19.09.02 · eco-evo-bio / population-ecology

Metapopulation dynamics: the Levins model, rescue effect, and habitat fragmentation

stub3 tiersLean: nonepending prereqs

Anchor (Master): Hanski, I. — Metapopulation Ecology (1999)

Intuition Beginner

A metapopulation is a group of spatially separated populations of the same species that interact through occasional migration. Imagine a landscape of meadows separated by forest. Butterflies live in each meadow, but individuals occasionally fly between meadows. A single meadow's butterfly population might go extinct due to a bad storm or a predator outbreak — this is a local extinction. But the species does not disappear from the landscape because butterflies from other meadows recolonise the empty patch.

The key insight is that local extinctions are normal and expected. What matters for the species' survival is the balance between local extinctions and recolonisations. As long as colonisation keeps pace with extinction, the metapopulation persists across the landscape even though no single patch is permanently occupied.

Habitat fragmentation — breaking large, contiguous habitats into small isolated pieces — threatens metapopulations in two ways. Smaller patches support smaller populations that are more vulnerable to local extinction. Isolated patches are harder for migrants to reach, slowing recolonisation. When fragmentation pushes the extinction rate above the colonisation rate, the entire metapopulation collapses, even though individual patches remain habitable. This is why building roads, clearing land, or draining wetlands can cause regional species loss long after the last patch of habitat disappears: the metapopulation crosses a tipping point from persistence to collapse.

Visual Beginner

Imagine a landscape viewed from above: a mosaic of green habitat patches (suitable) surrounded by grey matrix (unsuitable). Some patches glow blue — they are occupied. Others are dark green — empty but habitable. Arrows show migrants moving between patches.

The fraction of occupied patches oscillates and then settles at an equilibrium value $p^{*}$ . If colonisation rate $c$ exceeds extinction rate $e$ , the equilibrium is positive and the metapopulation persists. If $e > c$ , the metapopulation declines to zero — regional extinction.

Worked example Beginner

Consider a metapopulation with colonisation rate $c = 0.4$ per year and local extinction rate $e = 0.1$ per year.

Step 1. The Levins model gives $d p / d t = c p (1 - p) - e p$ . At equilibrium, $d p / d t = 0$ , so $c p^{*} (1 - p^{*}) = e p^{*}$ .

Step 2. Solving: $p^{*} = 1 - e / c = 1 - 0.1/0.4 = 0.75$ . The metapopulation persists with 75% of patches occupied.

Step 3. If habitat fragmentation doubles the extinction rate to $e = 0.3$ , the new equilibrium is $p^{*} = 1 - 0.3/0.4 = 0.25$ . Occupancy drops from 75% to 25%. If fragmentation further increases $e$ to 0.5, then $p^{*} = 1 - 0.5/0.4 = - 0.25$ , which is biologically impossible — the metapopulation collapses to regional extinction.

What this tells us: there is a sharp threshold at $e = c$ . Below the threshold, the metapopulation persists. Above it, the species is lost from the landscape even though individual patches remain suitable.

Check your understanding Beginner

Formal definition Intermediate+

The Levins model

Levins (1969) introduced the classic metapopulation model. Let $p$ be the fraction of habitat patches occupied by the species at time $t$ . The model tracks only patch occupancy, not within-patch population dynamics:

\frac{d p}{d t} = c p (1 - p) - e p,

where $c$ is the colonisation rate per occupied patch per empty patch and $e$ is the local extinction rate per occupied patch.

Equilibria. Setting $d p / d t = 0$ : $c p (1 - p) - e p = 0 ⟹ p [c p (1 - p) - e p] = 0$ , giving $p^{*} = 0$ (always an equilibrium) and $p^{*} = 1 - e / c$ (positive when $c > e$ ).

Persistence criterion. The metapopulation persists if and only if $c > e$ . When this condition holds, $p^{*} = 1 - e / c \in (0, 1)$ is globally stable on $(0, 1)$ . When $c \leq e$ , the only feasible equilibrium is $p^{*} = 0$ and the metapopulation goes regionally extinct.

The rescue effect

Brown and Kodric-Brown (1977, Ecology 58, 445-449) observed that immigration reduces local extinction probability. In the rescue effect model, the extinction rate becomes a decreasing function of connectivity (itself proportional to $p$ ):

\frac{d p}{d t} = c p (1 - p) - e (1 - α p) p,

where $α \in [0, 1]$ measures the strength of the rescue effect. When $α = 0$ , this reduces to the standard Levins model. When $α > 0$ , higher occupancy reduces the effective extinction rate, shifting the equilibrium upward:

p^{*} = 1 - \frac{e ( 1 - α )}{c - e α} .

The rescue effect lowers the persistence threshold and increases the equilibrium occupancy. It also creates an Allee-like effect at the metapopulation level: when $p$ is small, extinction exceeds colonisation, but above a critical threshold the rescue effect kicks in and the metapopulation grows toward $p^{*}$ .

Incidence function model (Hanski)

Hanski (1994, J. Anim. Ecol. 63, 151-162) developed the incidence function model (IFM), which makes colonisation and extinction rates depend on patch-specific properties. For patch $i$ of area $A_{i}$ and isolation $I_{i}$ :

C_{i} = \frac{A _{i}^{x}}{A _{i}^{x} + y}, E_{i} = \frac{e}{A _{i}^{x}},

where $x$ scales area effects and $y$ sets the colonisation half-saturation. The probability that patch $i$ is occupied at the next time step is $J_{i} = C_{i} (1 - J_{i}^{current}) + (1 - E_{i}) J_{i}^{current}$ . At steady state, the incidence $J_{i}^{*}$ satisfies:

J_{i}^{*} = \frac{C _{i}}{C _{i} + E _{i}} .

The IFM can be parameterised from snapshot occupancy data and used to predict metapopulation viability under habitat change.

Source-sink dynamics

Pulliam (1988, Am. Nat. 132, 652-661) distinguished source patches (where local reproduction exceeds mortality, producing a surplus of emigrants) from sink patches (where local reproduction is insufficient, and the population persists only through immigration). In the source-sink extension of metapopulation theory, sources drive colonisation and sinks depend on continual rescue. The metapopulation collapses if source patches are lost, even if many sinks remain.

Mainland-island model

The simplest metapopulation model assumes a permanent mainland source that never goes extinct, supplying colonisers to offshore islands (habitat patches). Island $i$ has colonisation rate $c_{i}$ (proportional to proximity to mainland) and extinction rate $e_{i}$ (inversely proportional to island area). Each island is independently occupied with probability $p_{i}^{*} = c_{i} / (c_{i} + e_{i})$ . The mainland-island model ignores inter-island dispersal and is appropriate for systems with one dominant source population.

Counterexamples to common slips

The Levins model does not track population sizes within patches. It is a patch-occupancy model: each patch is either occupied or empty. Within-patch dynamics are collapsed into the parameters $c$ and $e$ . For species where within-patch dynamics matter (e.g., Allee effects within patches), structured metapopulation models are needed.
The persistence criterion $c > e$ applies to the deterministic Levins model. In stochastic versions (SPOMs), persistence depends on the number of patches: even when $c > e$ , a finite patch network has a nonzero probability of simultaneous extinction of all patches. The expected time to extinction grows exponentially with the number of patches.
Habitat fragmentation is not the same as habitat loss. Habitat loss reduces the total area of suitable patches. Fragmentation breaks remaining habitat into smaller, more isolated pieces without necessarily reducing total area. Both increase extinction risk, but fragmentation does so by reducing connectivity rather than reducing carrying capacity. Fahrig (2003, Oikos 103, 163-175) showed that habitat loss generally has a stronger effect than fragmentation per se, but fragmentation matters when patches fall below a minimum viable size.

Key theorem with proof Intermediate+

Theorem (Levins metapopulation persistence). For the Levins model $d p / d t = c p (1 - p) - e p$ with $c > 0$ and $e > 0$ , the equilibrium $p^{*} = 1 - e / c$ is globally asymptotically stable on $(0, 1)$ when $c > e$ , and $p^{*} = 0$ is globally asymptotically stable on $[0, 1]$ when $c \leq e$ .

Proof. Rewrite the Levins equation as:

\frac{d p}{d t} = c p (1 - p) - e p = p [c (1 - p) - e] .

Case 1: $c > e$ . The equilibrium $p^{*} = 1 - e / c$ lies in $(0, 1)$ . Define $V (p) = p - p^{*} - p^{*} ln (p / p^{*})$ for $p \in (0, 1)$ . Then $V (p^{*}) = 0$ and $V (p) > 0$ for $p \neq = p^{*}$ (by convexity of the logarithm). Compute:

\frac{d V}{d t} = (1 - \frac{p ^{*}}{p}) \frac{d p}{d t} = \frac{p - p ^{*}}{p} \cdot p [c (1 - p) - e] = (p - p^{*}) [c (1 - p) - e] .

Substituting $e = c (1 - p^{*})$ :

\frac{d V}{d t} = (p - p^{*}) [c (1 - p) - c (1 - p^{*})] = (p - p^{*}) \cdot c (p^{*} - p) = - c (p - p^{*})^{2} \leq 0.

Since $d V / d t = 0$ only at $p = p^{*}$ , LaSalle's invariance principle gives global asymptotic stability on $(0, 1)$ .

Case 2: $c \leq e$ . For any $p \in (0, 1)$ , $c (1 - p) < c \leq e$ , so $d p / d t = p [c (1 - p) - e] < 0$ . The occupancy $p$ decreases monotonically to zero. $□$

Bridge. The Lyapunov function used here parallels the logistic stability proof in 19.09.01, where $V (N) = N - K - K ln (N / K)$ establishes global stability of the carrying capacity equilibrium. Both proofs exploit the convexity of $ln$ to construct a positive-definite function whose derivative is non-positive. The Levins model is, in fact, a logistic equation in the variable $p$ with effective carrying capacity $1 - e / c$ and growth rate $c - e$ , so the mathematical structures are identical.

Exercises Intermediate+

Exercise 6 (hard, symbolic).

Consider a two-patch metapopulation where patch 1 is a source ( $λ_{1} > 1$ ) and patch 2 is a sink ( $λ_{2} < 1$ ). Emigration from patch 1 to patch 2 occurs at rate $m$ , and from patch 2 to patch 1 at rate $m^{'}$ . Write the coupled population equations and derive the condition for the sink population to persist.

Hint

Let $N_{1}$ , $N_{2}$ be patch populations. The source grows at rate $r_{1} = ln λ_{1} > 0$ and the sink declines at rate $r_{2} = ln λ_{2} < 0$ . The sink persists if immigration exceeds local decline.

Answer

The coupled equations are:

\frac{d N _{1}}{d t} = r_{1} N_{1} - m N_{1} + m^{'} N_{2}, \frac{d N _{2}}{d t} = r_{2} N_{2} + m N_{1} - m^{'} N_{2} .

At equilibrium, $d N_{2} / d t = 0$ : $(r_{2} - m^{'}) N_{2} + m N_{1} = 0$ , giving $N_{2}^{*} = m N_{1}^{*} / (m^{'} - r_{2})$ . The sink population is positive when $m^{'} - r_{2} > 0$ , i.e., $m^{'} > r_{2}$ . Since $r_{2} < 0$ , this is always satisfied for $m^{'} \geq 0$ . The sink population persists at level proportional to the emigration rate $m$ from the source and inversely proportional to $m^{'} - r_{2}$ . If the source is lost ( $N_{1} = 0$ ), the sink declines exponentially at rate $r_{2} - m^{'} < 0$ , confirming that sink persistence depends entirely on the source.

Stochastic patch occupancy models and metapopulation capacity Master

Stochastic patch occupancy models (SPOM)

The deterministic Levins model assumes an infinite number of patches, so stochastic extinction of all patches has zero probability. Real metapopulations have a finite number of patches $n$ , and simultaneous extinction of all occupied patches — however unlikely — is possible.

A stochastic patch occupancy model (SPOM) represents the metapopulation as a binary vector $z = (z_{1}, \dots, z_{n})$ where $z_{i} \in {0, 1}$ indicates whether patch $i$ is occupied. At each time step, each patch transitions independently with probabilities:

P (z_{i} \to 1 ∣ z_{i} = 0) = C_{i} (z), P (z_{i} \to 0 ∣ z_{i} = 1) = E_{i} (z),

where $C_{i}$ depends on connectivity to occupied patches and $E_{i}$ depends on patch area. The full state space has $2^{n}$ states; the metapopulation is extinct when $z = 0$ , an absorbing state.

The expected time to extinction $T_{E}$ scales as $T_{E} \sim exp (α n)$ for $n$ identical patches when $c > e$ : each additional patch roughly multiplies the time to extinction by a constant factor. This exponential scaling means that large networks are effectively persistent on ecological timescales, but no finite network is immortal.

Metapopulation capacity

Hanski and Ovaskainen (2000, J. Anim. Ecol. 69, 1-10) introduced metapopulation capacity $λ_{M}$ , the leading eigenvalue of a landscape matrix $M$ whose entries $m_{ij}$ encode the connectivity between patches $i$ and $j$ weighted by patch areas. The metapopulation persists if and only if $λ_{M} > λ_{c}$ , where $λ_{c} = e / c$ is a critical threshold determined by species-specific extinction and colonisation rates.

The landscape matrix is:

M_{ij} = {A_{i}^{c_{ext}} A_{j}^{c_{col}} exp (- α d_{ij}) 0 i \neq = j, i = j,

where $A_{i}$ is the area of patch $i$ , $d_{ij}$ is the distance between patches, $α$ is a dispersal decay parameter, and $c_{ext}$ , $c_{col}$ scale area effects on extinction and colonisation.

Metapopulation capacity provides a rigorous ranking of landscape configurations: adding a patch, enlarging a patch, or reducing inter-patch distance all increase $λ_{M}$ . Conservation planners can compare alternative reserve designs by computing $λ_{M}$ for each and selecting the configuration that maximises $λ_{M}$ relative to $λ_{c}$ .

Minimum viable metapopulation size

Analogous to the minimum viable population (MVP) concept, the minimum viable metapopulation (MVM) is the smallest patch network that provides an acceptably low extinction probability over a specified time horizon. The MVM depends on the number and configuration of patches, not just total habitat area. A single large patch may have lower extinction risk than several small patches of the same total area if the small patches are too isolated to allow effective recolonisation — but the reverse holds if the single patch is vulnerable to a catastrophic disturbance that simultaneously eliminates the entire population.

Spatially explicit metapopulation models

Spatially explicit models assign each patch a geographic location and compute colonisation rates as a function of inter-patch distance. The colonisation rate from patch $j$ to empty patch $i$ typically follows a dispersal kernel:

c_{ij} = c_{0} exp (- α d_{ij}),

where $d_{ij}$ is the Euclidean distance and $α$ sets the dispersal scale. Total colonisation of patch $i$ is $C_{i} = 1 - \prod_{j \in occupied} (1 - c_{ij})$ , treating each potential coloniser independently. These models predict which specific patches are most critical for metapopulation persistence — typically those with high connectivity (central position) and large area (low extinction rate, high colonisation output).

Landscape connectivity and conservation applications Master

Graph-theoretic connectivity metrics

Landscape connectivity can be analysed using graph theory. Each habitat patch is a node; pairs of patches connected by dispersal are linked by edges weighted by the probability of successful movement. Key metrics include:

Connectance: the fraction of possible links present, $C = L / [n (n - 1) /2]$ where $L$ is the number of links.
Betweenness centrality: for patch $i$ , the number of shortest paths between all pairs of patches that pass through $i$ . High-betweenness patches act as stepping stones whose removal fragments the network.
Component structure: the number and size of connected components. A metapopulation in a single large component has much higher persistence than one split into several isolated components.

Urban and Keitt (2001, Ecology 82, 1203-1218) applied graph theory to the habitat network of the Mexican spotted owl, identifying patches whose removal would disconnect the landscape and prioritising them for protection.

Circuit theory

McRae et al. (2008, Ecology 89, 2712-2724) modelled landscape connectivity using circuit theory, treating the landscape as an electrical network where habitat quality determines conductance. Random walkers (analogous to dispersers) move through the network, and the effective resistance between two patches measures how isolated they are. Circuit theory accounts for all possible dispersal paths simultaneously, not just the shortest path, and identifies pinch points where all paths converge — critical bottlenecks for movement.

Corridor design

Habitat corridors — strips of suitable habitat connecting otherwise isolated patches — increase metapopulation connectivity and persistence. The optimal corridor design depends on the species' dispersal ability and the matrix quality between patches. For species with limited dispersal, corridors must be continuous (no gaps exceed the organism's movement range). For better dispersers, stepping stone corridors — small habitat patches spaced at intervals — may suffice.

The Banff wildlife overpasses in Alberta, Canada, demonstrate the principle at large scales: highway overpasses vegetated with native plants connect habitat on both sides, reducing road mortality and maintaining gene flow for bears, wolves, and elk. Monitoring data show that crossing structures reduce the genetic isolation of populations on opposite sides of the Trans-Canada Highway by approximately 30% (Sawaya et al. 2014, Proc. R. Soc. B 281, 20140029).

Climate change and range shifts as metapopulation processes

Climate change forces species to track suitable conditions by shifting their geographic ranges. Range shifts are metapopulation processes: populations at the trailing edge go extinct as conditions become unsuitable, while populations at the leading edge are founded by colonisers reaching newly suitable habitat. The rate of range shift depends on the metapopulation parameters $c$ (dispersal ability) and $e$ (local extinction risk from novel climate). Species with low dispersal and high climate sensitivity have $c < e$ in the shifting landscape and cannot keep pace with climate velocity, resulting in range contraction and potential extinction. This metapopulation framing predicts which species are most vulnerable to climate change and identifies where assisted migration (translocating individuals to new patches) may prevent extinction.

The glanville fritillary as a model system

Hanski's research group has studied the glanville fritillary butterfly (Melitaea cinxia) on the Åland Islands in Finland since 1991, creating the most detailed empirical metapopulation dataset in ecology. The Åland landscape contains approximately 4,000 dry meadow patches in a 50 km $\times$ 80 km area, of which 300-500 are occupied in any given year. Annual surveys track occupancy, population size, and extinction-colonisation events across the entire network.

Key findings include: (1) patch occupancy follows the IFM prediction, with small isolated patches having lower incidence; (2) the rescue effect is detectable — patches near occupied neighbours have lower extinction rates; (3) inbreeding depression reduces colonisation ability in small populations, creating an eco-evolutionary feedback; (4) metapopulation dynamics are sensitive to weather, with drought years causing widespread local extinctions that require years of recolonisation to recover; and (5) the network structure matters — removing a few well-connected patches causes disproportionate occupancy loss across the network.

Connections Master

Population ecology and Lotka-Volterra 19.09.01. The Levins model is a logistic equation in disguise: $d p / d t = (c - e) p - c p^{2} = (c - e) p [1 - p / (1 - e / c)]$ . All results from logistic theory — the Lyapunov stability proof, the maximum growth rate, the approach to equilibrium — transfer directly. The metapopulation framework extends the single-patch models of 19.09.01 to spatially structured landscapes.
Genetic drift 19.04.01. Metapopulation structure creates conditions for strong genetic drift. Small local populations experience drift within patches, and local extinction followed by recolonisation generates founder effects. The effective population size of a metapopulation is typically much lower than the census size, because most patches are small and turnover is frequent. Whitlock and Barton (1997, Genetics 146, 427-441) showed that metapopulation $N_{e}$ depends on the variance in reproductive success among demes, not just within demes.
Migration and gene flow 19.02.04 pending. Metapopulation connectivity determines gene flow between patches. High colonisation rates maintain genetic homogeneity across the metapopulation; low colonisation rates allow genetic differentiation and potential speciation. Island biogeography theory (MacArthur and Wilson 1967) is the metapopulation model applied to species richness rather than single-species occupancy, with immigration rate replacing colonisation rate and extinction rate playing the same role.
Conservation biology 19.14.01. Metapopulation theory directly informs reserve design. The SLOSS debate (Single Large or Several Small reserves) is resolved by metapopulation analysis: the optimal configuration depends on the species' dispersal ability, extinction risk, and the landscape matrix. Several small reserves outperform one large reserve when inter-patch connectivity is maintained and each patch exceeds a minimum viable size; one large reserve is superior when the matrix is hostile and dispersal is negligible.
Community ecology 19.10.01. Metacommunity theory extends metapopulation dynamics to multi-species assemblages. The four archetypes of Leibold et al. (2004, Ecology Letters 7, 601-613) — patch dynamics, species sorting, mass effects, and neutral — describe how spatial structure interacts with species interactions to shape community composition. Each archetype makes different assumptions about dispersal rate relative to the timescale of local dynamics.

Historical & philosophical context Master

Richard Levins introduced the metapopulation concept in 1969 in "Some demographic and genetic consequences of environmental heterogeneity for biological control" (Bull. Entomol. Soc. Am. 15, 237-240). Levins was a mathematical ecologist at the University of Chicago whose work on strategy-building in population biology emphasised the use of simple, robust models that capture the qualitative behaviour of complex systems. The Levins model — a single differential equation tracking patch occupancy — is the epitome of this philosophy: it abstracts away all within-patch detail to focus on the colonisation-extinction balance that determines regional persistence ^{[Levins 1969]}.

The metapopulation concept lay dormant for two decades before Ilkka Hanski at the University of Helsinki revived and developed it into a research programme. Hanski's 1999 monograph Metapopulation Ecology (Oxford University Press) synthesised theoretical models with empirical data, particularly from the glanville fritillary butterfly system ^{[Hanski 1999]}. Hanski's key contribution was the incidence function model (1994), which made the abstract Levins model operationally testable: by estimating patch-specific colonisation and extinction rates from field data, one could predict metapopulation viability under alternative management scenarios. Hanski was elected to the Finnish Academy of Science in 1998 and received the Balzan Prize in 2000 for his contributions to population ecology.

The rescue effect was named by Brown and Kodric-Brown (1977, Ecology 58, 445-449), who observed that island plant populations near the mainland had lower extinction rates than isolated islands, not because island conditions were better but because continual immigration "rescued" small populations from demographic stochasticity. This observation challenged the assumption that extinction rate is a fixed property of a patch and motivated the development of occupancy-dependent extinction models ^{[Brown & Kodric-Brown 1977]}.

H. Ronald Pulliam (1988, Am. Nat. 132, 652-661) introduced the source-sink distinction, showing that populations in marginal habitats (sinks) could be maintained by immigration from productive habitats (sources) even when sinks were demographic traps where reproduction fell below replacement ^{[Pulliam 1988]}. This result had profound implications for conservation: censuses showing large populations in sink habitats could mask an underlying dependence on small but crucial source populations. Protecting sinks without maintaining sources is ineffective.

The concept of extinction debt — species committed to extinction by past habitat loss but not yet extinct — was formalised by Tilman et al. (1994, Nature 371, 65-66). They showed that habitat destruction creates a transient period during which the metapopulation appears stable but is declining toward a new, lower equilibrium. Species with slow life histories (low $c$ , low $e$ ) carry the largest extinction debts, taking decades or centuries to reach their post-fragmentation equilibrium. This provides a mechanistic explanation for the time lag between deforestation and species loss observed in many tropical landscapes ^{[Tilman et al. 1994]}.

The application of graph theory to landscape connectivity was pioneered by Urban and Keitt (2001, Ecology 82, 1203-1218), who showed that the graph-theoretic representation of patch networks — far simpler than spatially explicit simulation models — captures the essential connectivity properties needed for conservation planning. Circuit theory was introduced to landscape ecology by McRae (2006, Evolution 60, 1551-1561) and extended by McRae et al. (2008), providing a method that accounts for all possible movement paths rather than just the shortest one ^{[Urban & Keitt 2001]}.

Bibliography Master

@article{Levins1969,
  author = {Levins, Richard},
  title = {Some demographic and genetic consequences of environmental heterogeneity for biological control},
  journal = {Bull. Entomol. Soc. Am.},
  volume = {15},
  pages = {237--240},
  year = {1969},
}

@article{BrownKodricBrown1977,
  author = {Brown, James H. and Kodric-Brown, Astrid},
  title = {Turnover rates in insular biogeography: effect of immigration on extinction},
  journal = {Ecology},
  volume = {58},
  pages = {445--449},
  year = {1977},
}

@article{Pulliam1988,
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  year = {1988},
}

@article{Hanski1994,
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}

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}

@article{HanskiOvaskainen2000,
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}

@article{Tilman1994,
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}

@article{UrbanKeitt2001,
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}

@article{McRae2006,
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  year = {2014},
}

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}

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}

@article{WhitlockBarton1997,
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}

Prerequisites

19.09.01 pending

Used in

19.09.03

Tier anchors

beginner: Ricklefs & Relyea, Ecology: The Economy of Nature, 8th ed. (2014), Ch. 10
intermediate: Hartl & Clark, Principles of Population Genetics, 4th ed. (2007), Ch. 5
master: Hanski, I. — Metapopulation Ecology (1999)

References

Ricklefs, R. E. & Relyea, R. — Ecology: The Economy of Nature, 8th ed. (W. H. Freeman, 2014) · Ch. 10 Population dynamics
Hartl, D. L. & Clark, A. G. — Principles of Population Genetics, 4th ed. (Sinauer, 2007) · Ch. 5 Spatial structure
Hanski, I. — Metapopulation Ecology (Oxford, 1999) · Ch. 1-4 Levins model, incidence function
Levins, R. — Some demographic and genetic consequences of environmental heterogeneity · Bull. Entomol. Soc. Am. 15 (1969) 237-240

Estimated time

beginner: 15m
intermediate: 30m
master: 45m