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Competition for resources: complicated dynamics in the simple Tilman model.

van Opheusden JH, Hemerik L, van Opheusden M, van der Werf W - Springerplus (2015)

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.

View Article: 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.


For stable coexistence of the consumers not only the supply of the resource must suffice, but also each consumer should consume resources in a ratio that favours the competitor. If such is not the case, the coexistence point is a saddle. With the supply point inside the wedge there are two stable states, each with only one consumer present. Which consumer survives depends entirely on the initial state. In a, b after moving to the trivial state and the coexistence point only the B survives. For a different initial state the system develops to a situation in which A eventually prevails. The trajectories as shown seem to coincide partially. This occurs because only the phase plot projected onto the PR-plane is given. In the full phase space, including the state variables A and B, the basins of the two stable equilibria do not overlap
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Fig7: For stable coexistence of the consumers not only the supply of the resource must suffice, but also each consumer should consume resources in a ratio that favours the competitor. If such is not the case, the coexistence point is a saddle. With the supply point inside the wedge there are two stable states, each with only one consumer present. Which consumer survives depends entirely on the initial state. In a, b after moving to the trivial state and the coexistence point only the B survives. For a different initial state the system develops to a situation in which A eventually prevails. The trajectories as shown seem to coincide partially. This occurs because only the phase plot projected onto the PR-plane is given. In the full phase space, including the state variables A and B, the basins of the two stable equilibria do not overlap

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


Competition for resources: complicated dynamics in the simple Tilman model.

van Opheusden JH, Hemerik L, van Opheusden M, van der Werf W - Springerplus (2015)

For stable coexistence of the consumers not only the supply of the resource must suffice, but also each consumer should consume resources in a ratio that favours the competitor. If such is not the case, the coexistence point is a saddle. With the supply point inside the wedge there are two stable states, each with only one consumer present. Which consumer survives depends entirely on the initial state. In a, b after moving to the trivial state and the coexistence point only the B survives. For a different initial state the system develops to a situation in which A eventually prevails. The trajectories as shown seem to coincide partially. This occurs because only the phase plot projected onto the PR-plane is given. In the full phase space, including the state variables A and B, the basins of the two stable equilibria do not overlap
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4559563&req=5

Fig7: For stable coexistence of the consumers not only the supply of the resource must suffice, but also each consumer should consume resources in a ratio that favours the competitor. If such is not the case, the coexistence point is a saddle. With the supply point inside the wedge there are two stable states, each with only one consumer present. Which consumer survives depends entirely on the initial state. In a, b after moving to the trivial state and the coexistence point only the B survives. For a different initial state the system develops to a situation in which A eventually prevails. The trajectories as shown seem to coincide partially. This occurs because only the phase plot projected onto the PR-plane is given. In the full phase space, including the state variables A and B, the basins of the two stable equilibria do not overlap
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

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.

View Article: 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.