17 Flashcards
population size symbol
N
population density symbol
N/area
why do we care about understanding population size, N?
- natural resources management (eg size of fish stocks in the ocean, abundance of outbreaking insect pests in forests)
- conservation: population decline of a species
- health: monitoring populations of viruses or bacteria in humans
- understanding and predicting human population growth
- basic science question of what limits population growth
Population declines in Myotis lucifugous bats due to white nose syndrome (WNS)
- novel pathogen (fungus) emerged in bat population, causing a disease caused white nosed syndrome and really high mortality.
- steep decline in no of over-wintering bats
HIV population dynamics in humans - draw graph of CD4 cells over time
Malthus’ essay on population growth
in 1798, Malthus published an essay on the principle of population, arguing that the human population cannot grow faster than food production
Paul Ehlrich
published The Population Bomb, arguing that explosive growth in the human population would have catastrophic social and environmental consequences
how is the human population expected to change in the future?
demographers project that human population is soon going to peak, then fall dramatically (depopulation)
goals of most population models
- predict the trajectory of population growth through time, i.e. N as a function of t
- how many individuals are in the population now? Nt
- how many individuals are in the population one step later? N(t+1)
- so the general model is N(t+1) = fN(f)
- challenge: choosing simple but realistic parameters for f
when using differential equations, time steps are
infinitesimally small: use concept of limits and calculus; growth is smooth; best suited for species with continuous reproduction
when using difference equations, time steps are
discrete units (days, years, etc); use iterated recursion equations; growth is stepwise and bumpy; best suited for episodic reproduction
two types of time step approaches
continuous-time and discrete-time
how do we pick between the two time-step approaches?
different organisms might be better fit by one or the other
simple bookkeeping model: how can N change from Nt to Nt+1
D = number who die during one time step
B = number born during one time step
E - number who emigrate during one time step
I = number who immigrate during one time step
Nt+1 = Nt - D + B - E + 1
what variables can we consider to be equivalent?
- birth and immigration (ie individuals added to the population)
- death and emigration (ie individuals that disappear from the population)
geometric growth model
- assume no immigration or emigration
- treat birth and death during one time step as per capita rates that are fixed constants
- then, population changes by a constant factor each time step: N(t+1) = λNt
- λ is a multiplicative factor by which population changes over one time unit = ‘finite rate of increase’
- λ = Nt+1/Nt
if λ>1,
birth exceed deaths and population grows
if λ<1
deaths exceed births and population shrinks
N1 =
λN0
N2 =
λN1 = λλN0
N3 =
λN2 = λλN1 = λλλN0
so how can geometric growth be generalised
Nt = N0λ^t
exponential growth
- instantaneous, fixed per-capita rates of birth and death (b and d)
- instantaneous, per-capita rate of population change = b-d=r (a constant)
- r = intrinsic rate of increase
- differential equation is dN/dt = rN
- this model is exponential growth
draw a table comparing discrete-time and continuous time growth models
find the relationship between r and λ
lnλ=r
regardless of which model is adopted, the important consequence is the same
- in both models, the growth rate (λ or r) is a constant that simply reflects biology
- but a constant positive growth rate produces a population growth size that is not constant, but rather exploding in an exponential way
all species…
- have the potential for positive population growth under good conditions (λ>1.0, births exceed deaths)
- have the potential for negative population growth under bad conditions (λ<1.0, deaths exceed births)
- but no species has ever sustained λ>1 or λ<1 for a long period
why is exponential growth a bad model of reality over the long term?
- some factors use tend to keep populations from exploding or going extinct
- two kinds of factors may be acting: density dependent regulation (growth depends on N) or density-independent reduction
how can we model the classically, density-dependent growth?
the logistic equation; an exponential growth with a new term added for brakes
dN/dt = rN(1-N/k)
use bacteria as an example of two types of growth
The logistic braking term models… (draw graph)
the simplest form of density dependence
K
carrying capacity of the environment
logistic trajectories are truly
S-shaped only when
starting from low numbers
label an N vs t graph
logistic model pros
- Mathematically tractable model of intraspecific competition for resources
- Simple (only one extra parameter, K, beyond exponential)
- Can be expanded to consider multispecies competition
logistic podel cons
- Too simple: specifies one particular kind of
density dependence - Always a gradual approach to carrying
capacity - In reality, density-dependence is likely
to be non-linear, may see overshoots of K
