Movement and Spatial Populations

Visit a pond three times, hear no toads, and you still cannot say the pond is empty. Walk into a meadow full of butterflies and you still cannot say that population replaces itself. Both problems come from the same fact: many species live in patches, move between them, and are hard to see. This lesson asks how a species persists across patches that keep losing their populations, and how you count animals you mostly do not observe.

The idea: populations in patches

Picture a string of ponds across farmland. Each pond holds a local population, the individuals that live and breed there, and the fields between are hostile ground. A metapopulation is a set of such local populations in separate patches, linked by dispersal.

Dispersal is movement from where an individual was born, or last bred, to where it breeds next. It has three stages that can be measured separately: departure, transfer across the landscape, and settlement. They need not share causes. DiLeo and colleagues (2022) used genetic assignment to reconstruct dispersal of the Glanville fritillary butterfly across five years and 232 habitat patches in the Åland Islands of Finland. The factors that explained departure explained little of the variation in how far butterflies went, and they found no effect of distance on survival after settling. A single “dispersal rate” would have hidden both results.

Dispersal is not migration, the seasonal round trip that many birds, whales and insects make between the same areas. A migrating warbler returns; a dispersing one leaves for good. Dispersal also carries genes between patches, the gene flow treated in drift, gene flow and competing explanations, with the arithmetic in population genetics.

This lesson builds on demography, where survival estimates appear, and on population growth, where a growth rate lambda above or below 1 decides whether a population grows. Here the question moves up a level: can a region keep a species when no single pond does?

Levins: persistence as colonization balancing extinction

Richard Levins (1969) answered by ignoring how many animals each patch holds. Each patch is simply occupied or empty. Let p be the fraction of patches occupied.

dp/dt = c p (1 - p) - e p

The first term is colonization. Colonists come from occupied patches (p) and succeed only where a patch is empty (1 - p), at a rate c per year. The second term is local extinction: each occupied patch loses its population at rate e per year. Set dp/dt to zero and divide by p:

p = 1 - e/c, which is positive only when c is greater than e*

The analogy is a row of lamps that fail at random and get relit from lamps still burning. Lamps are always going out, yet the row stays partly lit as long as relighting outpaces failure. The model’s assumptions are strong: many identical patches, all equally reachable from every other, and constant rates. Real networks have big and small patches, near and far ones.

Worked problem 1 (illustrative numbers). Suppose a network has c = 0.4 per year and e = 0.1 per year.

1. Equilibrium. p* = 1 - 0.1/0.4 = 1 - 0.25 = 0.75. Three patches in four are occupied at any time, though which ones keeps changing.
2. Raise extinction. Now let e rise to 0.4 per year, equal to c. Then p* = 1 - 0.4/0.4 = 0, and the network is lost.
3. It is lost slowly. With c = e = 0.4, the equation becomes dp/dt = -0.4 p^2. Starting from 0.75, occupancy falls at 0.4 x 0.75^2 = 0.225 per year at first, and the solution 1/p = 1/0.75 + 0.4t gives p = 0.19 after 10 years.

That third step is the practical warning. A network past its threshold can keep occupied patches for years. Seeing animals in some patches does not show the region is persisting.

Colonization c 0.40 per year Extinction e 0.10 per year Show patches at year 10 Resample

Equilibrium p* = 1 − e/c

0.75

Expected p at this year

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Observed (this run)

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Colonizations in the year: expected | observed

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Extinctions in the year: expected | observed

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In this sketch: 100 identical patches, all equally reachable from every other, constant c and e, no rescue effect; the run starts with 75 patches occupied, and the random run is one draw (Resample for another). Rates apply in 0.05-year steps (a sketch choice); expected and observed counts differ by chance, since this is one random draw.

Keep the rule: occupancy settles at p* = 1 − e/c only when c is greater than e; patches keep changing state even when the fraction does not.

Spatial structure can also prolong interactions between species. C. B. Huffaker (1958) kept a predatory mite, Typhlodromus occidentalis, with its prey, the six-spotted mite Eotetranychus sexmaculatus, on arrays of oranges in the laboratory. When he spread out the food and added partial barriers to movement, the most complex arrangement sustained three successive oscillations before the run ended. That is a finite run in one preparation, not proof of permanent coexistence. Predator-prey cycles return in predation.

Sources, sinks and corridors

Levins treats every patch as equal. H. Ronald Pulliam (1988) let them differ. In a source, local births exceed local deaths, and the surplus emigrates. In a sink, local reproduction does not make up for local mortality, and the population persists only because immigrants keep arriving. So a sink can be full of animals. Abundance does not label a source; birth and death rates do. This is also why a species can be found outside the conditions it can sustain, a point taken up in niches and distributions.

Two ways that movement keeps a patch occupied are often confused. The rescue effect (a term from Brown and Kodric-Brown, 1977) is immigration into a population that is still present, which lowers its chance of dying out. Recolonization refounds a patch after its population has gone. Rescue makes e smaller; recolonization is the c in the Levins model.

Movement also brings a measurement trap. Traditional mark-recapture estimates apparent survival, the chance an animal is alive and still in the study area. An animal that emigrated permanently looks exactly like one that died.

If connection rescues populations, linking patches with corridors, strips of habitat between them, should reduce losses. Experiments support that in particular systems, as the design problem shows. Connectivity is not good in every model, though: in simulations of 21 competing bird-like species, Schippers and colleagues (2015) found that greater connectivity sped up diversity loss. That is a model result, a caution about extrapolation, not a field finding.

Counting what you cannot see: mark-recapture and occupancy

Every quantity above (c, e, births, deaths) needs counts of animals that hide.

Worked problem 2 (invented data). You catch and mark 50 animals on the first occasion (n1 = 50), release them, and later catch 90 (n2 = 90), of which 20 are marked (m2 = 20). Assume the population is closed (no births, deaths or movement between occasions), every animal is equally catchable, and marks are neither lost nor misread.

1. The logic. The marked fraction in the second sample, 20/90, estimates the marked fraction of the population, 50/N.
2. Lincoln-Petersen. N = n1 x n2 / m2 = 50 x 90 / 20 = 4,500 / 20 = 225 animals.
3. Chapman. The plain estimator runs high when samples are small, so Chapman’s version adds one to each count:

N = (n1 + 1)(n2 + 1) / (m2 + 1) - 1

N = (51 x 91) / 21 - 1 = 4,641 / 21 - 1 = 221 - 1 = 220 animals.

The overlap is 20, not 90. Using the whole second catch in the denominator is the commonest slip. And 220 is an estimate, not a head count. Each assumption can fail. Trap-happy animals inflate m2 and shrink N; trap-shy ones do the reverse. Photo identification, as in a 2018 survey of free-roaming dogs in Quito, Ecuador, avoids handling but can mismatch individuals.

For many species you cannot even mark individuals, only record whether a site has the species. Occupancy is the fraction of sites occupied. The trouble is that a non-detection is not an absence. MacKenzie and colleagues (2002) visited sites repeatedly while occupancy was assumed fixed, and recorded histories such as 010 (found on the second visit only) or 000.

Worked problem 3 (illustrative numbers). A pond is occupied, and each visit detects the species with probability 0.5, independently.

1. Three misses. P(000) = 0.5 x 0.5 x 0.5 = 0.5^3 = 0.125. One occupied pond in eight looks empty.
2. Harder to detect. At 0.3 per visit, each miss has probability 0.7, so P(000) = 0.7^3 = 0.343. One in three.
3. The meaning. A 000 history is consistent with an empty pond or an occupied one missed three times. The pattern of histories across many sites lets the model estimate both probabilities. In their Maryland data from 2000, MacKenzie and colleagues estimated that American toads occupied 0.49 of 32 wetlands. Treating 000 as absence would also count missed populations as local extinctions, inflating e.

Design problem

Gonzalez and colleagues (1998) worked with moss growing on limestone rocks in Derbyshire, England, home to tiny mites, springtails and other microarthropods. They cut the moss into patch systems of three kinds: patches joined by moss corridors; isolated patches; and patches with pseudocorridors, broken corridors with the same moss area as a real one but no continuous connection. After six months, species losses were 14.5% in connected landscapes and 41% in the isolated and pseudocorridor treatments. What can you conclude, what can you not, and how would you improve the study?

One way to do it

What the pseudocorridor buys. A corridor adds habitat as well as connection. The pseudocorridor adds the same moss area without the connection, and it lost species like the isolated patches. So the difference reflects connection, not area.

What it is consistent with. Fewer losses where patches connect fits a rescue effect: immigrants arriving through corridors propped up dwindling populations. But no animal’s movement was tracked, so recolonization after loss, or different conditions in connected moss, are not ruled out.

Improvements.

1. Mark animals, or use genetic assignment, to measure movement through corridors.
2. Record each species in each patch at several times, to separate fewer extinctions from more recolonizations.
3. Run it longer than six months, and replicate across rocks and seasons.
4. Keep the scale. This is moss on rocks, centimetres to metres, with animals of short generation time. It says nothing directly about forests or mammals.

A larger, longer contrast. Damschen and colleagues (2019) arranged restored longleaf pine savanna fragments at the Savannah River Site, South Carolina, in blocks, one fragment per block connected by a corridor and the others not, followed the plants for 18 years and analysed 239 species. They reported annual colonization 5% higher and annual extinction 2% lower in connected fragments, and nearly 14% more species after almost two decades. That design separates arrivals from losses directly, which the moss study could not, and its length shows that small yearly differences accumulate. It also stays in its system: one site, plants, one corridor geometry.