The paradox of enrichment is a term from population ecology coined by Michael Rosenzweig in 1971. He described an effect in six predator–prey models where increasing the food available to the prey caused the predator's population to destabilize. A common example is that if the food supply of a prey such as a rabbit is overabundant, its population will grow unbounded and cause the predator population (such as a lynx) to grow unsustainably large. That may result in a crash in the population of the predators and possibly lead to local eradication or even species extinction.
The term 'paradox' has been used since then to describe this effect in slightly conflicting ways. 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. 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 Rosenzweig 1972).
Many studies have been done on the paradox of enrichment since Rosenzweig. There is empirical support for the paradox of enrichment, mainly from small scale laboratory experiments, but limited support from field observations. as summarised by Roy and Chattopadhyay [1], 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.
The bifurcation can be obtained by modifying the Lotka–Volterra equation. 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.
Thus, one can write the following (normalized) equations:
| dx | |
| dt |
=x\left(1-
| x | |
| K |
\right)-y
| x | |
| 1+x |
| dy | |
| dt |
=\deltay
| x | |
| 1+x |
-\gammay
The term
x\left(1-
| x | |
| K |
\right)
| x | |
| 1+x |
The prey isoclines (points at which the prey population does not change, i.e. dx/dt = 0) are easily obtained as
x=0
y=(1+x)\left(1-x/K\right)
y=0
x=
| \alpha | |
| 1-\alpha |
\alpha=
| \gamma | |
| \delta |
x1=0, y1=0
x2=K, y2=0
x3=
| \alpha | |
| 1-\alpha |
, y3=(1+x3)\left(1-
| x3 | |
| K |
\right)
The first steady-state corresponds to the extinction of both predator and prey, the second one to the predator-free steady-state and the third to co-existence, which only exists when
\alpha
By the Hartman–Grobman theorem, one can determine the stability of the steady states by approximating the nonlinear system by a linear system. After differentiating each
f
g
x
y
(x3,y3)
| d | |
| dt |
\begin{bmatrix}x-x3\\y-y3\\\end{bmatrix} ≈ \begin{bmatrix}\alpha\left(1-(1+2x3)/K\right)&-\alpha\ \delta(1-\alpha)2y3&0\\\end{bmatrix}\begin{bmatrix}x-x3\\y-y3\\\end{bmatrix}
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 determinant is equal to the product of the eigenvalues and is positive, both eigenvalues have the same sign. Since the trace is equal to the sum of the eigenvalues, the co-existence steady-state is locally linearly stable if
\alpha\left(1-
| 1+2x3 | |
| K |
\right)<0,orK<1+2
| \alpha | |
| 1-\alpha |
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.