Tipping points emerge from weak mutualism in metacommunities
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.