## 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

### Citations:

Zbl 0843.60057; Zbl 0827.60050
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### References:

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