Lecture 13 notes

Population Dynamics Part 2 (Dave Allan)

Refer to Ricklefs Ch. 10, 12

Main points: adding necessary complications of age, spatial, and temporal distributions to our original model from last time (N_t = λ^tN₀) to make it more realistic

First, age:

• Age-specific rates
  • l_x – survivorship function – probability of a newborn surviving until age x, with convention that l₁=1
  • m_x / b_x / f_x – maternity/birth/fecundity function (all synonymous) – number of females born to a female of age x (Question: does that mean biologists should think it's crazy that most human populations are taught to prefer sons instead?)
  • s_x – survival rate – proportion of individuals of age x surviving the next year (x+1), or l_x+1/l_x= s_x
  • mortality rate = 1/s_x
  • These rates can be found using data (following a cohort in a certain population and seeing how many survive over the years, how many female offspring they produce, etc.)
• Structural implications
  • When you look at a larger picture (more time), these vital rates and age structure determine population change
  • R0 = net reproductive rate, or the number of daughters of a single individual in her lifetime = Σ(l_xm_x) from x=1 to max age
  • From this we can calculate a mean generation time T=Σ(xl_xm_x)/ Σ(l_xm_x), a female’s age when she has her median daughter
  • You can then estimate λ as well: λ=R₀^1/T
• Consider that if the mean generation time T=1, the reproductive rate R₀ would be equivalent to λ.
  • If a population is under constant environmental conditions then, a stable age structure will develop. (in real life this doesn’t happen much)
  • Populations with the same vital rates will always reach the same stable age distribution (SAD) regardless of where you start from.
• Alternative modeling method: Leslie Matrix to simplify running of demographic models → Nt=MtN0, where M is a Leslie matrix
• See http://en.wikipedia.org/wiki/Leslie_matrix for more
• The actual link to the site: Leslie_matrix
• 
• An application of the Leslie Matrix model: population viability analysis (PVA)
  • use it to assess and predict population dynamic for rare populations, where there is less margin for error – how should we manage them, conserve?
  • identify threats faced by species and finding likelihood of species survival for given time
  • rare species’ probability of extinction is measurable: p(E)
• demographic/deterministic factors could result in λ<1, which leads to extinction (birth rates < death rates)
• stochastic (chance variation – lack of food, or uneven sex ratio of offspring…) factors could also lead to extinction despite λ>1 or =1.
• What we need is to maintain several populations, each with a minimum population size
  • PVA uses both these factors too evaluate “extinction vortex”, a downward cycling towards extinction: small population size → increased inbreeding/genetic drift → decreased reproduction and survival → reduced λ → small population size…
  • Use PVA for planning research and data collection, assessing vulnerability on scarce resources, and ranking management options for species

Spatial structure and distribution of populations

• Species populations are made up of sub-populations, distributed in a larger total geographic occupied area, so that the population has spatial structure/spacing, which varies over time
• Fundamental niche – range of environmental conditions in which an organism can live
• Realized niche/habitat – where organism actually lives, subset of fundamental niche, determined by predators, competitors, pathogens, etc.
• Dave Allan quote: “Most invasive species are wimps, but some are Godzilla types” – referring to African tilapias (a Godzilla type)
• Migratory species are NOT species that undergo “great” migrations over long distances every year (Monarch butterflies or geese), but species who occupy large ranges, locally.
  • Require existence of spatially distant habitats and migratory pathway
• Metapopulation – patches of equal quality, but anything outside is unhabitable.
  • Size can vary
  • Viewed as a “population of populations” linked by immigration and emigration – how many patches are occupied? (instead of how many individuals per patch)
  • dP/dt = cP(1-P)-eP, where c is colonization rate and e is extinction rate. (regarding populated proportion of patches)
  • P*=1-e/c (fraction patch occupancy at equilibrium)
• If c<e, extinction; but if c>e, species will persist
• Source-sink dynamics – patches vary in quality, and in-between areas are still uninhabitable. But dispersal between these patches are critical to maintaining populations in them.
• Patch connectivity must be maintained – but if they are too connected, then they are basically one population; alternatively if they are not connected enough, thee will be no “rescue” effect between them
• Implications of population structure
  • Species formation – isolated patches are more likely to develop into species
  • Genetic variation and interchange – subpopulations could have different selection pressures, depending on how their own patch is different
  • Population viability – rescuing of other populations
  • Management of populations and habitat are linked