ode15s routine  (MathWorks Inc)

Journal: Journal of Mathematical Biology

Article Title: Deformations of acid-mediated invasive tumors in a model with Allee effect

doi: 10.1007/s00285-025-02209-w

Figure Lengend Snippet: Results of direct numerical simulations exhibiting different manifestations of the long wavelength instability. The simulations were performed in a co-moving frame corresponding to the wave speed of the initial front profile (so that in the absence of any instability, the fronts would appear stationary), with Neumann boundary conditions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} ξ , and periodic boundary conditions in y . Finite differences were used for spatial discretization, and MATLAB’s ode15s routine was used for time stepping. The corresponding v -profile is depicted, where yellow indicates high density of tumor cells, and blue depicts low density of tumor cells. (First row) Simulation for the parameter values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,\kappa , \delta _1, \delta _2, \delta _3, \rho , \varepsilon ) = (0.25,0.1,12.5,0.1,70.0,1.0, 0.0063)$$\end{document} ( a , κ , δ 1 , δ 2 , δ 3 , ρ , ε ) = ( 0.25 , 0.1 , 12.5 , 0.1 , 70.0 , 1.0 , 0.0063 ) at the times \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=5000,10000,15000,20000$$\end{document} t = 5000 , 10000 , 15000 , 20000 from left to right, with wave speed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.2211$$\end{document} c = 0.2211 . (Second row) Simulation for the parameter values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,\kappa , \delta _1, \delta _2, \delta _3, \rho , \varepsilon ) = (0.35,0.1,12.5,0.1,70.0,1.0, 0.0063)$$\end{document} ( a , κ , δ 1 , δ 2 , δ 3 , ρ , ε ) = ( 0.35 , 0.1 , 12.5 , 0.1 , 70.0 , 1.0 , 0.0063 ) at the times \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=10000,20000,30000,40000$$\end{document} t = 10000 , 20000 , 30000 , 40000 from left to right, with wave speed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.0401$$\end{document} c = 0.0401 . (Third row) Simulation for the parameter values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,\kappa , \delta _1, \delta _2, \delta _3, \rho , \varepsilon ) = (0.25,0.05,11.5,3,1,15, 0.05)$$\end{document} ( a , κ , δ 1 , δ 2 , δ 3 , ρ , ε ) = ( 0.25 , 0.05 , 11.5 , 3 , 1 , 15 , 0.05 ) at the times \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=250,350,450,550$$\end{document} t = 250 , 350 , 450 , 550 from left to right, with wave speed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.3296$$\end{document} c = 0.3296 . In each of the three simulations, initial conditions were constructed by trivially extending the 1D-stable profiles from Fig. (rows 2 through 4) in the y -direction and adding a small amount of positive noise

Article Snippet: The simulations were performed in a co-moving frame corresponding to the wave speed of the initial front profile (so that in the absence of any instability, the fronts would appear stationary), with Neumann boundary conditions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} ξ , and periodic boundary conditions in y . Finite differences were used for spatial discretization, and MATLAB’s ode15s routine was used for time stepping.

Techniques: Construct