We analyze a nonlocal partial differential equation (PDE) model describing the dynamics of adaptation of a phenotypically structured population, under the effects of mutation and selection, in a changing environment. Previous studies have analyzed the large-time behavior of such models, with particular forms of environmental changes---either linearly changing or periodically fluctuating. We use here a completely different mathematical approach, which allows us to consider very general forms of environmental variations and to give an analytic description of the full trajectories of adaptation, including the transient phase, before a stationary behavior is reached. The main idea behind our approach is to study a bivariate distribution of two “fitness components" that contains enough information to describe the distribution of fitness at any time. This distribution solves a degenerate parabolic equation that is dealt with by defining a multidimensional cumulant generating function associated with the distribution and solving the associated transport equation. We apply our results to several examples and check their accuracy using stochastic individual-based simulations as a benchmark. These examples illustrate the importance of being able to describe the transient dynamics of adaptation to understand the development of drug resistance in pathogens.


  1. nonlocal
  2. partial differential equation
  3. adaptation
  4. moving optimum
  5. mutation
  6. generating functions

MSC codes

  1. 35Q92
  2. 35R09
  3. 45G10
  4. 45K05
  5. 45M05
  6. 92D10
  7. 92D15

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Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Adaptation in general temporally changing environments

Authors: L. Roques, F. Patout, O. Bonnefon, and G. Martin

File: S1.pdf

Type: PDF

Contents: Description of the algorithm that was used for the numerical computation of the integro-differential model (4).

File: S2.pdf

Type: PDF

Contents: Additional computations corresponding to small mutation rates.

File: S3.zip

Type: Compressed file

Contents: C++ source code corresponding to the numerical solver of (4).

File: S4.zip

Type: Compressed file

Contents: Matlab source code corresponding to the individual-based model.


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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 2420 - 2447
ISSN (online): 1095-712X


Submitted: 2 March 2020
Accepted: 3 September 2020
Published online: 19 November 2020


  1. nonlocal
  2. partial differential equation
  3. adaptation
  4. moving optimum
  5. mutation
  6. generating functions

MSC codes

  1. 35Q92
  2. 35R09
  3. 45G10
  4. 45K05
  5. 45M05
  6. 92D10
  7. 92D15



Funding Information

Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-18-CE45-0019

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