Multispecies Metapopulation Models

An important question in ecology is whether species that cannot coexist locally because of competitive exclusion (i.e., within a single patch at the organism scale) can nevertheless coexist within a spatial mosaic or network

Metapopulation Models

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Figure 5 Transient chaos of five species competing for three essential resources. Aperiodic chaotic oscillations occur for a long time before the disappearance of two species and the installation of periodic oscillations among the three remaining species. The outcome of the transient chaos is unpredictable for it may be affected by small differences in initial population densities. Model of Huisman J and Weissing FJ (2001) Fundamental unpredictability in multispecies competition. American Naturalist 157: 488-494.

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Figure 5 Transient chaos of five species competing for three essential resources. Aperiodic chaotic oscillations occur for a long time before the disappearance of two species and the installation of periodic oscillations among the three remaining species. The outcome of the transient chaos is unpredictable for it may be affected by small differences in initial population densities. Model of Huisman J and Weissing FJ (2001) Fundamental unpredictability in multispecies competition. American Naturalist 157: 488-494.

of patches (i.e., within the whole community at the landscape scale). All organisms, especially terrestrial plants, interact mainly with their neighbors, but neighborhoods can differ in composition because of dispersal and local extinction. The easiest way to include these spatial processes in a competition model is the 'metapopulation framework' in which an infinite number of patches are linked by dispersal and competition. Every patch can be empty or occupied by a single local population (one single species of the community) and the dynamics is seen at the landscape level. In the classical spatially implicit formulation of metapopulation models, all patches are equally accessible to colonization, since no explicit spatial distances between patches are included. The strength of interspecific competition is proportional to the average density of interacting metapopulations in the landscape.

The competition-colonization tradeoff is a mechanism that is frequently invoked to explain coexistence, for instance in the Minnesota grasslands studied by Tilman's group. Some grass species allocate more biomass to their roots, which makes them better competitors for nitrogen, while others allocate more to seeds, which makes them better colonizers; and the different species coexist. The tradeoff can be illustrated in a multispecies metapopulation model in which it is assumed that superior resource competitors are not affected by competition with inferior competitors. Note that the model does not include the mechanisms of resource competition; rather, it summarizes the essential qualitative features of competition for a single limiting resource (see Tilman's nonspatial resource-reduction competition model in the previous section).

In this spatially implicit model, species are thus ranked from the best competitor (species 1 with the lowest R*) to the poorest (species n with the highest R*), and it is also assumed that propagules of each species disperse randomly among all patches. This simple deterministic model is therefore a system of n ordinary differential equations for n species. The general equation for the dynamics of the ith species is

= 1 - ¿a) - mP - XX jpp in which pi is the fraction of patches occupied by species of rank i (its proportional abundance, between 0 and 1), Ci its colonization rate, and mi its resource-independent mortality rate (probability of local extinction in absence of competition). The dynamics of each species depends on colonization (the first red term), on local extinction (the second blue term) and on competitive displacement by superior competitors (the last green term). This last term is zero for species 1, its proportional abundance following an S-shaped growth curve. Species 2 is replaced by offspring of species 1 if they land on patches that are occupied by species 2. Offspring of species n (the worst competitor) can land only on vacant patches, whereas offspring ofspecies 1 (the best competitor) can land on empty patches or on patches occupied by any other species.

The principal conclusion from this model is that an inferior competitor can coexist with a superior competitor as long as there is a sufficient tradeoff between their colonization rates and competitive abilities, plus some additional bounds on the extinction rates. The condition for this coexistence at landscape level is that the ratio between local colonization and extinction rates is higher for the inferior than for the superior competitor. In this model assuming an infinite number of patches and an infinite dispersal distance, the number of competitors that can stably coexist on a single resource is theoretically unlimited, even though the best competitor displaces all other species locally wherever it occurs.

An inferior competitor with a higher colonization rate is called a 'fugitive species'. Simulations with this model show that fugitive species may stably persist in metapopu-lations (Figure 6, species 3 and 4). Such species with sufficiently high dispersal rates persist in patches not occupied by superior competitors. At equilibrium, the best competitor (species 1) is not necessarily the dominant species with the highest proportional abundance (species 3). In this example, competitive exclusion occurs after a potential invader (species 2) has been introduced in the plant community and then outcompetes an established dominant lower competitor (species 3).

Metapopulation

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Figure 6 Dynamics of competition among four species (colored lines) in a mean-field metapopulation model. Species are ranked according to their resource competitive ability, with species 1 being the best competitor and species 4 the worst. The black dotted line represents the fraction of empty patches. Species 1, 3, and 4 are introduced at time 0 in the landscape. The invasive species 2, introduced at time 250, displaces the dominant species 3 and reduces the proportional abundance of the fugitive species 4. In this example, the best competitor is not the dominant species at equilibrium. Model of Tilman D (1994) Competition and biodiversity in spatially structured habitats. Ecology 75: 2-16.

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Figure 6 Dynamics of competition among four species (colored lines) in a mean-field metapopulation model. Species are ranked according to their resource competitive ability, with species 1 being the best competitor and species 4 the worst. The black dotted line represents the fraction of empty patches. Species 1, 3, and 4 are introduced at time 0 in the landscape. The invasive species 2, introduced at time 250, displaces the dominant species 3 and reduces the proportional abundance of the fugitive species 4. In this example, the best competitor is not the dominant species at equilibrium. Model of Tilman D (1994) Competition and biodiversity in spatially structured habitats. Ecology 75: 2-16.

Metapopulation models at the next level of complexity explicitly include distances between patches. Possible extensions are partial differential equations, such as reaction-diffusion equations, or integro-differential equations, which describe competition dynamics in an explicitly spatial setting.

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