The contact process is much easier to understand in one dimension than in two. The main reason is that:
Edge Speeds Characterize the Critical Value
If we start the contact process with all sites x > 0, and all
other sites occupied then there is a rightmost occupied site.
If we call its position r(t), then Durrett (1980) showed
and furthermore 
s3 Exercises. In one dimension, the critical value delta_c, which is about 0.3031. To get a feel for this, set the width of the array to be 400, the height equal to 250, and start from an initial interval of length 100 (choose the rectangle initialization).
1. If delta is larger than the delta_c, say 0.4, the system will die out very quickly.
2. When delta is smaller than delta_c, say 0.2, the system will grow linearly and have few holes. 3. Near delta_c, say 0.3, the leftmost and rightmost occupied sites behave like random walks with no drift, and the space-time picture has a fractal appearance.
For maximum simplicity we can consider the contact process in one dimension and in
Discrete Time
One way to formulate the dynamics is to declare that
(i) An occupied site x at time n will independently and with probability p send offspring at x+1 and at x-1 at time n+1, and will itself survive with probability 0.
(ii) If one or two offspring are sent to a site it will be occupied, otherwise it will be vacant.
(iii) Only the even integers to be occupied at time 0 and hence only odd integers can be occupied at time 1, etc.
The alternation in (iii) is a little awkward but allows us to make connection with the usual two dimensional oriented percolation model. To do this, we repeatedly flip a coin with probability p of heads to see if the arcs from (x,n) to (x+1,n+1) and from (x,n) to (x-1,n+1)) are open or closed. An open arc will corresponds to a birth if the lower endpoint is occupied. For example, consider the following picture where the thin arcs are closed and the thick ones are open.
Here the green arcs are the ones that will contain fluid if we imagine that there are sources at 0 and 2 at height (or time) 0 and tha fluid can only flow up through open bonds. It is easy to see that the sites which are wet in oriented percolation are the same as those that are occupied in oriented percolation.
Properties of the contact process are almost the same in discrete or continuous time, but usually things are easier to see in discrete time. For instance, duality reduces to the fact that the probability of an open path from A at time 0 to B at time n is the same as the probability of an open path from A at time 0 to B at time n. For more about oriented percolation, see Durrett (1984).
A More Realistic Discrete Time Model
With annual plants in mind we can think of the state at time n as indicating the sites occupied by young plants at the beginning of the growing season. With this interpretation it is natural to assume that:
(i) Plants die with probability gamma before they produce seeds.
(ii) A surviving plant, independently and with probability lambda will send seeds to x-1, x, and x+1. To avoid imposing a second death event we will suppose that such seeds survive to become young plants, but the reader is free to modify this rule as well.
The new formulation is equivalent to percolation on a new graph in which bonds from (x,n) to (x,n+1/2) are open with probability 1-gamma, and from (x,n+1/1) to (x+1,n+1), (x,n+1), and (x-1,n+1) are open with probability lambda.
Most of the results known for oriented percolation carry over to the new two parameter model. See Section 2 of Durrett and Levin (1994).
s3 Exercise. Run the discrete time model with its default values, gamma = 0.25 and lambda = 0.4. Here the system is close to dying out so fractal looking pictures result.
Brower, R.C., Furman, M.A., and Moshe, M. (1978) Critical exponents for the Reggeon quantum spin model. Phys. Lett. B. 76, 213-219
Durrett, R. (1980) On the growth of one dimensional contact processes. Ann. Prob. 8, 890-907
Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Prob. 12, 999-1040
Durrett, R., and Levin, S. (1994) Stochastic spatial models: A user's guide to ecological applications. Phil. Trans. Roy. Soc., B, 343, 329-350