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We consider a delay differential equation modeling production and destruction, and prove the presence of the paradoxial hydra effect. Namely, for the equation $$\dot{y}(t)=-\mu y(t)+f(y(t-1))$$y˙(t)=-μy(t)+f(y(t-1)) with $$\mu >0$$μ>0 and nonincreasing $$f:\mathbb {R}\rightarrow (0,\infty )$$f:R→(0,∞), it is shown that the mean value of certain solutions can be increased by increasing the value of the (destruction) parameter $$\mu $$μ. The nonlinearity f in the equation is a step function or a smooth function close to a step function. This particular form of f allows us to construct periodic solutions, and to evaluate the mean values of the periodic solutions. Our result explains how the global form of the nonlinearity f (the production term) induces the appearance of the hydra effect.

Periodic Solutions and Hydra Effect for Delay Differential Equations with Nonincreasing Feedback