In population ecology, the paradox of enrichment is the idea that increasing the food available to a prey species can cause the predator's population to destabilize. The term was coined by Michael Rosenzweig in 1971, where he described an effect in six predator–prey models. 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 prey carrying capacity in an ecosystem, the enriched environment can destabilise the biological populations. 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 an increase in the prey's carrying capacity, which can can render the equilibrium unstable, usually resulting in a limit cycle. The cycling behavior after destabilization was more thoroughly explored in a subsequent paper (May 1972) and discussion (Gilpin and Rosenzweig 1972).
Support and possible solutions to the paradox 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. Roy and Chattopadhyay list a number of mechanism for which the destabilisation need not occur:
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. Ratio dependent functional response. The presence of the paradox depends on the assumption of the prey dependence of the functional response. The Arditi–Ginzburg model, which uses a ratio-dependent functional response, does not show destabilisation. Spatial interactions or spatio-temporal chaos. The model for enrichment assumes that there is no spatial heterogeneity. Spatial versions of predator-prey models allow for spatial heterogeneity of predator and prey populations in different locations to build up, which can reduce the violent oscillations of the non-spatial model. If a spatiotemporally chaotic, heterogeneous environment is introduced, cyclic patterns may not arise. Inducible 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. Density dependent predator mortality: if the predator density cannot increase in proportion to that of the 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.
Link with Hopf bifurcation The paradox of enrichment can be accounted for by 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:
d x d t = x ( 1 − x K ) − y x 1 + x {\displaystyle {\frac {dx}{dt}}=x\left(1-{\frac {x}{K}}\right)-y{\frac {x}{1+x}}}
d y d t = δ y x 1 + x − γ y {\displaystyle {\frac {dy}{dt}}=\delta y{\frac {x}{1+x}}-\gamma y}
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