Lecture 13: Lecture notes

Population Dynamics – Part 2

  • Populations with age structure
  • Managing natural populations
  • Life histories
  • Spatial Structure

Remember….. Nt = λtN0

1. This Tells Us Nothing About the Age Distribution of Individuals in the Population

  • Why does that matter?
  • Most populations that we’re interested in managing have age structure that’s critical to their ecology and their population growth rate
  • Age-Specific Fecundities and Age-Specific Mortalities (and age specific rates of movement)

2. Age Structure of Human Populations

  • Constant: relatively constant proportion of young to old people. Birth rate is relatively constant.
  • Growing: large proportion of young people, high birth rate.
  • Shrinking: Population consists of a small proportion of young people, indicating low birth rates.

3.Age-specific rates

  • lx survivorship function
    • Probability that a newborn individual will survive to age x
    • By convention, l1 = 1
  • mx (or bx) maternity (birth) function
    • Number of female offspring born to a female of age x (between x – ½ and x + ½)
  • Sx survival rate
    • Proportion of individuals of age x surviving to age x+1 (so l2 = l1 s1)
    • Inverse is annual mortality rate

4. Age Structure is Critical to Many Ecological Processes Including Those Vital Rates That Determine Population Change

  • RO= Net reproductive rate of a single individual in her lifetime.
  • RO= The number of daughters born per female in her lifetime
  • RO= l1m1+l2m2+l3m3+…lxmx
  • RO= $\sum_{1}^{x} l(x)m(x)$

How can we convert this into a reproductive rate per year?

5. Mean Generation Time: A female’s age when she has her median daughter
T(gen)= $\frac{\sum xl_{x}m_{x}}{\sum l_{x}m_{x}}$

6. Managing Populations For Stable Age Structure

  • When age structure is stable, the proportion of individuals in each age class remains constant over time, even as population size varies
  • Stable age structure develops when populations grow under constant environmental conditions

7. Constant Environmental Conditions?

  • Environmental conditions vary all the time!
  • Ecologically, we can consider the environment to be constant if age-specific fecundities and mortalities remain unchanged
  • So, constant age-specific fecundities and mortalities promote stable age structure
  • We can manipulate (manage) these in an attempt to generate the stable age structure that we want – this is the basic purpose of hunting & fishing licenses

8. Stable age distribution

  • Will be reached regardless of what the starting age distribution was
  • And will be the same – hence a property of the sx (or lx) and bx terms
  • You’ll know it because:
    • l will be constant
    • Age distribution will be constant
  • If the population starts far from the stable age distribution, it will take longer to ‘settle down” on the SAD

9. Managing Bunnies

  • First, the population is still growing exponentially
  • If natural factors are too slow to operate we may have to impose our own density-dependent mortality
  • In other words, we should increase the number or decrease the price of hunting licenses as bunny density goes up
  • To get the age structure that we want, we might actually have to feed young bunnies to increase the proportion that survive to age 2…
  • …and then shoot them

10. Problem

  • How can we calculate the number of licenses we need, the number of young bunnies to feed, and what aged bunnies it’s really best to shoot?
  • We could do our demography calculations over and over again, using different age specific mortalities, but that’s a real chore for many organisms

11. Matrix Algebra

  • We can use some of the features of matrix algebra to simplify the running of demographic model
  • A Leslie Matrix is just a “box” that contains age-specific fecundities and mortalities
  • By manipulating a Leslie Matrix and its associated population vector, we can make predictions about populations

12. Assessing and Predicting Population Dynamics

  • Whether we’re managing resource populations (fish, timber, deer, etc), protecting endangered species, or studying exotic invasives, we need methods to assess their population dynamics
  • You’ve met some simple population models and explored the wonders of the Leslie Matrix
  • Population viability analysis is an application of the Leslie matrix for managing rare populations

13. Population Viability Analysis

  • A process of identifying the threats faced by a species and evaluating the likelihood that it will persist for a given time into the future
  • Oriented towards the conservation and management of rare and threatened species, with the goal of applying the principles of population ecology to improve their chances of survival

14. The Risks of Rarity

  • Human actions may cause a species to become rare, but may not directly cause the last individual to perish
  • Indeed, we often make a concerted effort to rescue rare species, with uncertain success
  • Rare species have a measurable probability of extinction (p[E]) due to deterministic and stochastic factors
    • demographic factors result in λ < 1 and will result in extinction
    • stochastic factors may lead to extinction even when λ > 1

15. Deterministic Extinction

  • When λ < 1, the population declines each year.
  • λ < 1 simply means birth rates are less than death rates.
    • An example would be poaching mortality in excess of natural reproduction.
    • Due to vehicle-caused mortality, the Florida panther has λ =~0.9. Time to extinction is ~ 30 years

16. Stochastic Extinction

  • Stochastic = chance variation. A rare population with λ > 1 may go extinct due to chance variation processes.
  • These include:
    • demographic stochasticity: chance variation in sex ratio of offspring
    • environmental stochasticity: chance variation in weather and food supply
    • catastrophes: extreme, rare (1- 5 per century environmental events
    • genetic stochasticity, including drift and in-breeding

17. PVA Approach

  • Population viability analysis is a probabilistic approach to assessing p(E) when λ > 1 but a population faces multiple threats and random (stochastic) variation, it might go extinct
  • PVA provides a means to evaluate multiple, varying, interacting factors of that might result in the “extinction vortex”
  • PVA takes into account
    • demographic risks
    • environmental variability
    • genetic risks (Ne-dependent) of drift and inbreeding
    • episodic catastrophes
  • PVA models require age-specific demographic data

18. The Extinction Vortex – small population size causes increased inbreeding and genetic drift, which causes decreased reproduction and survival, which causes reduced population growth rate, which causes smaller population size… and continues spiraling inward.

19. PVA Can Be Used For:

  • Planning research and data collection. PVA may reveal that population viability is insensitive to particular parameters. Research may be guided by targeting factors that may have an important impact on extinction probabilities or on the rank order of management options
  • Assessing vulnerability. Together with cultural priorities, economic imperatives and taxonomic uniqueness, PVA may be used to set policy and priorities for allocating scarce conservation resources
  • Ranking management options. PVA may be used to predict the likely response of species to reintroduction, captive breeding, prescribed burning, weed control, habitat rehabilitation, or different designs for nature reserves or corridor networks

The Distribution and Spatial Structure of Populations

  • Spatial patterns
  • Dispersal
  • Metapopulations

1. Some Key Terms and Concepts

  • A population consists of the individuals of a species within some area
  • Because the natural environment usually is a mosaic of suitable habitat, species populations often are made up of sub-populations
  • The distribution or geographic range of a species is the total geographic area occupied
  • Population size is the number of individuals in the total population or a sub-population
  • A population thus exhibits spatial structure in the number and spacing of its sub-populations
  • The size and spatial distribution of sub-populations (and thus the total population) vary over time

2. Habitat and Niche

  • Fundamental niche describes the range of environmental (temperature, moisture, pH, nutrients, vegetation types..) conditions in which an organism is physiologically able to persist
  • Realized niche is where the organism is actually found, and is a subset of the fundamental niche determined by predators, competitors, pathogens etc.
    • We often equate realized niche with habitat – a description of environmental conditions where a species occurs

3. Ecological niche modeling

  • Uses habitat or climatic features
  • Predict future range under a changing climate
    • May be larger, smaller, different locations
  • Predict the eventual distribution of an invading species

4. Modeling fish distributions in Belize

  • Species distribution model
  • African tilapias: a pan-tropical invader
  • Tilapia occurrence data
  • Habitat vulnerability
    • 7,510 km of river habitat vulnerable (24%) of total
    • 0 to 277 meters above sea level
    • 1 566 km2 mean watershed area
    • 0 to 446 km from sea
    • 1 Most important variables: watershed area, elevation

5. Migratory species

  • Occupy large ranges
  • Depend on existence of spatially distant habitat AND a secure migratory pathway

6. Dispersal integrates sub-populations

  • By chance, weather, etc
  • From high-density to low-density patches in response to crowding
  • From natal patches to find territories, eventual mating opportunities

7. Models of spatial structure

  • Metapopulations: patches are of similar quality but vary in size and distance. Intervening habitat is unsuitable.
  • Source-sink dynamics: Patches vary in quality, so that b > d in some, b < d in others. Reproductive success in high-quality patches and subsequent dispersal are critical to maintaining population
  • (and between-year variation in reproductive success in poor-quality patches may account for occasional outbreaks)

8. Conservation implications of spatial structure

  • Maintaining patch connectivity, either by direct corridors or by ensuring that habitat matrix is suitable for dispersing individuals
  • Once connectivity is lost, sub-populations are subject to the “risks of rarity”, and probabilistically will “wink out’, one by one
  • Dispersal underlies the “rescue effect”, in which a local population temporally is lost but then restored by re-colonization

9. Desert bighorn sheep in California. Larger populations persist longer than smaller populations. All populations less than 50 individuals went extinct within 50 years.

10. Metapopulations

  • Change in perspective
    • Population size vs. population persistence
    • Equilibrium manifested regionally
  • Number of populations filling landscape

11. The basic metapopulation model

  • p = fraction of occupied patches
  • 1-p = fraction unoccupied
  • e = probability of an occupied patch going extinct
  • Rate of colonization of empty patches depends on occupied patches, empty patches, and colonization rate c
    • dp/dt = cp(1-p) - ep
    • peqm = 1 – e/c
  • In reality, patches vary in size, quality and degree of isolation

12. Implications of Population Structure

  • Species formation
    • Isolates are more likely to develop into species
  • Genetic variation and interchange
    • Sub-populations may be subject to differing selection pressures
  • Population viability
    • Some sub-populations may survive various threats and thus be able to “rescue” lost sub-populations
  • The management of populations and habitat are linked
    • There are scientific reasons to protect multiple sub-populations (insurance, rescue)
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