Limits...
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.


Two consumers and a single resource. Parameters, given in the text, are the same for both plots, the only difference is the supply level of the resource. Also the starting point is the same, no food, much of B and a little bit of A. In both cases A successfully takes over from B. If the stable resource level is below the critical level for maintenance of B (a), this consumer simply disappears, and at some later time A grows to its stationary level. If the supply of resource is sufficient to support B (b), there is an interval where a finite population B survives on the available resource. The decline of the species B is in fact brought about by its competitor A eating away the required food. Eventually a higher population of A is reached because of a higher supply
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Fig3: Two consumers and a single resource. Parameters, given in the text, are the same for both plots, the only difference is the supply level of the resource. Also the starting point is the same, no food, much of B and a little bit of A. In both cases A successfully takes over from B. If the stable resource level is below the critical level for maintenance of B (a), this consumer simply disappears, and at some later time A grows to its stationary level. If the supply of resource is sufficient to support B (b), there is an interval where a finite population B survives on the available resource. The decline of the species B is in fact brought about by its competitor A eating away the required food. Eventually a higher population of A is reached because of a higher supply

Mentions: For sR < 0.5 the only biologically relevant stationary point is the trivial equilibrium, which is a stable node. Any initial state will evolve towards it, like for a single consumer. Both A and B die out, and finally R grows to its stable level. For sR = 0.8 (Fig. 3a), B dies out, be it rather slowly, R grows to its stable level, after which A picks up and grows to its stationary level, while the resource density drops. The trivial equilibrium is a saddle point, the coexistence of A and R is a stable node. For sR = 1.2 (Fig. 3b), R grows to a value slightly below unity, while B decreases to about the stationary level B * = 0.2, but eventually A takes over. The coexistence of B and R is a saddle point, the coexistence of A and R is a stable node. The size of the stable population of A is higher for the higher stable resource level, while the stationary resource level in the latter two plots is exactly the same. The reason is that although in the stationary point the abundance of the resource is the same, because of the higher stable level, the production rate of the resource is higher. Hence a higher consumer population level can be maintained.Fig. 3


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

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

Two consumers and a single resource. Parameters, given in the text, are the same for both plots, the only difference is the supply level of the resource. Also the starting point is the same, no food, much of B and a little bit of A. In both cases A successfully takes over from B. If the stable resource level is below the critical level for maintenance of B (a), this consumer simply disappears, and at some later time A grows to its stationary level. If the supply of resource is sufficient to support B (b), there is an interval where a finite population B survives on the available resource. The decline of the species B is in fact brought about by its competitor A eating away the required food. Eventually a higher population of A is reached because of a higher supply
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig3: Two consumers and a single resource. Parameters, given in the text, are the same for both plots, the only difference is the supply level of the resource. Also the starting point is the same, no food, much of B and a little bit of A. In both cases A successfully takes over from B. If the stable resource level is below the critical level for maintenance of B (a), this consumer simply disappears, and at some later time A grows to its stationary level. If the supply of resource is sufficient to support B (b), there is an interval where a finite population B survives on the available resource. The decline of the species B is in fact brought about by its competitor A eating away the required food. Eventually a higher population of A is reached because of a higher supply
Mentions: For sR < 0.5 the only biologically relevant stationary point is the trivial equilibrium, which is a stable node. Any initial state will evolve towards it, like for a single consumer. Both A and B die out, and finally R grows to its stable level. For sR = 0.8 (Fig. 3a), B dies out, be it rather slowly, R grows to its stable level, after which A picks up and grows to its stationary level, while the resource density drops. The trivial equilibrium is a saddle point, the coexistence of A and R is a stable node. For sR = 1.2 (Fig. 3b), R grows to a value slightly below unity, while B decreases to about the stationary level B * = 0.2, but eventually A takes over. The coexistence of B and R is a saddle point, the coexistence of A and R is a stable node. The size of the stable population of A is higher for the higher stable resource level, while the stationary resource level in the latter two plots is exactly the same. The reason is that although in the stationary point the abundance of the resource is the same, because of the higher stable level, the production rate of the resource is higher. Hence a higher consumer population level can be maintained.Fig. 3

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.