Fig 1.
The (well-mixed) dynamics of a population with strong Allee effect vs logistic growth.
A. If a population exhibits a strong Allee effect, its growth rate ∂tN is negative at low population sizes. The dynamics, therefore, heads toward extinction (N = 0) unless the initial population size exceeds a threshold, the Allee threshold A, upon which the population size rises to a limiting population size (N = K), referred to as carrying capacity. Full and open circles denote stable and unstable fixed points, respectively; arrows denote the flow of the dynamics. B. In contrast, a single-species population that undergoes regular logistic growth displays only one stable fixed point at the carrying capacity, while the extinct state is unstable. Additional mutualistic interactions, here tuned by a parameter α, can increase the effective carrying capacity K(α) without qualitatively changing the fixed point structure.
Fig 2.
Weak mutualism generates a tipping point.
A. Starting at small and large initial population sizes (triangles and circles, respectively), the mean deme population size 〈N〉 in our numerical solutions settles at a positive value or decays to zero. The long term steady state values are in very good agreement with our mean-field solution (lines). Solid and dashed lines denote stable and unstable manifolds, respectively. Colors denote different numbers S of species as indicated. The analytical result for λc is shown as black dotted vertical line and the S–dependent tipping point dispersal rates λt are indicated by vertical dotted lines of corresponding color. The deterministic steady states N*, given by Eq (2), are indicated by stars on the right next to the plot. B. Numerical and analytic solutions for the abundance distribution P[N] (circles and lines, respectively) for S = 100 with λ < λc (λ = 10−4, green open circles and solid line) and λ > λc (λ = 10−1, green full circles and dashed line) as well as for S = 1 with λ closely above λc (λ = 10−2.4, blue). C. Small changes in the species number, i.e. through perturbations, can lead to a collapse of the metacommunity (as indicated by the arrow), λ = 0.001. Parameters: r = 0.3, K = 10, α = 0.005, P = 500.
Fig 3.
Tipping point for metacommunities with random interactions.
A. Numerical solutions of Eq (3) for initially low (triangles) and high (circles) mean population sizes for three different sets of random interactions (denoted by three different colors) suggest bistability and hysteresis between the tipping point (gray dotted vertical line) and close to λc (black dotted vertical line). Gray solid line shows mean-field solution for identical interactions () B. Distributions of mean interactions with other species for all three sets of random interactions shown in A for λ = 10−2.8, λ = 10−2.2, λ = 10−1.5 from dark to light, respectively (λc ≈ 10−2.6). Remaining Parameters:
(on average competitive interactions).
Fig 4.
Density-dependent dispersal generates tipping point.
A. The mean population size in our numerical solutions can reach different values when starting at small and large initial population sizes (triangles and circles, respectively) in agreement with our mean-field solution (solid and dashed lines denote stable and unstable manifolds, respectively). The black dotted vertical line marks λc, the colored dotted vertical lines lines indicate λt based on numerical solutions. The inset illustrates density-dependent dispersal, where emigration increases with the abundance of individuals from other species on a patch. B. Mean-field solution of the mean abundance predicts a catastrophic shift as function of the species number, λ = 0.001. Remaining parameters: r = 0.3, K = 10, β = 0.02, P = 500.