The paradox of the pesticides is a paradox that states that by applying pesticide to a pest, one may in fact increase its abundance. This happens when the pesticide upsets natural predator-prey dynamics in the ecosystem.

The paradox can only occur when the target pest has a naturally occurring predator that is equally affected by the pesticide, and therefore presents a case for more specialized pesticide products.[citation needed]

## The Model: Lotka-Volterra equation

To describe the Paradox of the Pesticides mathematically, the Lotka-Volterra equation, a set of first-order, non-linear, differential equations that are frequently used to describe predator-prey interactions, can be modified to account for the additions of pesticides into the predator-prey interactions.

### Without pesticides

The variables represent the following:

\begin{align} H & = \text{the prey population at a given time} \\ P & = \text{the predator population at a given time} \\ c & = \text{the capture constant} \\ r & = \text{the rate of growth of the prey population} \\ a & = \text{the fraction of prey energy assimilated by the predator and turned into new predators} \\ m & = \text{predator mortality rate} \\ \end{align}

The following two equations are the original Lotka-Volterra equation that describe the rate of change of each respective population as a function of the other organism’s population.

\begin{align} \frac{dH}{dt} & = rH - cHP \\ \frac{dP}{dt} & = acHP - mP \\ \end{align}

By setting each equation to zero, and thus assuming a stable population, a graph of two lines (isoclines) can be made to find the equilibrium point, or the point at which both interacting populations are stable.

The isoclines for the two above equations are:

$P=\frac{r}{c} \quad \text{and} \quad H=\frac{m}{ac}$

### Accounting for pesticides

Predator-prey isoclines before and after pesticide application. Note that pest abundance has increased.

Now, to account for the difference in the population dynamics of the predator and prey that occurs with the addition of pesticides we add the variable of q to represent the per capita rate at which both species are killed by the pesticide. The original Lotka-Volterra equations change as follows:

\begin{align} \frac{dH}{dt} & = H(r-cP-q) \\ \frac{dP}{dt} & = P(acH-m-q) \\ \end{align}

Solving the isoclines as we did above we find the following equations to represent the two lines with the intersection that represents the new equilibrium point. The new isoclines for the populations are:

$P=\frac{r-q}{c} \quad \text{and} \quad H=\frac{m+q}{ac}$

As one can see from the new isoclines the new equilibrium will have a higher H value and a lower P value. This means that the number of prey will increase while the number of predator decreases. This means that the prey, which is normally the targeted by the pesticide, is actually being benefited instead of harmed by the pesticide.

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.[1] 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.[2]

## Empirical Evidence

The paradox has been documented repeatedly throughout the history of pest management. Predatory mites, for example, naturally prey upon phytophagous mites, which are common pests in apple orchards. Spraying the orchards kills both mites, but the effect of diminished predation is larger than the pesticide’s, and phytophagous mites increase in abundance.[3]

The effect has also been seen on rice, as documented by the International Rice Research Institute, which noted significant declines in pest populations when they stopped applying pesticide.[4]

## Related Phenomena

Recent studies suggest that such a paradox might not be necessarily caused by the reduction of the predator population due to harvesting itself, for example, by a pesticide. The host population is reduced at the moment of harvesting, and simultaneously the intraspecific density effect is weakened.[5] Intraspecific competition accounts for the competition between individuals of a same species. When the population density is high, and resources are consequently relatively scarce, each individual has less access to resources to invest energy in growth, survivorship and reproduction. This causes a decrease in the survival rate, or an increase in mortality.

Intraspecific competition increases with density. One could expect that a population decrease (due to harvesting, for example) will decrease the population density and reduce intraspecific competition, which would lead to a lower death rate among the prey population.

Studies show furthermore that direct effects on the predator population, through harvesting of the prey, are not necessary to observe the paradox.[5] Harvesting of prey has been shown to trigger a reduction in the predator’s reproduction rate, which lowers the equilibrium predator level. Thus, changes in life history strategy (patterns of growth, reproduction and survivorship) can also contribute to the paradox.

Seemingly then, the paradox can be accounted for by the indirect effects of harvesting on the native ecological interactions of prey and predator: reduction of intraspecific density effect for the prey, and reduction of the reproductive rate for the predator. The first effect increases the population recovery of the prey, and the second decreases the equilibrium population level for the predator.

## Implications

The Paradox of the Pesticides implies the need for more specialized pesticides that are tailored to the target pest. If the pesticide can effectively reduce only the prey population, the predator population will remain largely unaffected except for the change in its food supply. Broad spectrum pesticides are more likely to induce the Paradox and cause an increase in target pest population by killing its predators as well. In certain cases, however, where the predator is closely related to the target pest even narrow spectrum pesticides may be insufficient.

## Solutions

To deal with the Paradox of the Pesticides, growers may turn to Integrated Pest Management (IPM),[6] an ecological approach to pest control that accounts for the interactions between pests and their environment.[7] There is no one way to practice IPM, but some methods include using mechanical trapping devices or increasing the abundance of natural predators.[8]

IPM is also often touted for its environmental and health benefits, as it avoids the use of chemical pesticides.