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Mechanisms of Species Coexistence involving stochasticity We have examined models with stochastic elements before: population models with random parameter variation metapopulation models with limited deme numbers However, addition of these random elements have not fundamentally altered model outcomes. Today, we discuss models where randomness does alter outcomes.

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Spatial heterogeneity: random variation in resource richness/habitat quality in space. Temporal heterogeneity: random variation in resource richness/habitat quality in time.

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Spatiotemporal heterogeneity: spatial patterns of resource richness/habitat quality vary through time.

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Criterion for stable coexistence in deterministic models: Dynamic properties of the multi-species equilibrium Criterion for stable coexistence in stochastic models: Invasibility: species can coexist if all species in the system can recover from low density. for all i. (The average low-density growth rate for all species is positive)

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How can this be? Assuming niche separation between species, each species is its own worst competitor. Thus, competition tends to be minimized for any species at low density.

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Variation-dependent coexistence is recognized by the different outcomes observed with and without variation: In the absence of environmental variation: competitive exclusion. With environmental variation: stochastically stable coexistence. Random variation in limiting resources theoretically allows an unlimited number of species to exist on one resource, whereas deterministic models (e.g. Timan’s Resource ratio Hypothesis) allows one more species to coexist for every additional limiting resource. Limits to variation-dependent coexistence are imposed only by limits to population size, as small populations have a greater extinction risk, even if their low density growth rate is positive on average.

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Example I: Coexistence by spatial heterogeneity. In the resource-ratio framework, the identity of the species with the lowest R* may vary spatially. pH Species 1 R* Species 2 S1 winsS2 wins

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(Werner and Platt 1976) Six species of goldenrod assort in space with respect to soil moisture conditions

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Mechanisms of Species Coexistence involving stochasticity - equalizing vs stabilizing mechanisms - Equalizing:minimize fitness differences between species Stabilizing:favoring population recovery from low density

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Example II: Coexistence by differences in dispersal ability. In the resource-ratio framework, the identity of the species with the lowest R* may vary spatially. pH Species 1 R* Species 2 S1 winsS2 wins

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R1R1 R2R2 R3R3 X1X1 X2X2 X3X3 Example: MacArthur’s consumer-resource model Fitness gradient: X 1 > X 2 > X 3

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Invasibility criterion: Species can coexist if all species in the system can recover from low density: R1R1 R2R2 R3R3 X1X1 X2X2 X3X3 ?

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Equalize fitnesses: e.g. equalize the death rates R1R1 R2R2 R3R3 X1X1 X2X2 X3X3

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Stabilizing: e.g. reduce inter-specific competition (niche overlap) R1R1 R2R2 R3R3 X1X1 X2X2 X3X3 (this will also equalize fitnesses some more)

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X 3 as invader now has a better chance to coexist: R1R1 R2R2 R3R3 X1X1 X2X2 X3X3

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Equalizing mechanism: alone, cannot make species stably coexist (niche separation is required) alone, can only prolong time to competitive or random exclusion however, less “stabilization” is needed when species have more equal fitness Stabilizing mechanism: required for stable coexistence includes any mechanism that increase negative intraspecific effects over negative interspecific effects systematically weakens competitive effects at low density (where intraspecific competition dominates) and promotes recovery.

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