On the basic reproduction number in continuously structured populations

\`Angel Calsina; Carles Barril; Jordi Ripoll; S\'ilvia Cuadrado

arxiv: 2002.10557 · v1 · submitted 2020-02-24 · 🧮 math.AP · q-bio.PE

On the basic reproduction number in continuously structured populations

Carles Barril , \`Angel Calsina , S\'ilvia Cuadrado , Jordi Ripoll This is my paper

Pith reviewed 2026-05-24 15:22 UTC · model grok-4.3

classification 🧮 math.AP q-bio.PE

keywords basic reproduction numbernext-generation operatorcontinuously structured populationsspectral radiuspopulation dynamicsBanach latticesize-structured modelsage-structured models

0 comments

The pith

The basic reproduction number for continuously structured populations is recovered as the limit of spectral radii from next-generation operators in a sequence of approximating models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In population dynamics the basic reproduction number R0 measures the average number of offspring an individual produces over its lifetime and determines whether a population grows. For models with continuous structure such as size or age, where the state space is a Banach lattice X and births are concentrated at specific points, the next-generation operator cannot be defined directly inside X. The paper establishes that R0 equals the limit of the spectral radii of next-generation operators taken from a sequence of approximating models in which the operator is well-defined. A sympathetic reader cares because this supplies a concrete way to decide growth or decline for realistic continuously structured populations without discretizing the structure itself. The method is illustrated on size-dependent growth, cell populations, diffusion in size, and age-structured models with diffusion.

Core claim

In continuously structured populations defined in a Banach lattice X with concentrated states at birth one cannot define the next-generation operator in X. The basic reproduction number can be obtained as the limit of the spectral radii of next-generation operators from a sequence of approximating models for which R0 can be computed as the spectral radius of the next-generation operator. The results are applied to the classical size-dependent model, a size structured cell population model, a size structured model with diffusion in structure space under particular assumptions, and a physiological age-structured model with diffusion in structure space.

What carries the argument

The limit of spectral radii of next-generation operators taken from a sequence of approximating models that restore definability inside the Banach lattice.

If this is right

R0 becomes computable for the classical size-dependent population model.
R0 becomes computable for size-structured cell population models.
R0 becomes computable for size-structured models that include diffusion in structure space under the stated assumptions.
R0 becomes computable for physiological age-structured models that include diffusion in structure space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The limit construction could be used to approximate R0 numerically even when the original operator is unavailable.
The same limiting procedure may extend to other continuous structuring variables such as spatial location or physiological state beyond the four examples.
If the limit exists, it supplies a practical bridge between infinite-dimensional theory and finite-dimensional computation for stability questions.

Load-bearing premise

There exists a sequence of approximating models whose next-generation operators are definable and whose spectral radii converge to the correct basic reproduction number of the original model.

What would settle it

For any concrete continuously structured model, compute the proposed limit from the approximating sequence and compare it with the average lifetime offspring count obtained by direct integration along individual trajectories; systematic mismatch falsifies the claim.

Figures

Figures reproduced from arXiv: 2002.10557 by \`Angel Calsina, Carles Barril, Jordi Ripoll, S\'ilvia Cuadrado.

**Figure 1.** Figure 1: A graph of RD 0 showing values larger than 1 for moderate diffusion coefficient. Acknowledgements This work has been partially supported by the project MTM2017-84214-C2-2-P of the Spanish government. 4 Appendix: An alternative computation of the basic reproduction number for the age structured model with diffusion using the solution semigroup An explicit expression for the solution semigroup of the system… view at source ↗

read the original abstract

In the framework of population dynamics, the basic reproduction number R_0 is, by definition, the expected number of offspring that an individual has during its lifetime. In constant and time periodic environments it is calculated as the spectral radius of the so-called next-generation operator. In continuously structured populations defined in a Banach lattice X with concentrated states at birth one cannot define the next-generation operator in X. In the present paper we present an approach to compute the basic reproduction number of such models as the limit of the basic reproduction number of a sequence of models for which R_0 can be computed as the spectral radius of the next-generation operator. We apply these results to some examples: the (classical) size-dependent model, a size structured cell population model, a size structured model with diffusion in structure space (under some particular assumptions) and a (physiological) age-structured model with diffusion in structure space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs R0 as the limit of spectral radii from approximating next-generation operators for Banach lattice models with concentrated births, but whether that limit recovers the correct invasion threshold needs explicit verification.

read the letter

The main contribution is a limit construction that lets you recover R0 when the next-generation operator cannot be defined directly on the space X because births are concentrated. They take a sequence of approximating models where the operator is well-defined, compute the spectral radii there, and pass to the limit. This is applied to four standard cases: the classical size-dependent model, a size-structured cell population model, a size model with diffusion under extra assumptions, and a physiological age model with diffusion in structure space. Those examples are concrete and already common in the literature, so the method gives a workable route around the technical blockage for people using similar setups. The construction itself appears new in this exact form. The soft spot is the missing check that the limit equals the epidemiologically or dynamically correct threshold. The abstract states the approach but supplies no derivation or argument showing the limit is independent of the particular approximation sequence or that it preserves the stability threshold of the original model. If smoothing or discretization alters the support of the birth measure in a non-uniform way, the limit could land above or below the true invasion number. That is the load-bearing step and it is not visible here. This is aimed at researchers already working inside mathematical population dynamics who hit the same operator-definition problem on Banach lattices. A reader focused on size- or age-structured models would find the examples useful. It deserves peer review because the technical device addresses a real gap that shows up in existing models, even though the convergence claim requires scrutiny in the full proofs.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a method to compute the basic reproduction number R_0 for continuously structured population models on Banach lattices X with concentrated birth states, where the next-generation operator cannot be directly defined in X. The approach constructs a sequence of approximating models for which the next-generation operator T_n is well-defined, computes r(T_n) for each, and takes the limit as the R_0 of the original model. The method is illustrated on four examples: the classical size-dependent model, a size-structured cell population model, a size-structured model with diffusion (under additional assumptions), and a physiological age-structured model with diffusion.

Significance. If the convergence result and threshold property are rigorously established, the method would extend the applicability of next-generation operator techniques to a broader class of structured models with singular birth measures, which are common in size- and age-structured population dynamics. This could facilitate threshold analysis in models that are currently handled only via ad-hoc approximations or numerical simulation.

major comments (3)

[Abstract and §4 (examples)] The central claim (R_0 = lim r(T_n)) is load-bearing for all applications. The abstract and example sections state that the limit recovers the epidemiologically meaningful quantity, but no general theorem is supplied showing that the limit exists for arbitrary Banach lattices with concentrated birth states and that it coincides with the invasion threshold (stability of the extinction equilibrium) of the original model rather than an artifact of the approximation scheme.
[§3] §3 (main construction): the existence of a sequence of approximating spaces and operators T_n whose spectral radii converge to the correct R_0 is asserted but not accompanied by a proof that the approximation preserves the support of the birth measure in a manner that does not systematically bias the threshold (e.g., by smoothing or discretization that alters the concentrated birth states non-uniformly).
[§4.3 and §4.4] In the diffusion examples (size-structured model with diffusion and age-structured model with diffusion), the additional assumptions required to define the approximating sequence are not shown to be necessary or sufficient for the limit to equal the biologically correct R_0; a counter-example or explicit verification that the limit satisfies the threshold property is missing.

minor comments (2)

[§3 and §4] Notation for the approximating spaces and operators is introduced without a clear diagram or table summarizing the sequence of embeddings or projections used in each example.
[§2] The manuscript would benefit from an explicit statement of the functional-analytic setting (e.g., the precise Banach lattice norm and the definition of concentrated birth states) at the beginning of §2.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

Referee: [Abstract and §4 (examples)] The central claim (R_0 = lim r(T_n)) is load-bearing for all applications. The abstract and example sections state that the limit recovers the epidemiologically meaningful quantity, but no general theorem is supplied showing that the limit exists for arbitrary Banach lattices with concentrated birth states and that it coincides with the invasion threshold (stability of the extinction equilibrium) of the original model rather than an artifact of the approximation scheme.

Authors: The manuscript presents a constructive method for computing R_0 via limits of spectral radii in approximating models, with the central claim validated explicitly in the four examples of §4 rather than asserted for arbitrary Banach lattices. The abstract accurately reflects the scope of the work as an approach illustrated on concrete cases where the limit matches the expected threshold. A fully general existence and threshold theorem for arbitrary lattices lies beyond the paper's focus on practical computation in models with concentrated birth states; we can revise the abstract and introduction to emphasize the example-driven validation. revision: partial
Referee: [§3] §3 (main construction): the existence of a sequence of approximating spaces and operators T_n whose spectral radii converge to the correct R_0 is asserted but not accompanied by a proof that the approximation preserves the support of the birth measure in a manner that does not systematically bias the threshold (e.g., by smoothing or discretization that alters the concentrated birth states non-uniformly).

Authors: Section 3 constructs the approximating sequence by regularizing the birth measure while ensuring the supports converge in the appropriate weak sense to the original concentrated measure; the convergence of r(T_n) is then established via continuity properties of the spectral radius under the given lattice assumptions. The construction is designed to avoid systematic bias precisely by preserving the location of the birth states in the limit. We can expand the exposition in a revision to include an explicit lemma on support preservation. revision: partial
Referee: [§4.3 and §4.4] In the diffusion examples (size-structured model with diffusion and age-structured model with diffusion), the additional assumptions required to define the approximating sequence are not shown to be necessary or sufficient for the limit to equal the biologically correct R_0; a counter-example or explicit verification that the limit satisfies the threshold property is missing.

Authors: In §4.3 and §4.4 the additional assumptions (e.g., sufficient regularity for the diffusion operator) are used to construct the sequence explicitly, after which the limit is computed in closed form and shown to coincide with the R_0 obtained from the non-diffusive case or from direct linearization at the extinction equilibrium. These calculations constitute the verification of the threshold property under the stated assumptions. A counter-example demonstrating necessity is not provided because the examples are intended as positive illustrations rather than an exhaustive characterization; we do not claim the assumptions are necessary in general. revision: no

standing simulated objections not resolved

A general theorem establishing existence of the limit and equivalence to the invasion threshold for arbitrary Banach lattices with concentrated birth states

Circularity Check

0 steps flagged

No circularity: R0 defined biologically, computed via independent limit construction

full rationale

The paper opens by stating the standard definition of R0 as expected lifetime offspring and notes that the next-generation operator cannot be defined directly on the given Banach lattice X. It then constructs an approximating sequence of models on which the operator is definable and proves (or assumes under the stated conditions) that the spectral radii converge to a value that recovers the threshold for the original model. This is a standard approximation argument in functional analysis; the target quantity is not defined in terms of the limit, no parameter is fitted to data and then relabeled as a prediction, and no load-bearing step reduces to a self-citation or ansatz smuggled from prior work by the same authors. The derivation chain therefore remains self-contained against external spectral-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard properties of Banach lattices, positivity, and spectral radius for next-generation operators; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math The next-generation operator is well-defined and its spectral radius equals R0 in the approximating models.
Invoked to justify computing R0 via spectral radius in the sequence of models.
domain assumption The limit of these spectral radii exists and equals the epidemiologically relevant R0 for the original model.
Central to the claim but not justified in the abstract.

pith-pipeline@v0.9.0 · 5691 in / 1329 out tokens · 28537 ms · 2026-05-24T15:22:52.937646+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

R0 := lim ρ(Bk M^{-1}) = lim L M^{-1} ϕk = L ψ∞ = L G(·,x0)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

sequence of type I models ... concentrating at x0 ... limit model ... boundary condition (8) or (9)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Abdellaoui, T.M

B. Abdellaoui, T.M. Touaoula . Decay solution for the renewal equation with dif- fusion. Nonlinear Diﬀerential Equations and Applications, 17: 271–288, 2010

work page 2010
[2]

Ackleh, J

A.S. Ackleh, J. Z. Farkas. On the net reproduction rate of continuous structured populations with distributed states at birth . Computers and Mathematics with Applica- tions, 66: 1685–1694, 2013

work page 2013
[3]

Baca ¨er, S

N. Baca ¨er, S. Guernaoui. , The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco . Journal of Mathematical Biology, 53 (3): 421–436, 2006

work page 2006
[4]

Barril, `A

C. Barril, `A. Calsina, J. Ripoll. A practical approach to R0 in continuous-time ecological models. Mathematical Methods in the Applied Sciences, 41 (18): 8432–8445, 2017

work page 2017
[5]

Barril, `A

C. Barril, `A. Calsina, J. Ripoll. On the reproduction number of a gut microbiota model. Bulletin of Mathematical Biology, 79: 2727–2746, 2017

work page 2017
[6]

Basse, B.C

B. Basse, B.C. Baguley, E.S. Marshall, W.R. Joseph, B. Van Brunt, G. Wake, D.J.N. Wall. A mathematical model for analysis of the cell cycle in cell lines derived from human tumors . Journal of Mathematical Biology, 47 (4): 295–312, 2003. 15

work page 2003
[7]

J.W. Brewer. The age-dependent eigenfunctions of certain Kolmogorov equations of engineering, economics, and biology. Applied Mathematical Modeling, 13:47–57, 1989

work page 1989
[8]

Cole, J.V

K.D. Cole, J.V. Beck, A. Haji-Sheikh, B. Litkouhi. Heat conduction using Green’s functions. Second edition. Series in Computational and Physical Processes in Mechanics and Thermal Sciences. CRC Press, Boca Raton, FL, 2011

work page 2011
[9]

Ducrot, P

J.Chu, A. Ducrot, P. Magal, S Ruan. Hopf bifurcation in a size-structured popu- lation dynamic model with random growth . J. Diﬀerential Equations 247 (3): 956–1000, 2009

work page 2009
[10]

Cushing, O

J.M. Cushing, O. Diekmann. The many guises of R0 (a didactic note). Journal of Theoretical Biology, 404: 295–302, 2016

work page 2016
[11]

Diekmann, J

O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts . The construction of next generation matrices for compartmental epidemic models . Journal of the Royal Society Interface, 7 (47): 873–85, 2010

work page 2010
[12]

Diekmann, J.A.P

O. Diekmann, J.A.P. Heesterbeek, J.A.J.Metz. On the deﬁnition and the com- putation of the basic reproduction ratio R0 in models for infectious diseases in hetero- geneous populations. Journal of Mathematical Biology, 28 (4): 365–382, 1990

work page 1990
[13]

Engel, R

K-J. Engel, R. Nagel , A short course on operator semigroups , Springer, 2006

work page 2006
[14]

J. Z. Farkas, P. Hinow , Physiologically structured populations with diﬀusion and dynamic boundary conditions. Mathematical Biosciences and Engineering, 8 (2): 503– 513, (2011)

work page 2011
[15]

Feller , The parabolic diﬀerential equations and the associated semi-groups of transformations

W. Feller , The parabolic diﬀerential equations and the associated semi-groups of transformations. Annals of Mathematics (2), 55: 468–519, 1952

work page 1952
[16]

K. P. Hadeler , Structured populations with diﬀusion in state space. Mathematical Biosciences and Engineering, 7 (1): 37–49, (2010)

work page 2010
[17]

Iannelli, A

M. Iannelli, A. Pugliese. An introduction to mathematical population dynamics. Along the trail of Volterra and Lotka. Unitext, 79. La Matematica per il 3+2. Springer, Cham, 2014

work page 2014
[18]

Inaba Age-structured population dynamics in demography and epidemiology

H. Inaba Age-structured population dynamics in demography and epidemiology. Springer, Singapore, 2017

work page 2017
[19]

Inaba , On a new perspective of the basic reproduction number in heterogeneous environments

H. Inaba , On a new perspective of the basic reproduction number in heterogeneous environments. Journal of Mathematical Biology, 65 (2): 309–348, 2012

work page 2012
[20]

Kakumani, S.K

B.K. Kakumani, S.K. Tumuluri. On a nonlinear renewal equation with diﬀusion , Mathematical Methods in the Applied Sciences, 39 (4): 697–708, 2016

work page 2016
[21]

Y.H. Lee, L. Sherbakov, J. Taber, J. Shi. Bifurcation diagrams of population models with nonlinear diﬀusion . Journal of Computational and Applied Mathematics, 194: 357–367, 2006

work page 2006
[22]

Michel, T.M

P. Michel, T.M. Touaoula. Asymptotic behavior for a class of the renewal nonlinear equation with diﬀusion. Mathematical Methods in the Applied Sciences, 36(3):323–335, 2012. 16

work page 2012
[23]

Pugliese, F

A. Pugliese, F. Milner, Fabio , A structured population model with diﬀusion in structure space. Journal of Mathematical Biology, 77 (6-7): 2079–2102, (2018)

work page 2079
[24]

Schaefer, Banach lattices and positive operators , Springer, 1974

H.H. Schaefer, Banach lattices and positive operators , Springer, 1974

work page 1974
[25]

Stakgold, M

I. Stakgold, M. Holst. Green’s functions and boundary value problems. Third edition. Pure and Applied Mathematics (Hoboken). John Wiley & Sons, Inc., Hoboken, NJ, 2011

work page 2011
[26]

Thieme, Spectral bound and reproduction number for inﬁnite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (1): 188– 211, 2009

H. Thieme, Spectral bound and reproduction number for inﬁnite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (1): 188– 211, 2009

work page 2009
[27]

Thieme, Discrete-time population dynamics on the state space of measures , Math- ematical Biosciences and Engineering, 17 (2): 1168–1217, 2020

H. Thieme, Discrete-time population dynamics on the state space of measures , Math- ematical Biosciences and Engineering, 17 (2): 1168–1217, 2020

work page 2020
[28]

G.F. Webb. Theory of nonlinear age-dependent population dynamics. Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc., New York, 1985. 17

work page 1985

[1] [1]

Abdellaoui, T.M

B. Abdellaoui, T.M. Touaoula . Decay solution for the renewal equation with dif- fusion. Nonlinear Diﬀerential Equations and Applications, 17: 271–288, 2010

work page 2010

[2] [2]

Ackleh, J

A.S. Ackleh, J. Z. Farkas. On the net reproduction rate of continuous structured populations with distributed states at birth . Computers and Mathematics with Applica- tions, 66: 1685–1694, 2013

work page 2013

[3] [3]

Baca ¨er, S

N. Baca ¨er, S. Guernaoui. , The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco . Journal of Mathematical Biology, 53 (3): 421–436, 2006

work page 2006

[4] [4]

Barril, `A

C. Barril, `A. Calsina, J. Ripoll. A practical approach to R0 in continuous-time ecological models. Mathematical Methods in the Applied Sciences, 41 (18): 8432–8445, 2017

work page 2017

[5] [5]

Barril, `A

C. Barril, `A. Calsina, J. Ripoll. On the reproduction number of a gut microbiota model. Bulletin of Mathematical Biology, 79: 2727–2746, 2017

work page 2017

[6] [6]

Basse, B.C

B. Basse, B.C. Baguley, E.S. Marshall, W.R. Joseph, B. Van Brunt, G. Wake, D.J.N. Wall. A mathematical model for analysis of the cell cycle in cell lines derived from human tumors . Journal of Mathematical Biology, 47 (4): 295–312, 2003. 15

work page 2003

[7] [7]

J.W. Brewer. The age-dependent eigenfunctions of certain Kolmogorov equations of engineering, economics, and biology. Applied Mathematical Modeling, 13:47–57, 1989

work page 1989

[8] [8]

Cole, J.V

K.D. Cole, J.V. Beck, A. Haji-Sheikh, B. Litkouhi. Heat conduction using Green’s functions. Second edition. Series in Computational and Physical Processes in Mechanics and Thermal Sciences. CRC Press, Boca Raton, FL, 2011

work page 2011

[9] [9]

Ducrot, P

J.Chu, A. Ducrot, P. Magal, S Ruan. Hopf bifurcation in a size-structured popu- lation dynamic model with random growth . J. Diﬀerential Equations 247 (3): 956–1000, 2009

work page 2009

[10] [10]

Cushing, O

J.M. Cushing, O. Diekmann. The many guises of R0 (a didactic note). Journal of Theoretical Biology, 404: 295–302, 2016

work page 2016

[11] [11]

Diekmann, J

O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts . The construction of next generation matrices for compartmental epidemic models . Journal of the Royal Society Interface, 7 (47): 873–85, 2010

work page 2010

[12] [12]

Diekmann, J.A.P

O. Diekmann, J.A.P. Heesterbeek, J.A.J.Metz. On the deﬁnition and the com- putation of the basic reproduction ratio R0 in models for infectious diseases in hetero- geneous populations. Journal of Mathematical Biology, 28 (4): 365–382, 1990

work page 1990

[13] [13]

Engel, R

K-J. Engel, R. Nagel , A short course on operator semigroups , Springer, 2006

work page 2006

[14] [14]

J. Z. Farkas, P. Hinow , Physiologically structured populations with diﬀusion and dynamic boundary conditions. Mathematical Biosciences and Engineering, 8 (2): 503– 513, (2011)

work page 2011

[15] [15]

Feller , The parabolic diﬀerential equations and the associated semi-groups of transformations

W. Feller , The parabolic diﬀerential equations and the associated semi-groups of transformations. Annals of Mathematics (2), 55: 468–519, 1952

work page 1952

[16] [16]

K. P. Hadeler , Structured populations with diﬀusion in state space. Mathematical Biosciences and Engineering, 7 (1): 37–49, (2010)

work page 2010

[17] [17]

Iannelli, A

M. Iannelli, A. Pugliese. An introduction to mathematical population dynamics. Along the trail of Volterra and Lotka. Unitext, 79. La Matematica per il 3+2. Springer, Cham, 2014

work page 2014

[18] [18]

Inaba Age-structured population dynamics in demography and epidemiology

H. Inaba Age-structured population dynamics in demography and epidemiology. Springer, Singapore, 2017

work page 2017

[19] [19]

Inaba , On a new perspective of the basic reproduction number in heterogeneous environments

H. Inaba , On a new perspective of the basic reproduction number in heterogeneous environments. Journal of Mathematical Biology, 65 (2): 309–348, 2012

work page 2012

[20] [20]

Kakumani, S.K

B.K. Kakumani, S.K. Tumuluri. On a nonlinear renewal equation with diﬀusion , Mathematical Methods in the Applied Sciences, 39 (4): 697–708, 2016

work page 2016

[21] [21]

Y.H. Lee, L. Sherbakov, J. Taber, J. Shi. Bifurcation diagrams of population models with nonlinear diﬀusion . Journal of Computational and Applied Mathematics, 194: 357–367, 2006

work page 2006

[22] [22]

Michel, T.M

P. Michel, T.M. Touaoula. Asymptotic behavior for a class of the renewal nonlinear equation with diﬀusion. Mathematical Methods in the Applied Sciences, 36(3):323–335, 2012. 16

work page 2012

[23] [23]

Pugliese, F

A. Pugliese, F. Milner, Fabio , A structured population model with diﬀusion in structure space. Journal of Mathematical Biology, 77 (6-7): 2079–2102, (2018)

work page 2079

[24] [24]

Schaefer, Banach lattices and positive operators , Springer, 1974

H.H. Schaefer, Banach lattices and positive operators , Springer, 1974

work page 1974

[25] [25]

Stakgold, M

I. Stakgold, M. Holst. Green’s functions and boundary value problems. Third edition. Pure and Applied Mathematics (Hoboken). John Wiley & Sons, Inc., Hoboken, NJ, 2011

work page 2011

[26] [26]

Thieme, Spectral bound and reproduction number for inﬁnite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (1): 188– 211, 2009

H. Thieme, Spectral bound and reproduction number for inﬁnite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (1): 188– 211, 2009

work page 2009

[27] [27]

Thieme, Discrete-time population dynamics on the state space of measures , Math- ematical Biosciences and Engineering, 17 (2): 1168–1217, 2020

H. Thieme, Discrete-time population dynamics on the state space of measures , Math- ematical Biosciences and Engineering, 17 (2): 1168–1217, 2020

work page 2020

[28] [28]

G.F. Webb. Theory of nonlinear age-dependent population dynamics. Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc., New York, 1985. 17

work page 1985