Approximate probabilistic cellular automata for the dynamics of single-species populations under discrete logisticlike growth with and without weak Allee effects

We investigate one-dimensional elementary probabilistic cellular automata (PCA) whose dynamics in first-order mean-field approximation yields discrete logisticlike growth models for a single-species unstructured population with nonoverlapping generations. Beginning with a general six-parameter model, we find constraints on the transition probabilities of the PCA that guarantee that the ensuing approximations make sense in terms of population dynamics and classify the valid combinations thereof. Several possible models display a negative cubic term that can be interpreted as a weak Allee factor. We also investigate the conditions under which a one-parameter PCA derived from the more general six-parameter model can generate valid population growth dynamics. Numerical simulations illustrate the behavior of some of the PCA found.

Figure 1

Intrinsic population growth rate functions $g^{(+)}\left(x_t\right)$ (a) and $g^{(-)}\left(x_t\right)$ (b) given by (3). These functions provide models for the weak $\left[g^{(+)}\left(x_t\right)\right]$ and strong $\left[g^{(-)}\left(x_t\right)\right]$ Allee effects.

Figure 2

Projection on the $uv$ plane of the solution set $S_{\mathrm{I}}$ of (15) (hatched area). Given $(u,v)\in S_{\mathrm{I}}, f$ is given by the plane $f=v-u$ “hovering” over the hatched area. The hatched polygon has vertices at $(u,v)=(0,0), (1,2), (2,3)$, and $(3,3)$, with total area $2/9$. The dashed edge (0,0)-(1,2) as well as the lines external to the hatched area do not belong to $S_{\mathrm{I}}$.

Figure 3

Space-time diagram of PCA $(a,b,c,d,e,f)=(0,\frac{2}{3},\frac{1}{3},\frac{7}{8},\frac{1}{4},\frac{1}{3})$ exemplifying Case I. A total of 120 cells under periodic boundary conditions are evolved for 120 time steps (time runs downward) from an initially random state of density $1/2$.

Figure 4

Solution set $S_{\mathrm{II}}$ (shaded volume) of (17). The simplex has vertices at $(u,v,f)=(0,0,0), (\frac{3}{2},3,0), (0,3,0)$ (lower face), $(1,2,1), (\frac{3}{2},3,1), (0,3,1), (0,1,1)$ (upper face), with total volume $|S_{\mathrm{II}}|=25/12$. Dashed edges as well as faces bounded by dashed edges do not belong to $S_{\mathrm{II}}$.

Figure 5

Space-time diagram of PCA $(a,b,c,d,e,f)=(0,\frac{3}{8},\frac{2}{5},\frac{4}{5},\frac{3}{4},\frac{4}{5})$ exemplifying Case II. A total of 120 cells under periodic boundary conditions are evolved for 120 time steps (time runs downward) from an initially random state of density $1/2$.