Correction to: Finite dimensional state representation of physiologically structured populations
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Journal of Mathematical Biology https://doi.org/10.1007/s00285-020-01506-w CORRECTION
Correction to: Finite dimensional state representation of physiologically structured populations Odo Diekmann1 · Mats Gyllenberg2 · Johan A. J. Metz3,4 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Correction to: Journal of Mathematical Biology (2020) 80:205–273. https://doi.org/10.1007/s00285-019-01454-0 “In the original publication of the article, the Subsection 2.1.2 was published incorrectly. The corrected Subsection 2.1.2. is given below”.
2.1.2 The mathematical question Our starting point thus are models that can be represented as in the following diagram. U Ec (t,s)
O(E(t))
Y −−−−→ Y −−−−→ Rr
(A)
Here Y is the p-state space and Rr the output space. E is the time course of the environment and U Ec (t, s) the (positive) linear state transition map with s, t the initial
The original article can be found online at https://doi.org/10.1007/s00285-019-01454-0.
B
Mats Gyllenberg [email protected] Odo Diekmann [email protected] Johan A. J. Metz [email protected]
1
Department of Mathematics, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
2
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki, Finland
3
Mathematical Institute and Institute of Biology, Leiden University, 2333 CA Leiden, The Netherlands
4
Evolution and Ecology Program, International Institute of Applied Systems Analysis, 2361 Laxenburg, Austria
123
O. Diekmann et al.
and final time. (The upper index c here refers to the mathematical construction of the p-state, explained in Section 3, through the cumulation of subsequent generations.) Finally O(E(t)) is the linear output map. The mathematical question then is under which conditions on the model ingredients it is possible to extend diagram (A) (for all E, t, s) to the following diagram.
(B)
Here P is a linear map, Φ E (t, s) a linear state transition map (which should be differentiable with respect to t) and Q(E(t)) a linear output map. The dynamics of the output cannot be generated by an ODE when the space spanned by the output vectors at a given time is not finite dimensional. Hence ODE reducibility implies that there exists an r such that the outputs at a given time can be represented by Rr . (Below we drop the time arguments to diminish clutter, except in statements that make sense only for each value of the argument separately, or when we need to refer to those arguments.) Moreover, the biological interpretation dictates that O(E)m = m, Γ (E) :=
Ω
Γ (E)(x)m(d x),
where m is the p-state and the components of the vector Γ (E) functions γi (E): Ω → R. Thanks to the linearity of U Ec (t, s) and O(E(t)) we can without loss of generality assume P, Φ E (t, s) and Q(E(t)) to be linear. Moreover, ODE reducibility requires that P can be written as Pm = m, Ψ with Ψ = (ψ1 , . . . , ψk )T , ψi : Ω → R, where the ψi should be sufficiently smooth to allow d N /dt = K (E)N , with N := Pm and K (
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