Transcript Population fluctuations

Population fluctuations
Topics for this class:
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Population fluctuations in nature can result from changing
environment, i.e., extrinsic environmental factors
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Alternatively, population fluctuations can result from
intrinsic demographic factors, such as high growth rate
coupled with time delay allowing population to exceed
carrying capacity
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Under extreme conditions populations could in theory
behave chaotically, even in a constant environment!
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Both time delays and high population growth rate tend to
destabilize populations, leading to greater fluctuations
Population growth rate depends on
ecological conditions--e.g., two grain
beetle species (imp later, competition!)
Population biology helps ecologists
understand what factors stabilize or
destabilize populations
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Density-dependent population growth tends to
stabilize population size
 We
have just learned that logistic growth leads to
dynamically stable populations
 These always approach an asymptote (K = carrying
capacity) as long as N > 0
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If we look at populations in nature, however, they
are rarely constant: Dynamic (fluctuating)
populations are the norm
We can ask, then, what factors destabilize
populations?
A major cause of population
fluctuations is changing environments!
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Environments are rarely stable, especially at higher
latitudes
 Changes
in populations can result from changes in
food, temperatures, light levels, chemistry, and a
variety of other factors that influence birth and death
rates
 Populations can fluctuate due to spatially
heterogeneous environments, coupled with
emigration and immigration
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Ecologists refer to fluctuations brought about by
changes in the external environment as extrinsic
factors (they are outside a population, and
necessitate demographic adjustments)
Phytoplankton in lake Erie exhibit huge
fluctuations due to changing extrinsic
factors, e.g., temperature, light, food
Intrinsic factors can also cause
population fluctuations
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Sir Robert May was the first ecologist to
demonstrate, with models, how intrinsic population
factors can cause dramatic fluctuations
 May
was trained in Australia as a physicist, with
strong mathematical skills
 He became intrigued with biological problems at
least partly due to the theoretical work of Robert
MacArthur, who was at Princeton University
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Among other things, May showed that very simple
mathematical models of discrete time, densitydependent population growth could lead to an
extraordinary array of population dynamics-including limit cycles and chaos!
May’s model of population dynamics
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May used a difference equation analog of the logistic
model
N(t+1) = N(t)*e(r*[1-{Nt/K}])
 e,
r , K are constants, same as in prior models
 This equation is a discrete-time model, calculating a
new population based on the population one time
unit ago (e.g., one year)
 Notice also that when Nt is near zero {brackets},
right hand side of equation approaches N(t)*er, i.e.,
exponential growth!
 Conversely, when Nt approaches K, right hand side
of equation approaches N(t)*e0, = N(t); i.e., the
population ceases to grow, as in the logistic model
Behavior of May’s model easy to study
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Smooth approach to equilibrium (graph of N as a
function of t), if r < 1
Initial overshoot of K, damped oscillations around K, if
r between roughly 1 and 2
Stable limit cycles (continual oscillations, with fixed
periodicities) if r > 2
Chaos! I.e., one cannot predict population into future,
because of bizarre behavior, for r >> 2
Do any population behave in nature according to
these equations?
 Some
insects with high growth rates show limit cycles,
but none so far show chaotic growth
Why does discrete-time (difference)
equation lead to such fluctuations?
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One explanation is built-in (intrinsic) time-delay,
implicit in difference equation
 Population
can exceed K before negative feedback
occurs that tends to bring it back towards K
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Effect of time delay as a destabilizing factor can be
shown with models
dNt/dt
 Here
 This
= r*Nt*{(K - Nt-t)/K}
t is the time delay of the density-dependence
can be modeled easily:
N(t+1) = N(t) + r*N(t)*{(K - Nt-t)/K}
Nicholson’s lab study demonstrates
destabilizing effect of time-delay
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Classic lab experiment (1958) done with sheep
blowflies (Lucilia cuprina)
Time-delay treatment
 Larvae
provided 50 g liver to feed on per day
 Adults provided unlimited food
 Effect was that density-dependence experienced only
by larvae: When lots of adults present, they laid many
eggs resulting in so many larvae that they all failed to
pupate or produce adults-->population crash
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Elimination of time-delay by density-dependent adults
 Identical
to prior experiment, except that adults foodlimited (1 g liver per day)-->limited egg production
Blowflies growing with time delay: Green line
represents number of adult flies in population
cage; vertical black lines are number of adults
that eventually emerged from eggs laid on
days indicated by the lines
Blowflies grown without time-delay: Adults
food-limited (right hand side of top graph)
such thaf density-dependence occurs on
adults, not on larvae as in prior experiment
What’s the time delay in
Nicholson’s blowflies?
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Time delay was a period of about one week
 This
is equivalent to the time it takes for eggs to hatch
and larvae to develop to the size that they competed for
the limited (50 g) food
 The larvae were way too abundant for the food (densitydependence kicked in) because of the huge numbers of
eggs and larvae produced by the adults
 Adults were able to produce huge numbers of eggs in
the first experiment because adult food was unlimited in
abundance, providing protein for egg production
 Insects experienced “scramble” competition, in which
the larvae eventually had so little food per individual
that none could survive to pupation
Conclusions:
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Population fluctuations the norm in nature
In many cases populations vary in response to
extrinsic environmental factors such as changing
food, temperatures, light, chemicals, etc., that affect
reproduction and survival
In other cases, however, intrinsic dynamics including
time-delays can cause fluctuations, including limit
cycles and chaos--even though the environment is
constant (e.g., r, K do not change!)
Nicholson’s sheep blowfly experiments indicate that
a time-delay in the density-dependent feedback was
what likely caused the population fluctuations
(instability) in his laboratory system