User:Guillaume Filion

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I currently have a post-doctoral position at the NKI (the Netherlands Cancer Institute) in Amsterdam.

My work is mainly focused on (epi)genetics and informatics.

The Paradox of Enrichment is a term from population ecology coined by Michael Rosenzweig in 1973. He described an effect in six predator-prey models wherein increasing the food available to the prey caused the predator's population to destabilize.

[edit] Rosenzweig's result (Rosenzweig 1971)

Rosenzweig used ordinary differential equation models to simulate the prey population. Models only represented prey populations. Enrichment was taken to be increasing the prey carrying capacity and showing that the prey population destabilized, usually into a limit cycle.

The cycling behavior after destabilization was more thoroughly explored in a subsequent paper (May 1972) and discussion (Gilpin and Rozenzweig 1972).

[edit] Link with the Hopf bifurcation

The paradox of enrichment can be accounted for by the bifurcation theory. As the carrying capacity increases, the equilibrium of the dynamical system becomes unstable, whence the appearance of a limit cycle.

The bifurcation can be obtained by modifying the Lotka-Volterra equation. First, one assumes that prey's growth is determined by the logistic equation. Second, one assumes that predators have a non-linear functional response, typically of type II. The saturation in consumption may be caused by the time to handle the prey or satiety effects for instance.

Thus, one can write the following (normalized) equations:

$\frac{dx}{dt} = x\left(1 - \frac{x}{K}\right) - y \frac{x}{1 + x}$

$\frac{dy}{dt} = -y\left(\gamma - \delta \frac{x}{1 + x}\right)$

x is the prey density;
y is the predator density;
K is the prey population's carrying capacity;
γ and δ are predator population's parameters (rate of decay and benefits of consumption, respectively).

The term $x\left(1 - \frac{x}{K}\right)$ represents the prey's logistic growth, and $\frac{x}{1 + x}$ the predator's functional response.

The prey isoclines (points at which the prey population does not change, i.e. dx/dt = 0) are easily obtained as $\ x = 0$ and $y = (1 + x) \left(1 - x/K \right)$ . Likewise, the predator isoclines are obtained as $\ y = 0$ and $x = \frac{\alpha}{1-\alpha}$ , where $\alpha = \frac{\gamma}{\delta}$ . The intersections of the isoclines yields three equilibrium states:

$x_1 = 0,\; y_1 = 0$

$x_2 = K,\; y_2 = 0$

$x_3 = \frac{\alpha}{1-\alpha},\; y_3 = (1 + x_3) \left(1 - \frac{x_3}{K}\right)$

The first equilibrium corresponds to the extinction of both predator and prey, the second one to the extinction of the predator, and the third to co-existence.

[edit] References

Gilpin, Michael and Michael Rosenzweig. 1972. "Enriched Predator-Prey Systems: Theoretical Stability" Science Vol. 177, pp. 902-904.
May, Robert. 1972. "Limit Cycles in Predator-Prey Communities" Science Vol. 177, pp. 900-902.
Rosenzweig, Michael. 1971. "The Paradox of Enrichment" Science Vol. 171: pp. 385-387
Kot, Mark. 2001. "Elements of Mathematical Ecology" Cambridge University Press