The paradox of enrichment is a term from population ecology coined by Michael Rosenzweig in 1971.
[1] He described an effect in six predator–prey models where increasing the food available to the prey caused the predator's population to destabilize.
That may result in a crash in the population of the predators and possibly lead to local eradication or even species extinction.
The original sense was one of irony; by attempting to increase the carrying capacity in an ecosystem, one could fatally imbalance it.
Since then, some authors have used the word to describe the difference between modelled and real predator–prey interactions.
Rosenzweig used ordinary differential equation models to describe changes in prey populations.
The cycling behavior after destabilization was more thoroughly explored in a subsequent paper (May 1972)[2] and discussion (Gilpin and Rosenzweig 1972).
as summarised by Roy and Chattopadhyay [4] , such as these exceptions: 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.
First, one assumes that the growth of the prey population is determined by the logistic equation.
Then, one assumes that predators have a nonlinear functional response, typically of type II.
The saturation in consumption may be caused by the time to handle the prey or satiety effects.
The predator-free steady-state is locally linearly unstable if and only if the coexistence-steady-state exists.
, we get: It is possible to find the exact solution of this linear system, but here, the only interest is in the qualitative behavior.
If both eigenvalues of the community matrix have negative real part, then by the stable manifold theorem the system converges to a limit point.
Since the trace is equal to the sum of the eigenvalues, the co-existence steady-state is locally linearly stable if At that critical value of the parameter K, the system undergoes a Hopf bifurcation.
It comes as counterintuitive (hence the term 'paradox') because increasing the carrying capacity of the ecological system beyond a certain value leads to dynamic instability and extinction of the predator species.