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Diffusion limits for a nonlinear density dependent space-time population model. (English) Zbl 0868.60078

The author considers a reaction-diffusion model and constructs a population density process by using approximately \(Nl\) particles performing rate \(N^2\) random walks between \(N\) cells distributed on the unit interval. Within a cell each particle gives birth at rate \(b_1+ \gamma Nl/2\) and dies at rate \(d_2n_k/l+ d_1+\gamma Nl/2\), where \(n_k\) \((1\leq k\leq N)\) is the number of particles in the occupied cell and \(b_1\), \(d_1\), \(d_2\), \(\gamma\) are fixed nonnegative parameters. Particles also immigrate into the system according to a rate \(b_0 Nl\) Poisson process and are then uniformly distributed among the \(N\) cells. For the case \(\gamma=0\), it is well-known that with suitable scaling, deterministic limits satisfy the PDE \[ {\partial\psi\over\partial t}= \Delta\psi(t)- d_2\psi^2(t)+ c\psi(t)+ b_0,\tag{1} \] where \(c=b_1-d_1-\varepsilon_0 d_2l^{-1}\) with \(\varepsilon_0=0\) or 1. Let \(W\) denote a cylindrical Brownian motion on \(L_2([0,1])\). For the case of particle interaction where \(d_2>0\) and \(\gamma>0\), the limit by passage to \(l\to\infty\) as \(N\to\infty\) leads to the SPDE \[ d\psi(t)= \{\Delta\psi- d_2\psi^2+ (b_1-d_1)\psi+ b_0\}dt+ \sqrt{\gamma\psi(t)}dW(t),\tag{2} \] and the other limit with \(l\) being held constant as \(N\to\infty\) leads to the SPDE \[ d\psi(t)= \{\Delta\psi- d_2\psi^2+ (b_1- d_1-d_2l^{-1})\psi+b_0\}dt+ \sqrt{\gamma\psi}dW(t).\tag{3} \] Both of them are nonlinear perturbations of the equation (1) satisfied by the density process of super Brownian motion. Convergence in distribution holds in \(D([0,T];L_2([0,1]))\) with the Skorokhod topology if \(l\to\infty\) as \(N\to\infty\), and convergence holds in \(D([0,T];H_{-\alpha})\) for any \(\alpha>0\) if \(l\) is constant as \(N\to\infty\). Holding \(l\) constant requires that the cells be averaged together before obtaining the diffusion process limit. If \(l\to\infty\) as \(N\to\infty\), then a diffusion limit occurs in each cell. In this case, one obtains the same limit as first letting \(l\to\infty\), giving a system of \(N\) coupled diffusion processes, and then letting \(N\to\infty\) to obtain a limit satisfying an SPDE. The extra linear term \(d_2l^{-1}\psi\) in (3) can be seen from the case of the deterministic limit (1) when \(\gamma=0\) and \(l\) is constant. In this case, the random walk jumps occur so much faster than reaction jumps that one may expect that cell numbers at a fixed time \(t>0\) will be approximately distributed as independent Poisson random variables. The proof is technically based on the result done by the author [ibid. 22, No. 4, 2040-2070 (1994; Zbl 0843.60057)]. An analogue of the SPDE (2) has been obtained by C. Mueller and R. Tribe [Probab. Theory Relat. Fields 102, No. 4, 519-545 (1995; Zbl 0827.60050)].
Reviewer: I.Dôku (Urawa)

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60B10 Convergence of probability measures
60J60 Diffusion processes
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