Two particular stochastic parabolic equations are identified as weak limits of \(N\)-dimensional systems of stochastic differential equations (that, in turn, arise as diffusion limits of stochastic population models) as \(N\) tends to infinity. To be more precise, let us describe one of the models: Denote \(N^{-1}\mathbf Z\cap [0,1[\) by \(\mathbf S_{N}\) and the quotient space \(\mathbf R/\mathbf Z\) by \(\mathbf S\). Let \(R\) be a polynomial with a negative leading coefficient, \(R(x) = \sum ^{p}_{i=0} c_{i}x^{i}\), \(x\in \mathbf R\), \(c_{0} \geq 0\), \(c_{p}0\), \(\{N^{-d/2}W_{N} (t,r), \;r\in \mathbf S_{N}\}\) are independent standard Wiener processes and \(\Delta _{N}\) stands for the discrete Laplacian, \(\Delta _{N}f(r) = N^{2}[f(r+N^{-1})+f(r-N^{-1})-2f(r)]\). The weak existence and uniqueness of nonnegative solutions to this system is established by means of the theory of measure-valued processes. Let us consider \(\gamma _{N}\), \(X_{N}(t,\cdot )\) as piecewise constant functions on \([0,1[\); suppose that \(\gamma _{N} \to \gamma >0\) on \([0,1[\) and that \(X_{N}(0)\) converges weakly to an \(L^{2}(\mathbf S)\)-valued random variable \(\xi \). Under some integrability assumptions on \(X_{N}(0)\), the processes \(X_{N}\) are shown to converge weakly in \(C([0,\infty [; L^{2}(\mathbf S))\) to the mild solution \(X\) of the equation \[ dX(t) = \left \{\frac {\partial ^{2}X(t)}{\partial x^{2}} + R(X(t))\right \}dt + dZ(t), \quad X(0)=\xi , \] where \(Z\) is an \(W^{-\alpha ,2}\)-valued martingale with continuous paths such that the quadratic variation of \(\langle Z(t), f\rangle \) is \(\int ^{t}_{0} \langle \gamma X(s),f^{2}\rangle ds\) for any \(f\in W^{\alpha ,2}\), \(\alpha >\frac 12\) (\(W^{s,2}\) denotes the usual Sobolev-Slobodetskij space on \(\mathbf S\)).