Competition for resources: complicated dynamics in the simple Tilman model.
Bottom Line:
We argue that the relative stability of saddle points is more important for the actual observed transient dynamics in realistic systems than the predicted asymptotic behaviour towards the stable equilibria.For the mathematical analysis this implies that not only the signs, but also the magnitudes of the eigenvalues of the Jacobi matrix at the stationary points, the rates at which the system evolves, must be considered.We present the underlying mathematics of the Tilman model in a way that should be accessible to any ecologist with a basic mathematical background.
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PubMed Central - PubMed
Affiliation: Biometris, Wageningen University, Droevendaalsesteeg 1, 6708PB Wageningen, The Netherlands.
ABSTRACT
Graphical analysis and computer simulations have become the preferred tools to present Tilman's model of resource competition to new generations of ecologists. To really understand the full dynamic behaviour, a more rigorous mathematical analysis is required. We show that just a basic stability analysis is insufficient to describe the relevant dynamics of this deceptively simple model. To investigate realistic invasion and succession processes, not only the stable state is relevant, but also the time scales at which the system moves away from the unstable situation. We argue that the relative stability of saddle points is more important for the actual observed transient dynamics in realistic systems than the predicted asymptotic behaviour towards the stable equilibria. For the mathematical analysis this implies that not only the signs, but also the magnitudes of the eigenvalues of the Jacobi matrix at the stationary points, the rates at which the system evolves, must be considered. We present the underlying mathematics of the Tilman model in a way that should be accessible to any ecologist with a basic mathematical background. No MeSH data available. |
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Mentions: A quite different situation is found when we take qPA = 0.8, qRA = 1, qPB = 1, and qRB = 0.8 (Fig. 7). For sP = 1, sR = 1 the coexistence point and the trivial equilibrium are saddle points, the other two equilibria are stable nodes. All stationary densities are positive, so all stationary points are biologically relevant. For initial values P(0) = 0, R(0) = 0.5, A(0) = 0.01, B(0) = 0.05 the systems travels via the trivial and coexistence point to the stable B-point (Fig. 7a, b). If we start in P(0) = 0.8, R(0) = 0, A(0) = 0.01, B(0) = 0.03, a similar detour brings us to the stable A-point (Fig. 7c, d). The two stable points each have their own basin of attraction, the choice of the starting point completely determines where the system will end up. Note that it looks as if the two trajectories in Fig. 7b, d intersect. In fact they are fully separated, any apparent intersection occurs because the projection of the orbit upon the PR-plane is plotted.Fig. 7 |
View Article: PubMed Central - PubMed
Affiliation: Biometris, Wageningen University, Droevendaalsesteeg 1, 6708PB Wageningen, The Netherlands.
No MeSH data available.