Chapter 15: Temporal and Spatial Dynamics of Populations

1. Robert E. Ricklefs The Economy of Nature, Fifth Edition Chapter 15: Temporal and Spatial Dynamics of Populations

2. Some populations exhibit regular fluctuations. • Charles Elton first called attention to regular population cycles in 1924: • such cycles were known to earlier naturalists, but Elton brought the matter more widely to the attention of biologists • Elton also called attention to parallel fluctuations in populations of predators and their prey

3. Evidence for Cycles in Natural Populations • Records of the Hudson’s Bay Company yield important data on fluctuations of animals trapped in northern Canada: • data for the snowshoe hare (prey) and the lynx (predator) have been particularly useful • thousand-fold fluctuations are evident in these records • Records of gyrfalcons exported from Iceland in the mid-eighteenth century also provide evidence for dramatic natural population fluctuations.

4. Fluctuations in Populations • Populations are driven by density-dependent factors toward equilibrium numbers. • However, populations also fluctuate about such equilibria because: • populations respond to changes in environmental conditions: • direct effects of temperature, moisture, etc. • indirect environmental effects (on food supply, for example) • populations may be inherently unstable

5. Domestic sheep on Tasmania – relatively stable population after becoming established (variation due to env. factors)

6. Phytoplankton pop.

7. Fluctuation is the rule for natural populations. • Tasmanian sheep and Lake Erie phytoplankton both exhibit different degrees of variability in population size: • the sheep population is inherently stable: • sheep are large and have greater capacity for homeostasis • the sheep population consists of many overlapping generations • phytoplankton populations are inherently unstable: • phytoplankton have reduced capacity for homeostatic regulation • populations turn over rapidly

8. Periodic cycles may or may not coincide for many species. • Periodic cycles: period between successive highs or lows is remarkably regular • Populations of similar species may not exhibit synchrony in their fluctuations: • four moth species feeding on the same plant materials in a German forest showed little synchrony in population fluctuations • 4-5 year population cycles of small mammals in northern Finland were regular and synchronized across species

9. Temporal variation affects the age structure of populations. • Sizes of different age classes provide a history of past population changes: • a good year for spawning and recruitment may result in a cohort that dominates progressively older classes for years to come • The age structure in stands of forest trees may reflect differences in recruitment patterns: • some species (such as pine) recruit well only after a disturbance • other species (such as beech) are shade-tolerant and recruit almost continuously

10. Commercial whitefish: excellent spawning and recruitment in 1944

11. Population cycles result from time delays. • A paradox: • environmental fluctuations occur randomly: • frequencies of intervals between peaks in tree-ring width are distributed randomly (they vary in direct proportion to temperature and rainfall) • populations of many species cycle in a non-random fashion: • frequencies of intervals between population peaks in red fox are distributed non-randomly

12. A Mechanism for Population Cycles? • Inherent dynamic properties associated with density-dependent regulation of population size • Populations acquire “momentum” when high birth rates at low densities cause the populations to overshoot their carrying capacities. • Populations then overcompensate with low survival rates and fall well below their carrying capacities. • The main intrinsic causes of population cycling are time delaysin the responses of birth and death rates to environmental change.

13. Time Delays and Oscillations: Discrete-Time Models • Discrete-time models of population dynamics have a built-in time delay: • response of population to conditions at one time is not expressed until the next time interval • continuous readjustment to changing conditions is not possible • population will thus oscillate as it continually over- and undershoots its carrying capacity

14. Oscillation Patterns - Discrete Models • Populations with discrete growth can exhibit one of three patterns: • r0 small: • population approaches K and stabilizes • r0 exceeds 1 but is less than 2: • population exhibits damped oscillations • r0 exceeds 2: • population may exhibit limit cycles or (for high r0) chaos

15. 3 oscillation patterns

16. Time Delays and Oscillations: Continuous-Time Models • Continuous-time models have no built-in time delays: • time delays result from the developmental period that separates reproductive episodes between generations • a population thus responds to its density at some time in the past, rather than the present • the explicit time delay term added to the logistic equation is tau (t)

17. Oscillation Patterns - Continuous Models • Populations with continuous growth can exhibit one of three patterns, depending on the product of r and τ: • rτ <e-1 (about 0.37): • population approaches K and stabilizes • rτ < π/2 (about 1.6): • population exhibits damped oscillations • rτ > π/2: • population exhibits limits cycles, with period 4τ - 5τ

18. Cycles in Laboratory Populations • Water fleas, Daphnia, can be induced to cycle: • at higher temperature (25oC), Daphnia magna exhibits oscillations: • period of oscillation is 60 days, suggesting a time delay of 12-15 days • this is explained as follows: when the population approaches high density, reproduction ceases; the population declines, leaving mostly senescent individuals; a new cycle requires recruitment of young, fecund individuals • at lower temperature (18oC), the population fails to cycle, because of little or no time delay of responses

19. Storage can promote time delays. • The water flea Daphnia galeata stores lipid droplets and can transfer these to offspring: • stored energy introduces a delay in response to reduced food supplies at high densities • Daphnia galeata exhibits pronounced limit cycles with a period of 15-20 days • another water flea, Bosmina longirostris, stores smaller amount of lipids and does not exhibit oscillations under similar conditions

20. Overview of Cyclic Behavior • Density dependent effects may be delayed by development time and by storage of nutrients. • Density-dependent effects can act with little delay when adults produce eggs quickly from resources stored over short periods. • Once displaced from an equilibrium at K, behavior of any population will depend on the nature of time delay in its response.

21. Metapopulations are discrete subpopulations. • Some definitions: • areas of habitat with necessary resources and conditions for population persistence are called habitat patches, or simply patches • individuals living in a habitat patch constitute a subpopulation • a set of subpopulations interconnected by occasional movement between them is called a metapopulation

22. Metapopulation models help managers. • As natural populations become increasingly fragmented by human activities, ecologists have turned increasingly to the metapopulationconcept. • Two kinds of processes contribute to dynamics of metapopulations: • growth and regulation of subpopulations within patches • colonization to form new subpopulations and extinction of existing subpopulations

23. Connectivity determines metapopulation dynamics. • When individuals move frequently between subpopulations, local fluctuations are damped out. • At intermediate levels of movement: • the metapopulation behaves as a shifting mosaic of occupied and unoccupied patches • At low levels of movement: • the subpopulations behave independently • as small subpopulations go extinct, they cannot be reestablished, and the entire population eventually goes extinct

24. The Basic Model of Metapopulation Dynamics • The basic model of metapopulation dynamics predicts the equilibrium proportion of occupied patches, ŝ: ŝ = 1 - e/c where e = probability of a subpopulation going extinct c = rate constant for colonization • The model predicts a stable equilibrium because when p (proportion of patches occupied) is below the equilibrium point, colonization exceeds extinction, and vice versa.

25. Is the metapopulation model realistic? • Several unrealistic assumptions are made: • all patches are equal • rates of colonization and extinction for all patches are the same • In natural settings: • patches vary in size, habitat quality, and degree of isolation • larger subpopulations have lower probabilities of extinction

26. The Rescue Effect • Immigration from a large, productive subpopulation can keep a declining subpopulation from going extinct: • this is known as the rescue effect • the rescue effect is incorporated into metapopulation models by making the rate of extinction (e) decline as the fraction of occupied patches increases • the rescue effect can produce positive density dependence, in which survival of subpopulations increases with more numerous subpopulations

27. Chance events may cause small populations to go extinct. • Deterministic models assume large populations and no variation in the average values of birth and death rates. • Randomness may affect populations in the real world, however: • populations may be subjected to catastrophes • other factors may exert continual influences on rates of population growth and carrying capacity • stochastic (random sampling) processes can also result in variation, even in a constant environment

28. Understanding Stochasticity • Consider a coin-tossing experiment: • on average, a coin tossed 10 times will turn up 5 heads and 5 tails, but other possibilities exist: • a run with all heads occurs 1 in 1,024 trials • if we equate a “tail” as a death in a population where each individual has a 0.5 chance of dying, there is a 1 in 1,024 chance of the population going extinct • for a population of 5 individuals, the probability of going extinct is 1 in 32

29. Stochastic Extinction of Small Populations • Theoretical models exist for predicting the probability of extinction of populations because of stochastic events. • For a simple model in which birth and death rates are equal, the probability of extinction increases with: • smaller population size • larger b (and d) • time

30. Probability of extinction increases over time (t) but decreases as a f of initial population size (N)

31. Stochastic Extinction with Density Dependence • Most stochastic models do not include density-dependent changes in birth and death rates. Is this reasonable? • density-dependence of birth and death rates would greatly improve the probability that a population would persist • however, density-independent stochastic models may be realistic for several reasons...

32. Density-independent stochastic models are relevant. • The more conservative density-independent stochastic models are relevant to present-day fragmented populations for several reasons: • most subpopulations are now severely isolated • changing environments are likely to reduce fecundity • when populations are low, the individuals still compete for resources with larger populations of other species • small populations may exhibit positive density-dependence because of inbreeding effects and problems in locating mates

33. Size and Extinction of Natural Populations • Evidence for the relationship between population size and the likelihood of extinction comes from studies of avifauna on the California Channel Islands: • smaller islands lost a greater proportion of species than larger islands over a 51-year period • proportions of populations disappearing over this interval were also related to population size

34. Summary 1 • Populations of most species fluctuate over time, although the degree of fluctuation varies considerably by species. Some species exhibit regular cyclic fluctuations. • Both discrete and continuous population models show how species populations may oscillate.

35. Summary 2 • Population oscillations predicted by models are caused by time delays in the responses of individuals to density. Such delays are also responsible for oscillations in natural populations. • Metapopulations are divided into discrete subpopulations, whose dynamics depend in part on migration of individuals between patches. • The dynamics of small populations depend to a large degree on stochastic events.