Population fluctuations Topics for this class: n Population fluctuations in nature can result from changing environment, i.e., extrinsic environmental factors n Alternatively, population fluctuations can result from intrinsic demographic factors, such as high growth rate coupled with time delay allowing population to exceed carrying capacity n Under extreme conditions populations could in theory behave chaotically, even in a constant environment! n 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 n Density-dependent population growth tends to stabilize population size u We have just learned that logistic growth leads to dynamically stable populations u These always approach an asymptote (K = carrying capacity) as long as N > 0 n If we look at populations in nature, however, they are rarely constant: Dynamic (fluctuating) populations are the norm n We can ask, then, what factors destabilize populations?

A major cause of population fluctuations is changing environments! n Environments are rarely stable, especially at higher latitudes u 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 u Populations can fluctuate due to spatially heterogeneous environments, coupled with emigration and immigration n 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 n Sir Robert May was the first ecologist to demonstrate, with models, how intrinsic population factors can cause dramatic fluctuations u May was trained in Australia as a physicist, with strong mathematical skills u He became intrigued with biological problems at least partly due to the theoretical work of Robert MacArthur, who was at Princeton University n 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 n May used a difference equation analog of the logistic model n N(t+1) = N(t)*e (r*[1-{N t /K}]) u e, r, K are constants, same as in prior models u This equation is a discrete-time model, calculating a new population based on the population one time unit ago (e.g., one year) u Notice also that when N t is near zero {brackets}, right hand side of equation approaches N(t)*e r, i.e., exponential growth! u Conversely, when N t approaches K, right hand side of equation approaches N(t)*e 0, = N(t); i.e., the population ceases to grow, as in the logistic model

Behavior of May’s model easy to study n Smooth approach to equilibrium (graph of N as a function of t), if r < 1 n Initial overshoot of K, damped oscillations around K, if r between roughly 1 and 2 n Stable limit cycles (continual oscillations, with fixed periodicities) if r > 2 n Chaos! I.e., one cannot predict population into future, because of bizarre behavior, for r >> 2 n Do any population behave in nature according to these equations? u 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? n One explanation is built-in (intrinsic) time-delay, implicit in difference equation u Population can exceed K before negative feedback occurs that tends to bring it back towards K n Effect of time delay as a destabilizing factor can be shown with models  dN t /dt = r*N t *{(K - N t-  )/K}  Here  is the time delay of the density-dependence  This can be modeled easily: N(t+1) = N(t) + r*N(t)*{(K - N t-  )/K}

Nicholson’s lab study demonstrates destabilizing effect of time-delay n Classic lab experiment (1958) done with sheep blowflies (Lucilia cuprina) n Time-delay treatment u Larvae provided 50 g liver to feed on per day u Adults provided unlimited food u 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 n Elimination of time-delay by density-dependent adults u 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? n Time delay was a period of about one week u 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 u 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 u 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 u Insects experienced “scramble” competition, in which the larvae eventually had so little food per individual that none could survive to pupation

Conclusions: n Population fluctuations the norm in nature n In many cases populations vary in response to extrinsic environmental factors such as changing food, temperatures, light, chemicals, etc., that affect reproduction and survival n 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!) n 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