Correction to: Evolution of states in a continuum migration model


1 Correction to: Anal.Math.Phys. (2018) 8:93–121

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).


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

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fakutät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Interdisciplinary Center for Complex SystemsDragomanov UniversityKyivUkraine
  3. 3.Instytut MatematykiUniwersytet Marii Curie-SkłodowskiejLublinPoland

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