Population fluctuationsTopics 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

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

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 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!

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)

Phytoplankton in lake Erie exhibit huge fluctuations due to changing extrinsic factors, e.g., temperature, light, food

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!

May’s model of population dynamics 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

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?

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 = r*Nt*{(K - Nt-)/K} Here is the time delay of the density-dependence

This can be modeled easily:

N(t+1) = N(t) + r*N(t)*{(K - Nt-)/K}

Nicholson’s 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-->population crash

Elimination of time-delay by density-dependent adults Identical to prior experiment, except that adults food-

limited (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?

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

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!)

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