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

We consider a delay differential equation modeling production and destruction, and prove the presence of the paradoxial hydra effect. Namely, for the equation ẏ(t)=-μ y(t)+f(y(t-1)) with μ >0 and nonincreasing f:ℝ→ (0,∞ ) , it is shown that the mean value of certain solutions can be increased by increasing the value of the (destruction) parameter 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.