= Solution
The <reaction–diffusion system> for amoebae contains local reproduction $\alpha n_0^2n$ and density-dependent loss $-\alpha n^3$, giving carrying density $n_0$. The term $\nabla^2n$ describes <diffusion> with coefficient one. The <chemotaxis> flux is $J_{\rm chem}=\chi(n)n\nabla a$, so its contribution to the conservation equation is $-\nabla\cdot J_{\rm chem}$. Attraction towards greater chemical concentration requires $\chi(n)>0$ at the relevant positive densities; negative values would mean repulsion, and zero removes the chemotactic response.
In the chemical equation, $\beta n$ is secretion proportional to amoeba density, $-\gamma a$ is first-order chemical degradation, and $D\nabla^2a$ is chemical diffusion. The cell reaction term permits births and deaths, so the total number of amoebae is not conserved.
$$
\boxed{\chi(n)>0\text{ for attraction},\qquad J_{\rm chem}=\chi(n)n\nabla a.}
$$
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