Paradox of enrichment

Many studies have been done on the paradox of enrichment since Rosenzweig, and some have shown that the model initially proposed does not hold in all circumstances, as summarised by Roy and Chattopadhyay in 2007, such as these exceptions:

• Inedible prey: if there are multiple prey species and not all are edible, some may absorb nutrients and stabilise cyclicity.
• Invulnerable prey: even with a single prey species, if there is a degree of temporal or spatial refuge (the prey can hide from the predator), destabilisation may not happen.
• Unpalatable prey: if prey do not fulfil the nutritional preferences of the predator to as great an extent at higher densities, as with some algae and grazers, there may be a stabilising effect.
• Heterogeneous environment: the model for enrichment follows an assumption of environmental homogeneity. If a spatiotemporally chaotic, heterogeneous environment is introduced, cyclic patterns may not arise.
• Induced defense: if there is a predation-dependent response from prey species, it may act to decelerate the downward swing of population caused by the boom in predator population. An example is of Daphnia and fish predators.
• Autotoxins and other predator density-dependent effects: if predator density cannot increase in proportion to that of prey, destabilising periodicities may not develop.
• Prey toxicity: if there is a significant cost to the predator of consuming the (now very dense) prey species, predator numbers may not increase sufficiently to give periodicity.

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:

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

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

• 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 ${\displaystyle x\left(1-{\frac {x}{K}}\right)}$ represents the prey's logistic growth, and ${\displaystyle {\frac {x}{1+x}}}$ 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 ${\displaystyle \ x=0}$ and ${\displaystyle y=(1+x)\left(1-x/K\right)}$. Likewise, the predator isoclines are obtained as ${\displaystyle \ y=0}$ and ${\displaystyle x={\frac {\alpha }{1-\alpha }}}$, where ${\displaystyle \alpha ={\frac {\gamma }{\delta }}}$. The intersections of the isoclines yields three equilibrium states:

${\displaystyle x_{1}=0,\;y_{1}=0}$

${\displaystyle x_{2}=K,\;y_{2}=0}$

${\displaystyle x_{3}={\frac {\alpha }{1-\alpha }},\;y_{3}=(1+x_{3})\left(1-{\frac {x_{3}}{K}}\right)}$

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.

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 ${\displaystyle f}$ and ${\displaystyle g}$ with respect to ${\displaystyle x}$ and ${\displaystyle y}$ in a neighborhood of ${\displaystyle (x_{3},y_{3})}$, we get:

${\displaystyle {\frac {d}{dt}}{\begin{bmatrix}x-x_{3}\\y-y_{3}\\\end{bmatrix}}\approx {\begin{bmatrix}\alpha \left(1-(1+2x_{3})/K\right)&-\alpha \\\delta (1-\alpha )^{2}y_{3}&0\\\end{bmatrix}}{\begin{bmatrix}x-x_{3}\\y-y_{3}\\\end{bmatrix}}}$

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 system is stable if

${\displaystyle \alpha \left(1-{\frac {1+2x_{3}}{K}}\right)<0,{\text{ or }}K<1+2{\frac {\alpha }{1-\alpha }}}$

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.

A credible, simple alternative to the Lotka–Volterra predator–prey model and its common prey dependent generalizations is the ratio dependent or Arditi–Ginzburg model.[4] The two are the extremes of the spectrum of predator interference models. According to the authors of the alternative view, the data show that true interactions in nature are so far from the Lotka–Volterra extreme on the interference spectrum that the model can simply be discounted as wrong. They are much closer to the ratio dependent extreme so if a simple model is needed one can use the Arditi–Ginzburg model as the first approximation.[5]

The presence of the paradox is strongly dependent on the assumption of the prey dependence of the functional response; because of this the ratio dependent Arditi–Ginzburg model does not have the paradoxical behavior. The authors' claim that the paradox is absent in nature (simple laboratory systems may be the exception) is in fact a strong argument for their alternative view of the basic equations.[6]

• Braess' paradox: Adding extra capacity to a network may reduce overall performance.
• Paradox of the pesticides: Applying pesticide may increase the pest population.

• Gilpin, Michael; Rosenzweig, Michael (1972). "Enriched Predator–Prey Systems: Theoretical Stability". Science. 177 (4052): 902–904. Bibcode:1972Sci...177..902G. doi:10.1126/science.177.4052.902. PMID 17780992.
• May, Robert (1972). "Limit Cycles in Predator–Prey Communities". Science. 177 (4052): 900–902. Bibcode:1972Sci...177..900M. doi:10.1126/science.177.4052.900. PMID 17780991. S2CID 33149522.
• Rosenzweig, Michael (1971). "The Paradox of Enrichment". Science. 171 (3969): 385–387. doi:10.1126/science.171.3969.385. PMID 5538935. S2CID 43699627.
• Kot, Mark (2001). Elements of Mathematical Ecology. Cambridge University Press. ISBN 0-521-80213-X.
• Roy, Shovonlal; Chattopadhyay, J. (2007). "The stability of ecosystems: A brief overview of the paradox of enrichment" (PDF). Journal of Biosciences. 32 (2): 421–428. doi:10.1007/s12038-007-0040-1. ISSN 0250-5991. PMID 17435332. S2CID 16526292.