Supplementary Mathematica notebook for "Distributing defenses: How resource defendability shapes the optimal response to risk"

Many organisms divide limited defenses among heterogeneous assets. Plants allocate defensive chemicals among tissues differing in value, cost of defense, and risk of herbivory. Some ant colonies allocate specialized defenders among multiple nests differing in volume, entrance size, and risk of attack. We develop a general mathematical model (here implemented and analyzed in Mathematica) to determine the optimal strategy for dividing defenses among assets depending on their value, defendability, and risk of attack. We build upon plant defense theory by focusing on defendability, which we define as the functional relationship between defensive investment and successful defense. We show that if hard-to-defend assets cost more to defend, as assumed in resource defense theory, the optimal strategy allocates more defenses to those assets, regardless of risk. Inspired by cavity-nesting ants, we also consider the possibility that hard-to-defend assets have a lower chance to be successfully defended, even when defensive investment is high. Under this assumption, the optimal response to elevated risk is to reduce defensive allocation to hard-to-defend assets, a conservative strategy previously observed in turtle ants (Cephalotes). This new perspective on defendability suggests that, in systems where assets differ in the chance of successful defense, defensive strategies may evolve to be sensitive to risk in surprising ways.