Ecology Flashcards
Population
Group of plants, animals, or other organisms, all of the same species, that live together and reproduce.
Gotelli p. 2
Population size (Exponential population growth)
N
Gotelli p. 2
Exponential population growth
N(t+1) = N(t) + B - D + I - E
Del-N = B - D + I - E, ignoring immigration and emigration this becomes
Del-N = B - D, when assuming that population growth is continuous this would make Del-N infinitesimally small. This allows us to model population population growth rate (dN/dt)
dN/dt = B - D, and now B and D represent birth and death RATES, where B = bN and D = dN, where b and d are the constant per capita birth and death rates. Therefore this model is density dependent.
dN/dt = (b-d)N, where b-d can equal r, the instantaneous rate of increase. Therefore,
dN/dt=rN is the exponential population growth model.
Gotelli p. 3-5
Malthusian parameter
r is also known as Malthusian parameter, after Reverend Thomas Robert Malthus because of “Essay on the Principle of Population”
How to project population size from exponential growth model
N(t) = N(0) e^rt
Gotelli p.6
Calculating doubling time
Constant doubling time is an important feature of a population that is growing exponentially (i.e. no matter how large or small the population, it will always double in size after a fixed time period)
N(tdouble) = 2N(0), when substituting this back into the population size model we get
2N(0) = N(0) e^rt
Dividing by N(0) results in 2=e^rt, and then taking the natural log gives
t(double) = ln(2)/r which is the equation for doubling time
Population size by time for exponential growth model
ENTER FIG 1.1
Assumptions of the exponential growth model N(t) = N(0) e^rt
No I or E (closed population)
Constant b and d (no carrying capacity)
No genetic structure (all individuals have the same birth and death rate)
No age or size structure
Continuous growth with no time lags (individuals are constantly born and dying)
Variations to exponential growth model N(t) = N(0) e^rt, when population grows by finite amount each year
1) Non-overlapping generations with discrete difference equation (for example, animals with seasonality in reproduction that reproduce only once and then die, and the surviving offspring make up the population for the next year). Therefore, the population grows by the discrete growth factor, r(d), and 1 + r(d) is the finite rate of increase, lambda. Then, the recursion equation for population at time t is N(t) = lambda ^ t * N(0) (because growth is not per capita, it is a set increase per year).
Gotelli p. 11-12
Population size by time for N(t) = lambda ^ t * N(0)
ENTER FIG. 1.2
Change discrete model of N(t) = lambda ^ t * N(0) to continuous
If time step becomes infinitesimally and we solve for the limit of lambda (1 + r(d)) we get e^r = lambda, OR r = ln(lamba)
Gotelli p. 13
Relationship between r and lambda
r> 0, lambda > 1 (population increases)
r= 0, lambda = 1 (population stays the same)
r< 0, 0
Variations to exponential growth model N(t) = N(0) e^rt, with environmental stochasticity
Suppose that the instantaneous rate of increase is no longer a constant, than we use an AVERAGE of R to calculate the AVERAGE expected population size at a given time.
N_(t) = N(0) e^r_t
sigma^2(N(t)) = N(0)^2 * e^2r_t * (e^sigma^2(r)*t - 1)
Gotelli p. 15
Exponential growth with environmental stochasticity
ENTER FIG 1.3
With environmental stochasticity, what is the limit of the variance in r to prevent extinction?
Extinction from environmental stochasticity will almost certainly happen if the variance in r is greater than twice the average or r
sigma^2(r) > 2*r_
Gotelli p. 16
Variations to exponential growth model N(t) = N(0) e^rt, with demographic stochasticity
Demographic stochasticity arises because most organisms reproduce themselves as discrete units.
In a deterministic world, the sequence of births and deaths may look like BBDBBD, but with demographic stochasticity this may look like BBBDDB.
Demographic stochasticity is analogous to genetic drift.
In a model of demographic stochasticity, the probability of a birth or death depends on the relative magnitudes of b and d
P(birth) = b/(b+d)
P(death) = d/(b+d)
If b=.55 and d = .5, then r= .05
N_(t) = N(0) e^rt (r is no longer an average but N bar still is) sigma^2(N(t)) = 2* N(0) * b * t if b and d are equal and sigma^2(N(t)) = (N(0)*(b+d) * e^r*t * (e^r*t - 1))/r
P(extinction) = (d/b)^N(0), if d>b then P(extinction) increases
What model should be used when birth and death rates are density dependent (and resource dependent)?
Logistic growth model
Gotelli p. 26
Logistic growth model
Start with the basic equation dN/dt = (b’-d’)N, but make b’ and d’ density dependent by specifying that b’ = b-aN and b’ = b-aN and d’=d-cN, where a and c are are constants. When substituting this back into dN/dt=rN this yields dN/dt=rN[1-N(a+c)/(b-d)].
K=(b-d)/(a+c), which is essentially the ratio of number of individuals that increase the population to the relative density dependence of birth and death
This yields dN/dt=rN(1-N/K),
where rN is the rate of population growth and (1-N/K) is the unused carrying capacity
For example, if K=100 and N=7, the unused carrying capacity is 93%, so the rate of increase will be utilized to 93%
Gotelli p. 27-28
Carrying capacity
Maximum population size that can be supported (K)
To determine the unused carrying capacity, 1-N/K
Gotelli p. 28
Density dependent birth and death rates by population size
ENTER FIG 2.1
Logistic population growth population size
N(t) = K/(1+[(K-N(0))/N(0)]e^-rt)
yields stereotypical S shaped curve when population size starts below K
Population size by time for logistic growth model
ENTER FIG. 2.2
Assumptions of the logistic growth model N(t) = K/(1+[(K-N(0))/N(0)]e^-rt)
1) Constant carrying capacity
2) Linear density dependence (i.e. each individual added to the population causes an incremental decrease in the per capita rate of population growth).
Per capita growth rates as a function of population size with logistic growth and exponential growth
ENTER FIG. 2.4
Variations to logistic growth model with time lags in density-dependent response
Population size at time t is controlled by what happened at time t-tau, when tau is the associated time lag, so dN/dt = rN(1-(N(t-tau)/K)), also known as the delay differential equation.
This equation depends on the time lag and the response time of the population.
Response time is inversely proportional to r (fast growth, short response time).
If rtau is small, between 0 and .368, the population increases smoothly to carrying capacity.
If rtau is medium, between .368 and 1.570, the population first overshoots and then undershoots in damped oscillations until K is reached.
If rtau is larger than 1.570 the population enters a stable limit cycle, periodically rising and falling about K, but never settling on an equilibrium point.
If rtau is larger than 2.570 system breaks down into chaos
Gotelli p. 32-33
Figure of logistic growth curves with time lags.
ENTER FIG. 2.5
Behavior of discrete logistic growth
ENTER FIG 2.6
Variations to logistic growth model with random variation in carrying capacity
N_=K_-sigma^2(K)/2
Gotelli p. 38
Variations to logistic growth model with periodic variation in carrying capacity
K(t) = k(0)+k(1)[cos(2pit/c)]
Under these conditions, the population size is
N_ = sqrt(k(0)^2 - k(1)^2)
Gotelli p. 39
