HOME

TheInfoList



OR:

The paradox of enrichment is a term from
population ecology Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment (biophysical), environment, such as birth rate, birth and death rates, and by immigration an ...
coined by
Michael Rosenzweig Michael L. Rosenzweig (born 1941) is a professor of ecology and evolutionary biology at the University of Arizona. He developed and popularized the concept of reconciliation ecology. He received his Ph.D. in zoology at the University of Pennsylv ...
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 In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
models to describe changes in prey populations. Enrichment was taken to be increasing the prey
carrying capacity The carrying capacity of an ecosystem is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as the ...
and showing that the prey population destabilized, usually into a
limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
. 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. as summarised by Roy and Chattopadhyay , 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. *Ratio dependent functional response. The presence of the paradox is 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 the paradoxical behaviour. *Spatial interactions or spatio-temporal chaos. The model for enrichment assumes that there is no spatial heterogeneity. Spatial version predator-prey models allow for spatial heterogeneity of predator and prey populations in different locations 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 ''Daphnia'' is a genus of small planktonic crustaceans, in length. ''Daphnia'' are members of the Order (biology), order Anomopoda, and are one of the several small aquatic crustaceans commonly called water fleas because their Saltation (gait), ...
'' and fish predators. *Density dependent predator mortality: 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.


Link with Hopf bifurcation

The paradox of enrichment can be accounted for by the
bifurcation theory Bifurcation theory is the Mathematics, mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential e ...
. As the
carrying capacity The carrying capacity of an ecosystem is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as the ...
increases, the equilibrium of the
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
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 A functional response in ecology is the intake rate of a consumer as a function of food density (the amount of food available in a given ecotope). It is associated with the numerical response, which is the reproduction rate of a consumer as a fu ...
, 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: :\frac = x\left(1 - \frac\right) - y \frac :\frac = \delta y \frac - \gamma y *''x'' is the
prey Predation is a biological interaction in which one organism, the predator, kills and eats another organism, its prey. It is one of a family of common feeding behaviours that includes parasitism and micropredation (which usually do not ki ...
density; *''y'' is the
predator Predation is a biological interaction in which one organism, the predator, kills and eats another organism, its prey. It is one of a family of common List of feeding behaviours, feeding behaviours that includes parasitism and micropredation ...
density; *''K'' is the prey population's
carrying capacity The carrying capacity of an ecosystem is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as the ...
; *''γ'' and ''δ'' are predator population's parameters (rate of decay and benefits of consumption, respectively). The term x\left(1 - \frac\right) represents the prey's logistic growth, and \frac the predator's functional response. The prey
isocline 300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. The solution curves are y = C e^. Given a family of curves, assumed to be differentiable, an isocline for that family is formed b ...
s (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, where \alpha = \frac. The intersections of the isoclines yields three steady-states: :x_1 = 0,\; y_1 = 0 :x_2 = K,\; y_2 = 0 :x_3 = \frac,\; y_3 = (1 + x_3) \left(1 - \frac\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 is sufficiently small. The predator-free steady-state is locally linearly unstable if and only if the coexistence-steady-state exists. By the
Hartman–Grobman theorem In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisa ...
, one can determine the stability of the steady states by approximating the nonlinear system by a linear system. After differentiating each f and g with respect to x and y in a neighborhood of (x_3, y_3), we get: :\frac\beginx - x_3\\y - y_3\\\end \approx \begin\alpha\left( 1 - (1 + 2 x_3)/K \right)&- \alpha\\ \delta (1 - \alpha)^2 y_3 & 0\\\end \beginx - x_3\\y - y_3\\\end 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 In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of the
community matrix In mathematical biology, the community matrix is the linearization of a generalized Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equilibrium point. For example, the Lo ...
have negative real part, then by the stable manifold theorem the system converges to a limit point. Since the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
is equal to the product of the eigenvalues and is positive, both eigenvalues have the same sign. Since the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album), by Nell Other uses in arts and entertainment * ...
is equal to the sum of the eigenvalues, the co-existence steady-state is locally linearly stable if :\alpha\left(1 - \frac\right) < 0, \text K < 1 + 2\frac At that critical value of the parameter K, the system undergoes a
Hopf bifurcation In the mathematics of dynamical systems and differential equations, a Hopf bifurcation is said to occur when varying a parameter of the system causes the set of solutions (trajectories) to change from being attracted to (or repelled by) a fixed ...
. 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.


See also

*
Braess's paradox Braess's paradox is the observation that adding one or more roads to a road network can slow down overall road traffic, traffic flow through it. The paradox was first discovered by Arthur Cecil Pigou, Arthur Pigou in 1920, and Stigler's law of ep ...
: Adding extra capacity to a network may reduce overall performance. * Paradox of the pesticides: Applying pesticide may increase the pest population.


References


Other reading

* * * * * {{DEFAULTSORT:Paradox Of Enrichment Mathematical and theoretical biology Predation