Table 1.
Transition rate for the lattice model.
Fig 1.
Interaction between organism-environment positive feedback and Allee effect caused by other ecological mechanisms.
Organism-environment feedback results in weak, strong, and fatal Allee effect as well as the situation of no Allee effect (denoted by green, red, blue, and cyan area, respectively). Allee effect caused by other ecological mechanisms is strong (left of black line) and fatal (right of black line) Allee effect. The interaction of the two kinds of Allee effects leads to strong (below yellow line in (a), above yellow line in (b) and (c)) and fatal (above yellow line in (a), below yellow line in (b) and (c)) Allee effect. For the parameters, We take: μ = 0.03 and e = 0.1 in (a)-(c); c = 0.5 in (a) and (b); d = 0.05 in (b) and (c); λ = 0.8 in (a); c = 0.3 in (c); Point A-F in (a) correspond to panel A-F in Fig 3.
Fig 2.
Definitions of various Allee effects.
Note: function f(p) is per capita growth rate of population, and p* > 0 its maximum point. In schematic diagram, the horizontal axis is population size or density, and vertical axis is per capita growth rate of population.
Fig 3.
Relationship between population size and per capita growth rate.
Green and red lines represent organism-environment feedback (a = 0) and Allee effect caused by other mechanism (d = 0), respectively; blue lines represent the combination of the two types of Allee effects. Panel A-F corresponds to point A-F on Fig 1a.
Fig 4.
The curves indicate zero isoclines of corresponding system. Filled and empty circles represent stable and unstable equilibriums, respectively. The left column correspond to Fig 3A and 3B (also point A and B in Fig 1a).
Table 2.
Condition for Allee effect caused by organism-environment positive feedback.
Fig 5.
The dependence of population dynamics on its initial state when population dispersal locally.
(A) and (B) based on probability transition model, (C) and (D) based on cellular automata model (neighborhood size n = 4 for A and C, n = 8 for B and D). Red areas indicate the initial values leading to the population extinction, while green areas represent the initial values from which the population persists stably. The simulations ran on 100 × 100 lattices with periodic boundary condition. Parameter λ = 0.4, μ = 0.1, d = 0.1; and other parameters are the same as in Fig 1a.
Fig 6.
The distribution of population subjected to Allee effects on two-dimensional space, which are snapshots of 100 × 100 lattice when simulations arrive at 3000 time step.
The color hot indicates the probability of patch occupancy (A) and suitable patch (B). Parameters: d = 0.1 and other parameters are the same as in Fig 1a.
Fig 7.
Spatial distribution of population subjected to Allee effect.
Red, green, and blue colors indicate occupied cells, suitable but no occupied cells, and destroyed (unsuitable) cells. The simulations ran on 100 × 100 lattices with periodic boundary condition and 4 neighborhood. Parameters are taken as a = 0, 0.1, and 0.15 from left to right, λ = 0.3, 0.5, and 0.8 from bottom to top, respectively. Other parameters values are the same as in Fig 4.