Population fluctuations

1
Population fluctuations

• Topics for this class
• Population fluctuations in nature can result from
  changing environment, i.e., extrinsic
  environmental factors
• Alternatively, population fluctuations can result
  from intrinsic demographic factors, such as high
  growth rate coupled with time delay allowing
  population to exceed carrying capacity
• Under extreme conditions populations could in
  theory behave chaotically, even in a constant
  environment!
• Both time delays and high population growth rate
  tend to destabilize populations, leading to
  greater fluctuations

2
Population growth rate depends on ecological
conditions--e.g., two grain beetle species (imp
later, competition!)
3
Population biology helps ecologists understand
what factors stabilize or destabilize populations

• 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 gt 0
• 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?

4
A major cause of population fluctuations is
changing environments!

• 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
• Ecologists refer to fluctuations brought about by
  changes in the external environment as extrinsic
  factors (they are outside a population, and
  necessitate demographic adjustments)

5
Phytoplankton in lake Erie exhibit huge
fluctuations due to changing extrinsic factors,
e.g., temperature, light, food
6
Intrinsic factors can also cause population
fluctuations

• 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
• Among other things, May showed that very simple
  mathematical models of discrete time,
  density-dependent population growth could lead to
  an extraordinary array of population
  dynamics--including limit cycles and chaos!

7
Mays model of population dynamics

• May used a difference equation analog of the
  logistic model
• N(t1) N(t)e(r1-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

8
Behavior of Mays model easy to study

• Smooth approach to equilibrium (graph of N as a
  function of t), if r lt 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 gt 2
• Chaos! I.e., one cannot predict population into
  future, because of bizarre behavior, for r gtgt 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

9
Why does discrete-time (difference) equation lead
to such fluctuations?

• 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
• Effect of time delay as a destabilizing factor
  can be shown with models
• dNt/dt rNt(K - Nt-t)/K
• Here t is the time delay of the
  density-dependence
• This can be modeled easily
  N(t1) N(t) rN(t)(K - Nt-t)/K

10
Nicholsons lab study demonstrates destabilizing
effect of time-delay

• 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--gtpopulation crash
• Elimination of time-delay by density-dependent
  adults
• Identical to prior experiment, except that adults
  food-limited (1 g liver per day)--gtlimited egg
  production

11
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
12
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
13
Whats the time delay in Nicholsons blowflies?

• 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
  (density-dependence 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

14
Conclusions

• 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!)
• Nicholsons 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