Abstract
We consider a class of singularly perturbed parabolic equations for which the degenerate equations obtained by setting the small parameter equal to zero are algebraic equations that have several roots. We study boundary layer type solutions that, as time increases, periodically go through two fairly long lasting stages with extremely fast transitions in between. During one of these stages the solution outside the boundary layer is close to one of the roots of the degenerate (reduced) equation, while during the other stage the solution is close to the other root. Such equations may be used as models for bio-switches where the transitions between various stationary states of biological systems are initiated by comparatively slow changes within the systems.
Highlights
Parabolic equations with several possible locally stable stationary solutions are often used for modeling of biological switches; see, for example, Okubo and Levin [5], Keener and Sneyd [2], Murray [3]
We introduce a new variable, u = φ3 + Π
The description of a moving front corresponding to transition, for example, from the upper to the lower boundary layer type solution (shown in Figure 3.6(c)) is similar to that presented for the case a = 0
Summary
Parabolic equations with several possible locally stable stationary solutions are often used for modeling of biological switches (i.e., biological systems where transitions may occur between various biologically meaningful stationary states); see, for example, Okubo and Levin [5], Keener and Sneyd [2], Murray [3]. Our main goals are (1) to show how such solutions appear in the cases where the non-linearity in the equation does not contain explicit dependence on spatial variable, x, and (2) to clarify the effect of the presence of a slow convectionlike term on the initiation of fast transition between the two longer lasting states of the periodic solution. During these stages the solution outside the boundary layer is close to different roots of the nonlinearity.
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