Cellular Automata (CA) Model

In the cellular automata model space is represented explicitly as a rectangular grid of cells, where each cell has one unit of area and A is the total number of cells. Each cell represents a potential site of occupation by an individual prey. Since only one prey can occupy a cell at any time, we explicitly assume that there is no intraspecific competition for space among growing prey.

The recruitment, growth, and death of prey are modeled as transitions among “cell states”. Each cell is denoted by its x,y coordinates in the grid and the state of a cell is represented by an discrete variable S_xy(t). We use S_xy(t) = 0 to indicate an empty cell, S_xy(t) = s₀ for a cell with a new recruit, and S_xy(t) = s_∞ for a cell with a prey at the maximum size. The size range [s₀, s_∞] is divided into m discrete increments, where m is chosen large enough to approximate continuous growth. (We use m =200). Time is advanced in discrete steps Dt and all cell transitions occur simultaneously. Prey recruitment is represented as a transition from S_xy(t) = 0 to S_xy(t) = s₀. Prey growth is represented as a transition from S_xy(t) = s to S_xy(t) = s + Ds , where Ds ≥ 0. Prey death is represented as a transition from S_xy(t) = s to S_xy(t) = 0 for s > 0. An additional “global variable” P(t) represents the density of predators.

At each time step, cell transitions occur with probabilities that are derived using the rates specified in the ODE model (Table 1). For example, if Dt is sufficiently small, the probability that empty cell will receive a recruit is given by sDt. Similarly, the probability that an occupied cell x,y becomes empty due to prey death is given by

.

where

is the mean of the prey sizes in a neighborhood of (2h+1)x(2h+1) cells centered at x,y. (At the sides and corners of the lattice, S_xy,h(t) is the mean of the subset of neighboring cells within the system.) If h = 0, then S_xy,h(t) = S_xy(t) and the neighborhood is the cell x,y itself. When h = ∞, then S_xy,h(t) = S(t) and we have the mean field approximation of the ODE and SBD models.

Prey growth is also treated as a stochastic process. For a prey of size s, the expected increase in size l_s is given by the differential form of the von Bertalanffy model:

.

The expectation l_s is then used as the mean parameter in a Poisson probability function to determine the actual increase in size Ds for each prey. In rare cases when s + Ds > s_∞, we set Ds = s_∞ – s.

Predator immigration and emigration are treated as a random birth-death process. The number of predators that enter the system at each time step is chosen from a Poisson distribution with a mean value AIDt. Each predator in the system at time t either leaves the system with probability e_P(t)Dt or remains with probability 1 – e_P(t)Dt, where e_P(t) is computed using the same equation as in the ODE model.

Figure 5 compares output from the CA model when two of the assumptions of the mean field ODE and SBD models are relaxed. Figure 6 shows output of the CA model over environmental gradients similar to what one would find in the intertidal zone.

CA Source Code

The CA model was implemented using the freely available Swarm 2.1.1 simulation software. The Swarm development environment must be installed before this source code can be compiled. Check out the web site www.swarm.org for more information on Swarm.

The CA model is still a work in progress. There have been several related versions implemented for different purposes. The version included in the compressed archive file ca.zip has parameter “switches” that turn various features on or off. These are too complicated to explain here in detail, but some comments are provided in the source code. Contact Bob Desharnais for more information.

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File ca.zip. Swarm Objective-C source code for implementation of the CA model with graphical output.