Density dependence with time delays can cause populations to be inherently cyclic

We have seen that some species have regular cycles of population increases and decreases. The challenge for us is to figure out the causes of these cycles. Some types of environmental variation are cyclical, including daily, lunar or tidal, and seasonal cycles. However, most environmental variation occurs irregularly over time. Prolonged periods of abundant rainfall or drought, extreme heat or cold, or natural disasters such as fires and hurricanes are not predictable. We must therefore conclude that irregular variation in the environment is not likely to be the underlying cause of regular population cycles. In this section, we will examine the inherent cycling behavior of populations and apply a modified version of the logistic growth equation to help us understand some of the causes.

The Inherent Cycling Behavior of Populations

In the 1920s and 1930s, population modelers discovered that populations can naturally oscillate above and below their carrying capacity when they are subjected to environmental variation, even if the environmental variation is random over time. Populations have an inherent periodicity and tend to fluctuate up and down, although the time required to complete a cycle differs among species.

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To help us understand such behavior, we can think of a population as being analogous to a swinging pendulum. We know that a pendulum is stable when hanging straight up and down. The pull of gravity will cause any movement of the pendulum to the right or left to move back toward the center. However, since this movement back to the center has momentum, the pendulum overshoots the stable, center position and swings to the other side. Gravity then pulls it back toward the center where momentum once again causes it to overshoot the stable, center position.

Populations behave like the pendulum; the momentum of increases and decreases in a population causes it to oscillate. Populations are stable at their carrying capacity. When reductions in the population’s size occur—because of events such as predation, disease, or a density-independent event—the population responds by growing. If the growth is sufficiently rapid, the population can grow beyond its carrying capacity. We see this phenomenon when there is a delay between the initiation of breeding and the time that offspring are added to the population. Populations that overshoot their carrying capacity subsequently experience a die-off that causes the population to swing back toward its carrying capacity. Because death rates are high and birth rates are low, the population can experience a large reduction and undershoot its carrying capacity.

Delayed Density Dependence

We can model population cycles by starting with the logistic growth model that we introduced in Chapter 12. You may recall that this model incorporates density dependence, which causes population growth rates to slow down as the population increases in size. The key to making density-dependent populations cycle in these models is to incorporate a delay between a change in environmental conditions and the time the population reproduces. When density dependence occurs based on a population density at some time in the past, we call it delayed density dependence.

Delayed density dependence When density dependence occurs based on a population density at some time in the past.

Delayed density dependence can be caused by any number of factors. For example, large herbivores such as moose often breed in the fall but do not give birth until the following spring. If food is abundant in the fall, the carrying capacity is high, but by the time the offspring are born in the spring, the carrying capacity of the habitat could be much lower. The offspring will still be born, but the population will now exceed the carrying capacity of the habitat because the amount of reproduction was based on the earlier environmental conditions. We can also think about time delays for predators. When predators experience an increase of prey, their carrying capacity increases. However, it may take weeks or months for the predators to convert abundant prey into higher reproductive rates. By this time, the prey may no longer be abundant. The lack of prey will cause the carrying capacity of the predator to decline just as the predator population is increasing. In both scenarios, the population experiences a time delay in density dependence.

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Modeling Delayed Density Dependence

We can modify the logistic growth model to demonstrate delayed density dependence. As you may recall, the logistic model uses the following equation:

where is the rate of change in the population size, r is the intrinsic growth rate, N is the current size of the population at time t, and K is the carrying capacity.

To incorporate a time delay, we begin by defining the amount of time delay as τ, which is the Greek letter tau. Now we can rewrite the logistic growth equation by making the density-dependent part of the equation based on the population’s size at τ time units in the past:

In words, this equation tells us that the population slows its growth when the population’s size, at τ time units in the past, approaches the carrying capacity.

Whether a population cycles above or below the carrying capacity depends both on the magnitude of the time delay and on the magnitude of the intrinsic growth rate. As the time delays increase, density dependence is further delayed, making the population more prone to both overshooting and undershooting the carrying capacity. In addition, having a high intrinsic rate of growth allows a population to grow more rapidly in a given amount of time, making an overshoot of the carrying capacity more likely.

Population modelers have determined that the amount of cycling in a population experiencing delayed density dependence depends on the product of r and τ, as illustrated in Figure 13.10. As you can see in Figure 13.10a, when this product is a low value ( < 0.37), the population approaches the carrying capacity without any oscillations. If this product is an intermediate value (0.37 < r τ < 1.57), as shown in Figure 13.10b, the population initially oscillates but the magnitude of the oscillations declines over time, a pattern known as damped oscillations. When the product is a high value ( > 1.57), as shown in Figure 13.10c, the population continues to exhibit large oscillations over time, a pattern known as a stable limit cycle.

Figure 13.10 Population cycles in models containing delayed density dependence. (a) In population models where the product of is a low value ( < 0.37), the population approaches the carrying capacity without any oscillations. (b) When the product of is an intermediate value (0.37 < < 1.57), the population will exhibit damped oscillations. (c) When the product is a high value ( > 1.57), the population will oscillate over time as a stable limit cycle.

Damped oscillations A pattern of population growth in which the population initially oscillates but the magnitude of the oscillations declines over time.

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ANALYZING ECOLOGY

Delayed Density Dependence in the Flixweed

Flixweed (Descurania sophia) is a weed that is native of Europe but has been introduced into North America. Researchers who have studied this weed have found that the number of plants per m2 of soil fluctuates in a cyclical manner over time. The population grows according to a delayed density dependence model where K = 100, r = 1.1, and τ = 1.

The flixweed, a common plant in Europe and North America.
Photo by Nigel Cattlin/Alamy.

From plant surveys, we know that there were 10 plants per m2 in year 1 and 20 plants per m2 in year 2. Based on these data, we can calculate the expected change in population size in year 3:

Rounded off to the nearest whole number, the flixweed will add 20 individuals to the population in year 3. Given that the population in year 2 is 20 individuals, adding 20 more individuals in year 3 will produce a total population size of 40 individuals.

We can continue the calculations to determine the population size in year 4:

As we can see, the flixweed population will increase by another 35 individuals in year 4, producing a total population size of 75 individuals.

Cycles in Laboratory Populations

Population models that incorporate delayed density dependence help us understand how time delays cause populations to oscillate in regular cycles. Because models do not identify specific mechanisms by which time delays occur, ecologists have investigated real populations using laboratory experiments to discover these mechanisms.

Stable limit cycle A pattern of population growth in which the population continues to exhibit large oscillations over time.

In some cases, delayed density dependence occurs because the organism can store energy and nutrient reserves. The water flea Daphnia galeata, for example, is a tiny zooplankton species that lives in lakes throughout the Northern Hemisphere. When the population is low and there is an abundance of food, individuals can store surplus energy in the form of lipid droplets. As the population grows over time to the carrying capacity and food becomes scarce, adults with stored energy can continue to reproduce. Daphnia mothers can also transfer some of these lipid droplets to their eggs, which allows their offspring to grow well even if the carrying capacity of the lake has been exceeded. Eventually, the stored energy is used up and the Daphnia population crashes to low numbers. When the population is low, the food can once again become abundant and the cycle begins once more. You can see these oscillations in Figure 13.11a.

Figure 13.11 The importance of energy reserves in causing population cycling. (a) Daphnia galeata water fleas can store high amounts of energy, which allows them to survive and reproduce even after reaching carrying capacity. When energy reserves run out, the population crashes to very low numbers and then rebounds and continues to oscillate. (b) Bosmina longirostris water fleas can only store a low amount of energy, so as the population nears carrying capacity, they experience reduced survival and reproduction. As a result, the population remains near its carrying capacity and oscillates much less.
Data from C. E. Goulden and L. L. Hornig, Population oscillations and energy reserves in planktonic cladocera and their consequences to competition, Proceedings of the National Academy of Sciences 77 (1980): 1716–1720.

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We can contrast the story of the Daphnia water flea with the story of another species of water flea, Bosmina longirostris. Bosmina do not store as many lipid droplets as Daphnia, so when their population approaches the lake’s carrying capacity, they have little energy to buffer the reduction in food. As a result, Bosmina populations do not exhibit large oscillations. Instead, as illustrated in Figure 13.11b, they grow to their carrying capacity and remain there.

Delayed density dependence can also occur when there is a time delay in development from one life stage to another. In a classic study of developmental delays, A. J. Nicholson examined the effect of time delays between the larval and adult stages of the sheep blowfly (Lucilia cuprina), an insect that feeds on the flesh of domestic sheep.

In his first experiment, Nicholson fed the larvae a fixed amount of food—thereby setting a carrying capacity for the larvae—but he fed the adults an unlimited amount of food. At the start of the experiment, the larvae began to metamorphose into the first set of adults, as shown in the orange line in Figure 13.12a. These adults then laid eggs that hatched into more larvae that eventually became adults. The adult fly population rapidly increased to more than 4,000 individuals.

Figure 13.12 Population cycling in sheep blowflies. (a) When researchers limited food for larvae but not adults, they observed a delay between the time that the adults produced a large number of eggs and the time these eggs hatched into larvae, died from high larval competition, and failed to produce new adults. As a result, the adult population experienced regular cycles. (b) When adults were initially raised with unlimited food, but then given limited food halfway through the experiment, they began to experience density dependence without a time delay. As a result, the adult population still fluctuated but no longer experienced regular cycles.
Data from A. J. Nicholson, The self-adjustment of populations to change, Cold Spring Harbor Symposia on Quantitative Biology 22 (1958): 153–173.

As the adult population increased, the unlimited food supply allowed them to continue laying eggs. The large number of eggs hatched into a large number of larvae, but, because the larvae had a limited food supply, they did not grow well enough to metamorphose into adults. The larvae died and no new adults were produced. Eventually the adult population crashed. However, before the last few adults died, they laid a small number of eggs. When these eggs hatched, the fixed food supply provided an abundance of food for the low number of larvae. As a result, the larvae experienced a high rate of survival and most of them metamorphosed into adults. These new adults then laid a large number of eggs and the cycle began again. In short, there was a delay between the time that the adults produced a large number of eggs and the time these eggs hatched into larvae, died from high larval competition, and failed to produce new adults. This time delay appeared to cause the adult population cycles.

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Nicholson then reasoned that if limited food for offspring caused the time delay, limiting the adult food should eliminate the time delay and reduce the extreme fluctuations in the adult population. To test this hypothesis, he ran the experiment again by starting with unlimited adult food, but halfway through the experiment he limited the adult food. Under these conditions, the adults experienced density dependence without delay. As you can see in Figure 13.12b, although the abundance of the adult population still fluctuated, it no longer exhibited regular population cycles. These laboratory studies confirmed that time delays between life stages caused population cycles.