus) is known to have an intrinsic growth rate of: 0.015 individual/individual*day Suppose your house is infested with 20 rats. How long will it be before the population doubles? How many rats would you expect to have after 2 months? Is the model more sensitive to N0 or r?

When Is Exponential Growth a Good Model? r-strategists Unlimited resources Vacant niche

Environmental Stochasticity Our exponential growth model is deterministic Outcome is determined only by model inputs Intrinsic growth rate varies with ‘good’ and ‘bad’ environmental conditions: Often we know the mean growth rate and the variance in the growth rate, These can be incorporated into our model!

Plover Population Model with Stochasticity Nur, Page and Stenzel: POPULATION VIABILITY ANALYSIS FOR PACIFIC COAST SNOWY PLOVERS

What Controls Population Size and Growth Rate (dN/dt)? Density-dependent factors: Intra-specific competition food Space contagious disease waste production Interspecific competition Other species interactions! Population Density: # of individuals of a certain species in a given area Density-independent factors: disturbance, environmental conditions hurricane flood colder than normal winter

Can the population really grow forever? What should this curve look like to be more realistic? Population size (N) Time (t)

# of individuals of a certain Population Growth Population Density: # of individuals of a certain species in a given area Logistic growth Assumes that density-dependent factors affect population Growth rate should decline when the population size gets large Symmetrical S-shaped curve with an upper asymptote

Population Growth How do you model logistic growth? How do you write an equation to fit that S-shaped curve? Start with exponential growth dN dt = r * N

Population Growth How do you model logistic growth? How do you write an equation to fit that S-shaped curve? Population growth rate (dN/dt) is limited by carrying capacity dN dt N K = r * N (1 – )

What does (1-N/K) mean? Unused Portion of K If green area represents carrying capacity, and yellow area represents current population size… K = 100 individuals N = 15 individuals (1-N/K) = 0.85 population is growing at 85% of the growth rate of an exponentially increasing population

Population Growth dN dt N K = r * N (1 – ) Logistic growth Lets look at 3 cases: N<<K (population is small compared to carrying capacity) Result? N=K (population size is at carrying capacity) N>>K (population exceeds carrying capacity)

Population Size as a Function of Time

Last Time… Metapopulation Theory Population Models Logistic growth Immigration and Emigration Source and Sink Populations Maintaining Genetic Diversity Population Models Exponential Assumptions Doubling time When should this model be used? Logistic growth How does it account for density dependent factors? What is the difference between dN/dt and r? 3 cases: N<<K (exponential growth) N=K (no growth) N>>K (exponential decline)

At What Population Size does the Population Grow Fastest? Population growth rate (dN/dt) is slope of the S-curve Maximum value occurs at ½ of K This value is often used to maximize sustainable yield (# of individuals harvested) /time Bush pg. 225

Fisheries Management: MSY (maximum sustainable yield) What is the maximum # of individuals that can be harvested, year after year, without lowering N? = rK/4 which is dN/dt at N= 1/2 K What happens if a fisherman ‘cheats’? What happens if environmental conditions fluctuate and it is a ‘bad year’ for the fishery?

Assumptions of Logistic Growth Model: Closed population (no immigration, emigration) No genetic structure No age/size structure Continuous growth with no time lags Constant carrying capacity Population growth governed by intraspecific competition “recruitment” depends on current population size

Lets Try It! Formulas: A fisheries biologist is maximizing her fishing yield by maintaining a population of lake trout at exactly 500 fish. Predict the initial population growth rate if the population is stocked with an additional 600 fish. Assume that the intrinsic growth rate for trout is 0.005 individuals/individual*day . How many fish will there be after 2 months?