Solving Differential Equations in R: Package deSolve

Research output: Contribution to journalResearch articleContributedpeer-review


  • Karline Soetaert - , Netherlands Inst Ecol NIOO, Royal Netherlands Academy of Arts & Sciences, Netherlands Institute of Ecology (NIOO-KNAW), Ctr Estuarine & Marine Ecol CEME (Author)
  • Thomas Petzoldt - , Chair of Limnology (Author)
  • R. Woodrow Setzer - , United States Environmental Protection Agency (Author)


In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines approach. The differential equations can be represented in R code or as compiled code. In the latter case, R is used as a tool to trigger the integration and post-process the results, which facilitates model development and application, whilst the compiled code significantly increases simulation speed. The methods implemented are efficient, robust, and well documented public-domain Fortran routines. They include four integrators from the ODEPACK package (LSODE, LSODES, LSODA, LSODAR), DVODE and DASPK2.0. In addition, a suite of Runge-Kutta integrators and special-purpose solvers to efficiently integrate 1-, 2- and 3-dimensional partial differential equations are available. The routines solve both stiff and non-stiff systems, and include many options, e. g., to deal in an efficient way with the sparsity of the Jacobian matrix, or finding the root of equations. In this article, our objectives are threefold: (1) to demonstrate the potential of using R for dynamic modeling, (2) to highlight typical uses of the different methods implemented and (3) to compare the performance of models specified in R code and in compiled code for a number of test cases. These comparisons demonstrate that, if the use of loops is avoided, R code can efficiently integrate problems comprising several thousands of state variables. Nevertheless, the same problem may be solved from 2 to more than 50 times faster by using compiled code compared to an implementation using only R code. Still, amongst the bene fits of R are a more flexible and interactive implementation, better readability of the code, and access to R's high-level procedures. deSolve is the successor of packageo desolve which will be deprecated in the future; it is free software and distributed under the GNU General Public License, as part of the R software project.


Original languageEnglish
Pages (from-to)1-25
Number of pages25
JournalJournal of statistical software
Issue number9
Publication statusPublished - Feb 2010

External IDs

WOS 000275204000001
Scopus 77953162975
ORCID /0000-0002-4951-6468/work/142256772



  • ordinary differential equations, partial differential equations, differential algebraic equations, initial value problems, R, Fortran, C, SYSTEMS