# Correction to: Evolution of states in a continuum migration model

• Yuri Kondratiev
• Yuri Kozitsky
Correction

## 2 Correcting Lemma 4.1 and Theorem 2.5

The proof of Lemma 4.1 of [1] has a certain inexactness which should be corrected. Namely, in proving the estimate in (4.18), one has to consider in (4.17) the case of $$l=1$$ separately from all other cases as $$F^{(l-1)}(\emptyset )=0$$ holds only for $$l\ge 2$$. For $$l=1$$, we have that $$F^{(l-1)}(\gamma \setminus x)=1$$ for all $$\gamma \ne \emptyset$$, including $$\gamma =\{x\}$$. Thus, starting from the second line in (4.17), we have, see the beginning of Sect. 3.2.2,
\begin{aligned} \frac{d}{dt} q^{(1)}_\Delta (t) \le b_\Delta - \int _{\Gamma _\Delta } \left( \sum _{x\in \gamma _\Delta } \sum _{y\in \gamma _\Delta \setminus x} a(x-y)\right) R^\Delta _{\mu _t}(\gamma _\Delta ) \lambda (\gamma _\Delta ), \end{aligned}
where $$R^\Delta _{\mu _t}$$ is the density of the projection of $$\mu _t$$ with respect to the Lebesgue–Poisson measure $$\lambda$$. By (2.5) and (3.32), this can be rewritten
\begin{aligned} \frac{d}{dt} q^{(1)}_\Delta (t)\le & {} b_\Delta - \sum _{n=2}^\infty \frac{a_\Delta }{(n-1)!} \int _{\Delta ^n}\left( R^\Delta _{\mu _t}\right) ^{(n)} (x_1, \dots , x_n) d x_1 \cdots d x_n\\= & {} b_\Delta - a_\Delta \int _\Delta k^{(1)}_{\mu _t} (x) d x + a_\Delta \mu _t (J_\Delta ) \le b_\Delta + a_\Delta - a_\Delta q^{(1)}_\Delta (t), \end{aligned}
where $$J_\Delta (\gamma ) = 1$$ if $$|\gamma _\Delta |=1$$ and $$J_\Delta (\gamma ) = 0$$ otherwise. That is,
\begin{aligned} \mu _t (J_\Delta ) = \int _\Delta (R^\Delta _{\mu _t})^{(1)}(x) d x \le 1, \end{aligned}
where the latter estimate follows by the fact that $$\mu _t$$ is a probability measure. The meaning of this correction is that the competition contributes to the disappearance from $$\Delta$$ (caused by entities located in $$\Delta$$) only if the number of entities in $$\Delta$$ is at least two. This fact had not been taken into account in the previous version. Then, the estimate in (4.16) holds true with
\begin{aligned} \kappa _\Delta = \max \{\mathrm{V}(\Delta )e^\vartheta ; 1+b_\Delta /a_\Delta \}, \end{aligned}
instead of that given in (4.12). However, for this $$\kappa _\Delta$$, we cannot get the limit of $$\kappa _\Delta /\mathrm{V}(\Delta )$$ as $$\mathrm{V}(\Delta )\rightarrow 0$$. Therefore, all the claims of Theorem 2.5 hold true except for the point-wise boundedness as in (1.8).

## Reference

1. 1.
Kondratiev, Y., Kozitsky, Y.: The evolution of states in a continuum migration model. Anal. Math. Phys. 8, 93–121 (2018).