Paradox of the pesticides

The paradox of the pesticides is a paradox that states that applying pesticide to a pest may end up increasing the abundance of the pest if the pesticide upsets natural predator–prey dynamics in the ecosystem.

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

To describe the paradox of the pesticides mathematically, the Lotka–Volterra equation, a set of first-order, nonlinear, differential equations, which are frequently used to describe predator–prey interactions, can be modified to account for the additions of pesticides into the predator–prey interactions.

The variables represent the following:

The following two equations are the original Lotka–Volterra equation, which describe the rate of change of each respective population as a function of the population of the other organism:

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, the point at which both interacting populations are stable.

These are the isoclines for the two above equations:

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

Solving the isoclines as was done above, the following equations represent the two lines with the intersection that represents the new equilibrium point. These are the new isoclines for the populations:

As one can see from the new isoclines, the new equilibrium will have a higher H value and a lower P value so the number of prey will increase while the number of predator decreases. Thus, 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. 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.

...
Wikipedia