Hierarchical competition models with Allee effects

Figures & data

Figure 1. A single-species model with no Allee effect. (a) We plot x_t+1=f(x_t). There are two fixed points: $x_1^{\ast }=0$ and $x_2^{\ast }=K$ (the carrying capacity) and (b) we plot the ‘overall’ fitness function x_t+1/x_t versus x_t. Notice that when x_t=K, the fitness function is equal to 1. Moreover, x_t+1/x_t>1.

Figure 2. A single-species model with Allee effect. (a) We plot x_t+1=f(x_t). We have two possible fixed points: A (threshold Allee point) and K (carrying capacity). If the population falls below the value of A, it will go extinct and (b) we plot the ‘overall’ fitness function x_t+1/x_t. Notice that x_t+1/x_t|_x=0=1 which is the hallmark of the strong Allee effect.

Figure 3. (a) We plot u(r, m, s)<2 represented by the region inside the above solid and (b) bifurcation diagram.

Figure 4. (a) Zero interior fixed point, (b) two interior fixed points, (c) four interior fixed points, (d) one interior fixed point and (e) three interior fixed points.