The joint evolution of defence and inducibility against natural enemies.

We develop and analyse a model of inducible defence where two traits--defence and its inducibility--jointly evolve. Inducibility reduces costs of defence in the absence of enemies thereby permitting higher defence levels when attacked. If the cost of inducibility is low, then inducibility and defence may reinforce one another, resulting in a runaway leading to a highly inducible and highly effective defence. When inducibility is more costly, a new joint-equilibrium in defence/inducibility emerges displaying intermediate levels of both traits, and the prior 'run-away' scenario (high defence, high inducibility) may disappear. In contrast to the cost of inducibility, the cost of defence has mixed effects. An increase in costs of defence generally diminishes the level of both defence and inducibility at the intermediate locally stable equilibrium, but can favour the existence of the 'run-away' scenario of high defence-high inducibility. The enemy encounter-rate also has mixed effects. At high encounter rates an increase in encounters can lead to a higher/maximal defence and a lower level of inducibility (defence being almost always useful), but at low rates, an increase in encounters can lead to both higher defence and higher inducibility. We finally consider potential enemy responses to defensive change, and illustrate that herd immunity (reduction of encounter rates due to population-level defence) can affect both individual defence and induction that can be, depending on conditions, increased or decreased.