has the same behavior. (Hint: u[1] in increasing in t and cannot converge to a limit < 1.)
In this model each site can be occupied by one of k species. An individual of type i produces a new offspring of its own type at rate 1, and sends it to a neighbor y chosen at random.
If the target site y is occupied by type j then y will switch to type j with probability p[i,j] and remain unchanged otherwise.
This model is a continuous time version of a system introduced by Silvertown et al (1992), who investigated a special case with
A Single Dominant Species. We say that type i dominates type j if p[i,j] > p[j,i]. Durrett and Levin (1997), using techniques of Grannan and Swindle (1991) and Mountford and Sudbury (1992), have shown
Theorem. If type 1 is dominant over each other type then it takes over the system. That is, if we start with infinitely many sites in state 1 then for each x P( state_t[x] = 1 ) converges to 1 as t tends to infinity.
Math Exercise. Show that the mean field ODE:
has the same behavior. (Hint: u[1] in increasing in t
and cannot converge to a limit < 1.)
s3 Exercise. Fill in parameters with p[1,2] > p[2,1] and p[1,3] > p[3,1] and watch the 1's win.
More interesting behavior occurs if we consider a
Cyclic Three Species System. Suppose p[1,3] = b[1], p[2,1] = b[2], p[3,2] = b[3], and the other p[i,j]=0. In this case
defines a fixed point for the ODE. Furthermore, the function
H(x[1], x[2], x[3]) = v[1] log x[1] + v[2] log x[2] + v[3] log x[3]
is constant along solutions of the ODE, so the fixed point is surrounded by periodic orbits. For example, when b[1] = 0.3, b[2] = 0.7, and b[3] = 1.0, the fixed point is at (0.5, 0.15, 0.35) and the ODE looks like:
Based on the behavior of the ODE, it is natural to
Conjecture. If all the b[i] > 0 then coexistence occurs.
s3 Exercise. Simulate the model using the parameter values b[1] = 0.3, b[2] = 0.7, b[3] = 1.0 to confirm that in this case there is an equilibrium state with an interesting spatial structure.
Related Work. If one considers the cyclic model with p[1,3] = p[2,1] = p[3,2] = 1 but attempted takeovers are successful with probability q and result in an empty site with probability 1-q, one gets a stochastic spatial analogue of ODE's considered by Gilpin (1975) and May and Leonard (1975). Orbits in the modified ODE spiral outward to the boundary but there is still coexistence in the spatial model. Other variations of the spatial model have been considered by Tainaka (1995).
Gilpin, M.E. (1975) Limit cycles in competition communities. Am. Nat. 108, 207-228
May, R.M. and Leonard, W.L. (1975) Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243-253
Grannan, E. and G. Swindle (1991). Rigorous results on mathematical models of catalyst surfaces. J. Stat. Phys. 61, 1085-1103
Mountford, T.S. and Sudbury, A. (1992). An extension of a result of Swindle and Grannan on the poisoning of catalyst surfaces. J. Stat. Phys. 67, 1219-1222
Silvertown, J., Holtier, S., Johnson, J. and Dale, P. (1992). Cellular automaton models of interspecific competition for space - the effect of pattern on process. J. Ecol. 80, 527-534
Tainaka, K. (1995) Indirect effect in cyclic voter model models. {\it Physics Letters A.} {\bf 207}, 53-57
Durrett, R. and Levin, S. (1997) Spatial aspects of interspecific competition. J. Theoret. Biol., to appear