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Adaptation in shifting and size-changing environments under selection

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Pith's one-line read The paper proves that a population diffusing in a shifting, size-changing habitat under periodic quadratic selection survives or dies according to the fixed-domain principal eigenvalue, with a critical linear shift speed $c^* =…

desk verdict Genuinely new combination of periodic selection and moving-domain effects, with a clean critical-speed criterion; the main theorem's initial-data condition is narrower than the corollaries imply and should be repaired before publication. read the letter →

arxiv 2506.03666 v1 pith:UMTCK36O submitted 2025-06-04 math.AP cs.NAmath.NA

Matthieu Alfaro (LMRS) , Adel Blouza (LMRS) , Nessim Dhaouadi (LMRS) This is my paper
classification math.APcs.NAmath.NA MSC 35K5735P1565M06
keywords reaction-diffusionmovinghabitatshiftingdomainperiodicselectionprincipaleigenvalueextinctionvspersistencespace-timefiniteelementphenotypeadaptation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a reaction-diffusion model in which a population lives in an interval that shifts and changes length while a periodic quadratic selection pressure pulls the optimum position around. The main result is that the solution is bounded for all times by explicit expressions built from the periodic principal eigenfunction of the same equation on a fixed interval. These bounds yield a survival/extinction classification: a superlinear shift guarantees extinction at an explicit rate, a sublinear shift leaves the fixed-domain behavior intact, and a linear shift has a critical speed $c^* = 2\sqrt{-\lambda d}$ below which the population persists and above which it dies out. The paper also proves eigenvalue estimates for the fixed-domain problem, showing that periodic fluctuations can help or hurt survival depending on the phase between selection intensity and optimum position, and it constructs a well-posed space-time finite element scheme that reproduces these regimes numerically.

What carries the argument

The carrying object is the periodic parabolic principal eigenfunction $\varphi(t,y)$, the positive $T$-periodic solution of the fixed-domain eigenproblem (2.2), together with its eigenvalue $\lambda$. The argument transforms the moving-interval problem into a fixed-domain equation by the change of variables (3.1)-(3.3), replaces the awkward time-dependent coefficients by their envelope bounds $\underline Q$, $\overline Q$, $\underline P$, $\overline P$, and then uses $\varphi$ evaluated at the rescaled time $\int_0^t L_0^2/L^2(s)\,ds$ as the spatial profile of explicit sub- and supersolutions. All long-time conclusions reduce to reading the sign of $\lambda$ and the polynomial growth of the exponential factors.

What would settle it

Run the moving-boundary problem with a linear shift at $c<c^*$ but with an initial datum that violates (3.16), for example a function that is positive at both moving boundaries, and see whether the solution still grows locally for large time; if it does not, the assertion that Corollary 3.6 covers all admissible nonnegative data would be falsified. The direct confirmation would be to compute $\lambda$ from the fixed-domain eigenproblem and check that survival for $c<c^*$ and extinction for $c>c^*$ occur exactly as predicted.

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Extended reading notes

Core claim

The central claim is that the long-time fate of the solution to (1.1) is controlled by the principal eigenvalue $\lambda$ of the fixed-domain periodic problem (2.2). After a change of variables that fixes the moving interval and removes the advection term, the transformed solution is sandwiched between two explicit functions built from the fixed-domain principal eigenfunction $\varphi$, multiplied by exponentials that encode the shift, the size change, and the periodic selection. From this sandwich the paper derives Corollaries 3.3, 3.4 and 3.6: superlinear shifts kill the population regardless of parameters, sublinear shifts preserve the fixed-domain persistence or extinction behavior, and for linear shifts the threshold is $c^* = 2\sqrt{-\lambda d}$. The eigenvalue analysis also shows that in a fixed domain with the optimum outside the interval, an arbitrarily large habitat can cause extinction, contrary to the homogeneous case, and that periodic fluctuations of selection intensity and optimum position help survival when they are in phase opposition and hurt it when they are in phase.

Load-bearing premise

The load-bearing premise is the initial-condition sandwich (3.16), which requires the transformed initial profile to lie between positive multiples of the fixed-domain eigenfunction $\varphi(0,\cdot)$; arbitrary nonnegative data, especially data that do not vanish at the moving boundary, need not satisfy it.

Editorial extensions

If this is right

  • In a fixed or sublinearly shifting habitat, survival in the fixed-domain sense ($\lambda<0$) implies the population locally grows without bound, and the critical case $\lambda=0$ still allows survival when the shift grows slower than $\sqrt{t}$.
  • A superlinear shift $A(t)=c(1+t)^a$, $a>1$, forces uniform extinction with the explicit decay rate $C_1 e^{-C_2 t^{2a-1}}$ no matter how favorable growth and selection are.
  • For a linear shift $A(t)=c(1+t)$, the threshold is $c^*=2\sqrt{-\lambda d}$: survival for $c<c^*$, survival without guaranteed growth at $c=c^*$, and extinction for $c>c^*$.
  • In a fixed domain, the principal eigenvalue lies between explicit bounds, so a sufficiently small habitat always causes extinction while a very large habitat can also cause extinction when the optimum lies outside the interval.
  • The space-time finite element scheme is well-posed and reproduces the predicted survival/extinction regimes, including the transient rebound of the population as the periodic optimum passes nearby.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same eigenfunction-sandwich construction should extend to rectangular or smoothly deforming higher-dimensional domains with Dirichlet boundaries, with the envelope coefficients adjusted; the extinction/persistence thresholds would then still be governed by the fixed-domain principal eigenvalue.
  • One could test the critical-speed prediction directly by measuring $\lambda$ from a fixed-domain simulation at each set of parameters and comparing the observed fate under a linear shift with the sign of $c-2\sqrt{-\lambda d}$.
  • The restriction (3.16) suggests that initial profiles that do not vanish at the moving boundary may lose a transient layer at the boundary, so their early-time behavior could deviate from the bounds even when the long-time fate matches the corollaries.
  • The paper leaves open the conjecture of a threshold $r^*$ separating systematic extinction from an extinction-survival-extinction pattern in $L$; a numerical scan over $r$ and $L$ would settle where that threshold lies.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper analyzes a one-dimensional reaction-diffusion equation with Dirichlet conditions on an interval whose endpoints move in time and whose length may change, subject to a time-periodic quadratic selection centered at a moving optimum. In the fixed-domain case it derives upper and lower bounds for the periodic principal eigenvalue (Theorem 2.1) and uses them for extinction/survival criteria (Corollaries 2.7 and 2.8). In the moving-domain case it changes variables to a fixed interval, constructs explicit sub- and supersolutions built from the fixed-domain principal eigenfunction, and obtains two-sided bounds for all times (Theorem 3.1). These bounds are applied to constant-length domains with power-like shifts: superlinear shifts force extinction (Corollary 3.3), sublinear shifts preserve the fixed-domain persistence behavior (Corollaries 3.4 and 3.5), and linear shifts have a critical speed c* = 2(-λd)^{1/2} (Corollary 3.6). The paper also develops a space-time finite element scheme for the moving-domain problem, proves well-posedness of the discrete problem, validates it against an exact solution, and presents simulations of the extinction/persistence regimes.

Significance. The analytic core is coherent: the eigenvalue estimates are obtained from monotonicity of the principal eigenvalue and from a self-contained Rayleigh-quotient computation, with no fitted parameters, and the corollaries are derived by explicit exponential estimates. The comparison bounds in Theorem 3.1 are of independent interest, and the critical-speed formula for linear shifts is a sharp, falsifiable prediction. The numerical section is a genuine contribution: the scheme is described in detail, its discrete well-posedness is proved, and the simulations confirm the qualitative predictions of Section 3. The main limitation is the initial-data comparability condition in Theorem 3.1, which is narrower than the statement suggests and needs a repair before the results cover the intended class of nonnegative L∞ data.

major comments (2)
  1. [Theorem 3.1 and Corollaries 3.3–3.6] Condition (3.16) is not automatic for the initial data advertised in Theorem 3.1. Since the principal eigenfunction φ(0,y) vanishes at y=0 and y=L0 with nonzero normal derivatives (Theorem A.1 and (A.1)), the inequalities aφ(0,y) ≤ w0(y) ≤ bφ(0,y) force w0 to vanish at the boundary and to be comparable to φ(0,·). For a typical nonnegative nontrivial u0∈L∞ that is positive up to the boundary, no such a and b exist, and the bounds (3.17)-(3.18), as well as Corollaries 3.3-3.6, do not apply. Because the solution becomes smooth, positive in the interior, and has nonzero boundary normal derivative at any t0>0, a restart argument should repair the gap, but it is not present in the manuscript. Please either add such an argument or state explicitly that all results are conditional on (3.16).
  2. [Section 3.2, Eqs. (3.9)–(3.15)] The supersolution/subsolution verification is not actually shown: the text says 'straightforward computations' lead to the choices (3.11)-(3.14), but the sign condition that P(s) and P(s) make (3.9) a supersolution of (3.7) and (3.15) a subsolution of (3.8) is the core of Theorem 3.1 and should be displayed. I checked the computation and it is correct, so this is a missing-derivation point rather than an error, but it should be written out for the reader.
minor comments (5)
  1. [Example 4.12] The phrase 'wit T = 15' should be 'with T = 15'.
  2. [References] Reference [35] appears misattributed: 'The Boundary Value Problems of Mathematical Physics' is by O.A. Ladyzhenskaya, and 'Lohwater' is the translator's name and should not be listed as a co-author.
  3. [Eqs. (3.12)–(3.14)] The dummy variable T inside the integrals is the same symbol as the period T; using another letter such as s′ would avoid confusion.
  4. [Corollary 3.4] The conclusion 'locally uniformly tends to infinity' follows from λ<0, but the word 'survival' is used for unbounded growth; a sentence clarifying that persistence means non-extinction would help the reader.
  5. [Theorem 3.1] The phrase 'or equivalently' links u, v, and w through a one-to-one change of unknown, but because L0 is a free parameter the choice of L0 affects the admissibility condition (3.16), which should be stated explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central bounds and survival/extinction criteria are derived from external periodic-eigenvalue theory and self-contained comparison estimates, with only a minor non-load-bearing self-citation.

full rationale

The central derivation chain is self-contained. Theorem 3.1 obtains two-sided bounds on v(t,y) by transforming the moving-domain problem to (3.4), constructing sub- and supersolutions of the comparison problems (3.7)-(3.8) with the explicit ansatz (3.9)/(3.15), and verifying the required inequalities pointwise. The quantities P, P, Q, Q defined in (3.12), (3.14), (3.5), (3.6) are explicit deterministic envelopes of the coefficients, not fitted parameters, and no prediction is obtained by renaming an input. The principal eigenvalue lambda and eigenfunction phi in the bounds come from the classical periodic parabolic problem (2.2), whose existence and uniqueness are quoted from Castro-Lazer [15, Theorem A.1]; the eigenvalue estimates in Theorem 2.1 use the exact eigenpair of Lemma 2.2, the comparison lemma quoted from Hess [24], and the averaging comparison from Hutson-Shen-Vickers [28]. These are external, independently established results, not the paper's own target statement. Corollaries 3.3-3.6 merely insert the explicit shift A(t)=c(1+t)^a into the bounds (3.17)-(3.18) and read off the dominant exponential/power terms; the critical speed c*=2*sqrt(-lambda d) is an explicit consequence of that asymptotic balance, not a fitted threshold. The only self-citation is [5], used for a technical concentration argument in the proof of Lemma 2.5(i), and this plays no role in the main comparison theorem or corollaries. The comparability hypothesis (3.16) is an explicit stated assumption, not a hidden equivalent of the conclusion, and the paper does not claim Theorem 3.1 for data failing it. The numerical section is validated against an exact solution and is not used to establish the analytical results. Overall, no prediction in the paper reduces by construction to its own inputs.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The analytical content is built on standard periodic-parabolic eigenvalue theory and comparison principles. There are no fitted parameters; the only numbers chosen by hand are the numerical stabilization parameters theta and delta and the arbitrary reference length L0. The most restrictive extra input is the eigenfunction-shaped initial data condition (3.16).

free parameters (3)
  • theta = 10^-5
    Stabilization parameter in the space-time FEM (Sections 4.2-4.3). Chosen by hand; the theorem requires theta small enough for coercivity (Lemma 4.7).
  • delta = 10^-6
    Penalty parameter for the jump of the time derivative across element interfaces in the FEM scheme (Sections 4.2-4.3). Chosen by hand.
  • L0
    Reference interval length used for the change of variables (3.1). Any positive value works; the bounds in Theorem 3.1 are independent of L0. Not fitted.
assumptions (8)
  • standard math Existence, uniqueness, positivity and boundary behavior of the T-periodic principal eigenpair for uniformly parabolic operators with Dirichlet condition on C^{2+nu} domains
    Quoted as Theorem A.1 from Castro-Lazer [15] and used throughout Section 2 to define and estimate lambda and phi.
  • standard math Strict monotonicity of the principal eigenvalue with respect to the reaction coefficient (if R1 <= R2, R1 not identical to R2, then lambda2 < lambda1)
    Used in Lemma 2.3 to derive the first eigenvalue bounds (2.6); cited from Hess [24, Lemma 15.5].
  • standard math For a time-periodic parabolic operator, the principal eigenvalue is strictly smaller than that of the time-averaged elliptic operator (lambda < lambda-hat)
    Used in Section 2.1 to improve the upper eigenvalue bound (2.13); cited from Hutson-Shen-Vickers [28, Theorem 2.1].
  • standard math Parabolic comparison principle on space-time domains with moving boundaries
    Underpins the sub- and supersolution argument in Theorem 3.1 (Section 3.2); used tacitly to transfer bounds from w to v to u.
  • standard math Variational (Rayleigh quotient) characterization of the principal eigenvalue of a self-adjoint elliptic operator
    Used in Lemma 2.5 to compute estimates on mu-hat and to prove Theorem 2.1.
  • standard math Trace and lifting theorems for Sobolev spaces on Lipschitz space-time domains (Geymonat-Krasucki [20], Grisvard [21])
    Used in Section 4.2 to justify the H^2 lifting of boundary data for the space-time FEM.
  • domain assumption Regularity assumptions A in C^2([0,infinity)), L in C^2([0,infinity); (0,infinity)), alpha, beta Holder continuous and T-periodic with alpha > 0
    Stated in Section 1 as the model class; all theorems are formulated under these assumptions.
  • ad hoc to paper Initial data comparison condition (3.16): a phi(0,y) <= w0(y) <= b phi(0,y) for some 0 < a < b
    This technical condition is introduced to run the comparison argument in Theorem 3.1. It is not guaranteed for arbitrary nonnegative L-infinity initial data and it restricts the applicability of the survival and extinction corollaries.

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Pith. "Pith review of Adaptation in shifting and size-changing environments under selection." pith.science (2026). https://pith.science/paper/UMTCK36O

@misc{pith2026250603666,
  author       = {Pith},
  title        = {Pith review of: Adaptation in shifting and size-changing environments under selection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMTCK36O}},
  note         = {Machine review of arXiv:2506.03666}
}
read the original abstract

We propose a model to characterize how a diffusing population adapts under a time periodic selection, while its environment undergoes shifts and size changes, leading to significant differences with classical results on fixed domains. After studying the underlying periodic parabolic principal eigenelements, we address the extinction vs. persistence issue, taking into account the interplay between the moving habitat and periodic selection. Subsequently, we employ a space-time finite element approach, establish the well-posedness of the approximation scheme, and conduct numerical simulations to explore these dynamics.

Figures

Figures reproduced from arXiv: 2506.03666 by the authors.

Figure 1
Figure 1. Sign of the eigenvalue as a function of (L, r). The parameters are as described in Example 2.9. In green, the survival zone (λ < 0). In red, the extinction zone (λ > 0). In white, the zone where our estimates are not enough to conclude. principal eigenvalue λ is not monotone with respect to L. Indeed, as already emphasized above, enlarging the domain may decrease the chances of survival. Also, considering for instan… view at source ↗
Figure 2
Figure 2. Evolution of the L 2 norm of the solution at final time T = 2 as a function of the length L of the domain. The parameters and the initial datum are as described in Example 2.9. 2.3 To fluctuate, or not to fluctuate Here we aim at comparing the chances of survival between a constant fitness and a fluctuating fitness, see [18, 19] and the references therein for related issues. First, we look for a situation where fluc… view at source ↗
Figure 3
Figure 3. Evolution of the consistency error (||u − uh||h) with respect to the mesh size h. The parameters are as set as the following, r = 6, L = 1.5, θ = 10−5 and δ = 10−6 . We now present several numerical tests with parameters θ = 10−5 , δ = 10−6 , and initial data u0(x) = sin( πx L ), to highlight some underlying properties of the model. To begin, we simulate the evolution of the solution to (2.1) to gain new insights in… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Evolution of the solution with a time-periodic optimum in a seasonal model. The parameters are as described in Example 4.8. In solid black lines, the evolution of the position of the optimum xopt(t) = β(t)L. Example 4.8 (Seasonal optimum’s position). Here we consider t…
Figure 5
Figure 5. Figure 5: Effect of a sub-linear domain shift and of the position of the optimum. The parameters are as described in Example 4.9. In solid black lines, the evolution of the position of the optimum xopt(t) = A(t) + β(t)L. at large times). This contrasts with the above sub-linear …
Figure 6
Figure 6. Figure 6: Population extinction due to a super-linear domain shift. The parameters are as described in Example 4.10. In solid black lines, the evolution of the position of the optimum xopt(t) = A(t) + β(t)L. Example 4.11 (Linear shift). We consider a domain that shifts linearly …
Figure 7
Figure 7. Figure 7: Effect of a linear domain shift and of the position of the optimum. The parameters are as described in Example 4.11. In solid black lines, the evolution of the position of the optimum xopt(t) = A(t) + β(t)L. Example 4.12 (Linear enlargement). We consider a domain that …
Figure 8
Figure 8. Figure 8: Effect of the speed of enlargement. The parameters are as described in Example 4.12. In solid black lines, the evolution of the position of the optimum xopt(t) = β(t)L(t). A Appendix: Periodic parabolic eigenproblems The following result on periodic parabolic eigenelem…

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Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    J. T. Ainsworth, M.and Oden , A posteriori error estimation in finite element anal- ysis, Wiley-Interscience, 2000

    work page 2000

  2. [2]

    Alf aro, H

    M. Alf aro, H. Berestycki, and G. Raoul , The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition, SIAM J. Math. Anal., 49 (2017), pp. 562–596

    work page 2017

  3. [3]

    Alf aro and R

    M. Alf aro and R. Carles , Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145 (2017), pp. 5315–5327

    work page 2017

  4. [4]

    Alf aro, F

    M. Alf aro, F. Hamel, F. Patout, and L. Roques , Adaptation in a heterogeneous environment II: to be three or not to be, J. Math. Biol., 87 (2023), pp. Paper No. 68, 53

    work page 2023

  5. [5]

    Alf aro and M

    M. Alf aro and M. Veruete, Evolutionary branching via replicator–mutator equations, Journal of Dynamics and Differential Equations, 31 (2019), pp. 2029–2052. 28

    work page 2019

  6. [6]

    J. All wright, Exact solutions and critical behaviour for a linear growth-diffusion equa- tion on a time-dependent domain, Proceedings of the Edinburgh Mathematical Society, 65 (2022), pp. 53–79

    work page 2022

  7. [7]

    , Reaction-diffusion problems on time-periodic domains , arXiv preprint arXiv:2210.11516, (2022)

    work page Pith review arXiv 2022

  8. [8]

    , Reaction-diffusion on a time-dependent interval: refining the notion of ‘critical length’, Commun. Contemp. Math., 25 (2023), pp. Paper No. 2250050, 11

    work page 2023

Show all 42 references
  1. [9]

    Berestycki, O

    H. Berestycki, O. Diekmann, C. J. Nagelkerke, and P. A. Zegeling , Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), pp. 399–429

    2009

  2. [10]

    Bouhours and T

    J. Bouhours and T. Giletti , Spreading and vanishing for a monostable reaction- diffusion equation with forced speed, J. Dynam. Differential Equations, 31 (2019), pp. 247– 286

    2019

  3. [11]

    Bouhours and G

    J. Bouhours and G. Nadin , A variational approach to reaction-diffusion equations with forced speed in dimension 1, Discrete Contin. Dyn. Syst., 35 (2015), pp. 1843–1872

    2015

  4. [12]

    A. N. Brooks and T. J. R. Hughes , Streamline upwind/Petrov-Galerkin formu- lations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), pp. 199–259. FENOMECH ”81, Part I (Stuttgart, 1981)

    1982

  5. [13]

    Bürger, The mathematical theory of selection, recombination, and mutation, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000

    R. Bürger, The mathematical theory of selection, recombination, and mutation, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000

    2000

  6. [14]

    R. S. Cantrell and C. Cosner , Practical persistence in ecological models via com- parison methods, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), pp. 247–272

    1996

  7. [15]

    Castro and A

    A. Castro and A. C. Lazer , Results on periodic solutions of parabolic equations sug- gested by elliptic theory, Boll. Unione Mat. Ital., VI. Ser., B, 1 (1982), pp. 1089–1104

    1982

  8. [16]

    Ern and J.-L

    A. Ern and J.-L. Guermond , Theory and Practice of Finite Elements, Springer, 2004

    2004

  9. [17]

    J. A. Ev ans and T. J. R. Hughes , Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements, Numerische Mathematik, 123 (2013), pp. 259– 290

    2013

  10. [18]

    Figueroa Iglesias and S

    S. Figueroa Iglesias and S. Mirrahimi , Long time evolutionary dynamics of pheno- typically structured populations in time-periodic environments, SIAM J. Math. Anal., 50 (2018), pp. 5537–5568

    2018

  11. [19]

    , Selection and mutation in a shifting and fluctuating environment, Commun. Math. Sci., 19 (2021), pp. 1761–1798

    2021

  12. [20]

    Geymonat and F

    G. Geymonat and F. c. Krasucki , On the existence of the Airy function in Lipschitz domains. Application to the traces ofH 2, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), pp. 355–360. 29

    2000

  13. [21]

    Grisv ard, Elliptic problems in nonsmooth domains, Pitman advanced Publishing Program, 1985

    P. Grisv ard, Elliptic problems in nonsmooth domains, Pitman advanced Publishing Program, 1985

    1985

  14. [22]

    Hamel, F

    F. Hamel, F. La vigne, G. Martin, and L. Roques , Dynamics of adaptation in an anisotropic phenotype-fitness landscape, Nonlinear Anal. Real World Appl., 54 (2020), pp. 103107, 33

    2020

  15. [23]

    Hamel, F

    F. Hamel, F. La vigne, and L. Roques , Adaptation in a heterogeneous environment I: persistence versus extinction, J. Math. Biol., 83 (2021), pp. Paper No. 14, 42

    2021

  16. [24]

    Hess, Periodic-parabolic boundary value problems and positivity, vol

    P. Hess, Periodic-parabolic boundary value problems and positivity, vol. 247 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copub- lished in the United States with John Wiley & Sons, Inc., New York, 1991

    1991

  17. [25]

    Hughes , The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, 1987

    T. Hughes , The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, 1987

    1987

  18. [26]

    Kelly’s condition

    T. J. R. Hughes, L. P. Franca, and M. Balestra , A new finite element formulation for computational fluid dynamics: V. circumventing the "Kelly’s condition" for stable and accurate solutions, Computer Methods in Applied Mechanics and Engineering, 58(1), 1- 23., (1986)

    1986

  19. [27]

    Hutson, J

    V. Hutson, J. López-Gómez, K. Mischaikow, and G. Vickers , Limit behaviour for a competing species problem with diffusion, in Dynamical systems and applications, vol. 4 of World Sci. Ser. Appl. Anal., World Sci. Publ., River Edge, NJ, 1995, pp. 343–358

    1995

  20. [28]

    Hutson, W

    V. Hutson, W. Shen, and G. T. Vickers , Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2001), pp. 1669–1679

    2001

  21. [29]

    Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987

    C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987

    1987

  22. [30]

    Kimura , A stochastic model concerning the maintenance of genetic variability in quantitative characters., Proc

    M. Kimura , A stochastic model concerning the maintenance of genetic variability in quantitative characters., Proc. Natl. Acad. Sci. USA, 54 (1965), pp. 731–736

    1965

  23. [31]

    Lande, The maintenance of genetic variability by mutation in a polygenic character with linked loci, Genetics Research, 26 (1975), pp

    R. Lande, The maintenance of genetic variability by mutation in a polygenic character with linked loci, Genetics Research, 26 (1975), pp. 221–235

    1975

  24. [32]

    La vigne, Adaptation of an asexual population with environmental changes, Math

    F. La vigne, Adaptation of an asexual population with environmental changes, Math. Model. Nat. Phenom., 18 (2023), pp. Paper No. 20, 21

    2023

  25. [33]

    S. Liu, Y. Lou, R. Peng, and M. Zhou , Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator, Proc. Amer. Math. Soc., 147 (2019), pp. 5291– 5302

    2019

  26. [34]

    , Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator II: Small diffusion, Trans. Amer. Math. Soc., 374 (2021), pp. 4895–4930

    2021

  27. [35]

    Lohw ater and O

    J. Lohw ater and O. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences, Springer New York, 2013. 30

    2013

  28. [36]

    Martin and T

    G. Martin and T. Lenormand , The fitness effect of mutations across environments: Fisher’s geometrical model with multiple optima, Evolution, 69 (2015), pp. 1433–1447

    2015

  29. [37]

    S. E. Moore, A stable space–time finite element method for parabolic evolution problems, Calcolo, 55 (2018), p. 18

    2018

  30. [38]

    Nadin , The principal eigenvalue of a space-time periodic parabolic operator, Ann

    G. Nadin , The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl. (4), 188 (2009), pp. 269–295

    2009

  31. [39]

    A. B. Potapov and M. A. Lewis , Climate and competition: the effect of moving range boundaries on habitat invasibility, Bull. Math. Biol., 66 (2004), pp. 975–1008

    2004

  32. [40]

    Roques, F

    L. Roques, F. Patout, O. Bonnefon, and G. Martin , Adaptation in general temporally changing environments, SIAM Journal on Applied Mathematics, 80 (2020), pp. 2420–2447

    2020

  33. [41]

    Roques, A

    L. Roques, A. Roques, H. Berestycki, and A. Kretzschmar , A population facing climate change: joint influences of allee effects and environmental boundary geometry, Population Ecology, 50 (2008), pp. 215–225

    2008

  34. [42]

    Tenaillon, The utility of Fisher’s geometric model in evolutionary genetics, Annual Review of Ecology, Evolution, and Systematics, 45 (2014), pp

    O. Tenaillon, The utility of Fisher’s geometric model in evolutionary genetics, Annual Review of Ecology, Evolution, and Systematics, 45 (2014), pp. 179–201. 31

    2014

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