Persistent unstable equilibria and the grace period in dynamic models of environmental change. (English) Zbl 1135.34327
Consider the three-dimensional autonomous system
\[ \begin{aligned} & {dx\over dt} =\varepsilon G(x,y,z),\\ & {dy\over dt}=\varepsilon H(y, z),\\ & {dz\over dt}= z(1- z) F(x,y,z),\end{aligned}\tag{\(*\)} \]
where \(\varepsilon\) is a small positive parameter. This multiscale system can be used to model the interaction of population \((x> 0)\), economy \((y> 0)\) and environment \((0\leq z\leq 1)\). Under the assumptions of the paper, system \((*)\) exhibits canard trajectories (representing the phenomenon of delayed exchange of stability). The authors call the delay as grace period. They derive expressions to approximate this delay as function of some parameters and perform a numerical study of a special system \((*)\).
\[ \begin{aligned} & {dx\over dt} =\varepsilon G(x,y,z),\\ & {dy\over dt}=\varepsilon H(y, z),\\ & {dz\over dt}= z(1- z) F(x,y,z),\end{aligned}\tag{\(*\)} \]
where \(\varepsilon\) is a small positive parameter. This multiscale system can be used to model the interaction of population \((x> 0)\), economy \((y> 0)\) and environment \((0\leq z\leq 1)\). Under the assumptions of the paper, system \((*)\) exhibits canard trajectories (representing the phenomenon of delayed exchange of stability). The authors call the delay as grace period. They derive expressions to approximate this delay as function of some parameters and perform a numerical study of a special system \((*)\).
Reviewer: Klaus R. Schneider (Berlin)
MSC:
| 34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 92D25 | Population dynamics (general) |
Software:
MathematicaReferences:
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