Incidence Function Model

The Incidence Function Model (IFM) is a spatially realistic mathematical model of metapopulation dynamics formulated by the Finnish ecologist Ilkka Hanski in 1994 to predict the long-term probability of species presence (incidence J_i) across networks of fragmented habitat patches of varying area A_i and spatial coordinates (x_i, y_i). Whereas Richard Levins's classical 1969 metapopulation differential equation (dp/dt = cp(1 − p) − ep) treated all habitat patches as identical in size and equally connected by global propagule dispersal, real landscapes shaped by natural heterogeneity or anthropogenic habitat fragmentation consist of discrete patches whose areas span several orders of magnitude and whose pairwise inter-patch Euclidean distances d_ij strongly constrain colonization. Hanski bridged the gap between abstract patch-occupancy theory and empirical conservation biology by modeling each habitat patch i as a two-state, first-order discrete-time Markov chain (occupied p_i(t) = 1 or empty p_i(t) = 0) whose local extinction probability E_i scales inversely with patch area A_i and whose colonization probability C_i increases with distance-weighted immigrant connectivity S_i. Notably, all parameters of the Incidence Function Model can be estimated statistically by maximum likelihood or Bayesian inference from a single snapshot survey of patch occupancy (p₁, p₂, ..., p_n) across a quasi-equilibrium landscape network.

Limitations of Levins's Model and MacArthur–Wilson Island Biogeography

Two foundational paradigms preceded the Incidence Function Model in spatial ecology:

1. MacArthur and Wilson's (1967) Theory of Island Biogeography: Introduced the concept of an incidence function J(A, d) (further developed by Jared Diamond in 1975 for avian archipelagos) representing the probability that an island of area A at distance d from a permanent mainland source is occupied at equilibrium. However, mainland–island models assume an unvarying external source pool, whereas in a true metapopulation no permanent mainland exists and every occupied patch serves as a transient source of emigrants.
2. Levins's (1969) Spatially Implicit Model: Captures internal recolonization feedbacks across a network of transient patches, but ignores patch geometry (A_i, d_ij) and cannot identify which specific patches in a reserve network contribute most to regional persistence.

First-Order Markov Chain Formulation for Patch i

In Hanski's (1994) Incidence Function Model, a landscape consists of n discrete, mapped habitat patches (i = 1, 2, ..., n) with measured areas A_i and pairwise distances d_ij. Time is discretized into annual or generational census intervals (t = 0, 1, 2, ...). During a single time step:

• An empty patch (p_i(t) = 0) becomes colonized at time t + 1 with patch-specific colonization probability C_i (and remains empty with probability 1 − C_i).
• An occupied patch (p_i(t) = 1) goes locally extinct by time t + 1 with patch-specific extinction probability E_i (or persists with probability 1 − E_i).

Under the quasi-stationary approximation where regional connectivity fluctuates slowly around its mean, the stationary probability—or incidence J_i—that patch i is occupied satisfies the detailed-balance equilibrium condition that the unconditional probability of an empty-to-occupied transition equals the unconditional probability of an occupied-to-empty transition:

( 1 − J_i ) C_i = J_i E_i ⇒ J_i = C_i / ( C_i + E_i ) (without rescue effect)

When immigrants arriving during the census interval can also rescue a declining population from local extinction within the same time step—the rescue effect of Brown and Kodric-Brown (1977)—the effective net extinction probability is reduced from E_i to E_i(1 − C_i). Substituting E_i(1 − C_i) into the stationary balance yields Hanski's canonical incidence equation:

J_i = C_i / ( C_i + E_i − C_i E_i ) = 1 / [ 1 + E_i ( 1 − C_i ) / C_i ]

1. Area-Dependent Extinction Probability (E_i) and Minimum Patch Area (A₀)

Stochastic population theory (Lande, 1993; Foley, 1994) demonstrates that the expected local carrying capacity K_i of a habitat patch is proportional to its area A_i, and that the probability of local extinction over a single generation scales as an inverse power law of local population size. Hanski parameterized the intrinsic local extinction probability E_i of patch i (in the absence of the rescue effect) as:

E_i = min ( 1 , e / A_i^x ) = min [ 1 , ( A₀ / A_i )^x ]

where:

• e is the baseline extinction probability of a unit-area patch (A_i = 1 ha).
• A₀ = e^1/x is the minimum threshold patch area: any patch with area A_i ≤ A₀ suffers local extinction (E_i = 1) within a single time step unless rescued by immigration.
• x is the environmental stochasticity scaling exponent governing how rapidly extinction risk declines with increasing patch area. When local extinctions are driven by uncorrelated demographic stochasticity, x is large (x > 2); when regional weather fluctuations or correlated environmental stochasticity dominate, x is smaller (typically x ≈ 0.5 to 1.5).

2. Distance-Weighted Connectivity (S_i) and Colonization (C_i)

The number of dispersing propagules arriving at patch i in generation t depends on the proximity and population sizes of all other occupied patches j ≠ i. Hanski defined patch connectivity S_i as the sum of distance-attenuated emigrant contributions across the patch network:

S_i = ∑_j≠i p_j exp ( − α d_ij ) A_j^b (or using J_j in place of p_j at steady state)

where p_j ∈ {0, 1} is the occupancy indicator of source patch j, d_ij is the inter-patch distance (km), α (km⁻¹) is the dispersal mortality parameter of the negative-exponential dispersal kernel exp(−αd_ij) such that 1 / α equals the mean lifetime dispersal distance, and A_j^b scales emigrant output (b ≈ 0.5–1.0).

When multiple independent immigrants arrive at empty patch i at a rate proportional to S_i, the probability C_i that an immigrant cohort establishes a local population follows a saturating rectangular hyperbola:

C_i = S_i² / ( S_i² + y² )

where y is the colonization efficiency parameter (C_i = 0.5 when S_i = y). Combining E_i and C_i with the rescue effect yields the explicit algebraic incidence formula for patch i:

J_i = 1 / [ 1 + ( e′ / S_i² A_i^x ) ] , where e′ = e y²

Separating e from y² requires either two consecutive survey years (measuring turnover events) or an independent biological estimate of the minimum patch area A₀ = e^1/x.

Hanski and Ovaskainen's (2000) Metapopulation Capacity λ_M

In 2000, Ilkka Hanski and Otso Ovaskainen derived a continuous-time analytical formulation of the spatially realistic model that condenses the geometric configuration of a fragmented landscape into a single scalar metric: the metapopulation capacity λ_M.

Define the n × n landscape matrix M with elements m_ij representing the colonization–extinction coupling between patch i and patch j:

m_ij = exp ( − α d_ij ) A_i^x A_j^b for i ≠ j , and m_ii = 0

By the Perron–Frobenius theorem, M possesses a unique positive leading eigenvalue λ_M with a corresponding positive eigenvector v = (v₁, v₂, ..., v_n)^T (normalized so ∑ v_i² = 1). Hanski and Ovaskainen proved that the spatially realistic metapopulation persists regionally if and only if the metapopulation capacity λ_M exceeds the species-specific extinction–colonization threshold δ = e / c:

λ_M > δ = e / c

Near the persistence threshold, the weighted regional equilibrium occupancy p_λ* satisfies a direct analogue of Levins's classical equilibrium equation:

p_λ* = 1 − ( δ / λ_M )

where the classical habitat fraction h is replaced by the leading eigenvalue λ_M of the landscape matrix M. Furthermore, the contribution of patch i to the overall metapopulation capacity of the reserve network is given by λ_i = v_i² λ_M, enabling conservation planners to rank individual habitat patches and corridors quantitatively across fragmented biodiversity networks and multi-species metacommunity landscapes.

Empirical Validation: The Glanville Fritillary (Melitaea cinxia)

The classic empirical testbed for the Incidence Function Model is the long-term study initiated by Ilkka Hanski in 1991 of the Glanville fritillary butterfly (Melitaea cinxia) across more than 4,000 dry-meadow habitat patches spanning 50 × 70 km on the Åland Islands in southwestern Finland. Every autumn since 1993, field teams have surveyed all 4,000+ meadows for the silk winter webs woven by gregarious larval families, recording ~400 to 800 occupied patches, ~100 to 250 local extinctions, and ~100 to 250 recolonizations each year. Parameterizing the IFM on single-year Åland survey snapshots accurately predicted multi-decade turnover rates, demonstrated extinction debt in fragmented sub-networks where λ_M had fallen below δ, and uncovered evolutionary coupling between patch connectivity S_i and allelic variation in the Pgi (phosphoglucose isomerase) metabolic flight-dispersal gene.

How can the Incidence Function Model estimate colonization and extinction rates from a single snapshot survey?

At quasi-stationary equilibrium, small patches have high incidences J_i only if they are well connected (large S_i), while isolated patches have high incidences only if they are large in area (large A_i). Fitting J_i = [1 + e′ / (S_i² A_i^x)]⁻¹ to observed binary presence–absence data (p_i ∈ {0, 1}) via maximum likelihood simultaneously identifies the area-scaling exponent x, the dispersal scale α, and the product e′ = e y², requiring only one additional anchor (such as A₀ = e^1/x or a single repeat survey) to separate e and y².

What is metapopulation capacity (λ_M) and why is it more informative than total habitat area?

Two landscapes with identical total habitat area (∑ A_i) can differ greatly in their ability to sustain a species depending on whether the patches are clustered within the species' mean dispersal distance (1/α) or scattered far apart. Metapopulation capacity (λ_M)—the leading eigenvalue of the connectivity matrix m_ij = exp(−αd_ij)A_i^xA_j^b—integrates both patch areas and spatial clustering into a single number that determines whether the metapopulation will persist (λ_M > e/c) or collapse (λ_M < e/c).

What is an "extinction debt" in the Incidence Function Model?

When habitat fragmentation removes or shrinks patches so that a landscape's metapopulation capacity λ_M drops below the persistence threshold δ = e/c, the species does not vanish overnight. Because Markov chain turnover takes many generations to relax to the new stationary distribution, the landscape temporarily harbors more occupied patches than it can sustain at equilibrium—a delayed deterministic loss known as extinction debt.

1. Hanski, I. (1994). "A practical model of metapopulation dynamics." Journal of Animal Ecology, 63(1), 151–162. https://doi.org/10.2307/5591
2. Hanski, I. (1998). "Metapopulation dynamics." Nature, 396(6706), 41–49. https://doi.org/10.1038/23876
3. Hanski, I., & Ovaskainen, O. (2000). "The metapopulation capacity of a fragmented landscape." Nature, 404(6779), 755–758. https://doi.org/10.1038/35008063
4. Moilanen, A. (1999). "Patch occupancy models of metapopulation dynamics: Efficient parameter estimation using implicit statistical inference." Ecology, 80(3), 1031–1043. [https://doi.org/10.1890/0012-9658(1999)080[1031:POMOMD]2.0.CO;2](https://doi.org/10.1890/0012-9658(1999)080[1031:POMOMD]2.0.CO;2)
5. Ovaskainen, O., & Hanski, I. (2001). "Spatially structured metapopulation models: Global and local assessment of metapopulation capacity." Theoretical Population Biology, 60(4), 281–302. https://doi.org/10.1006/tpbi.2001.1548