Under certain conditions, predation acts as a selective pressure that drives prey adaptation. It follows then that a necessary piece of understanding population dynamics is to understand the underlying genetic forces that drive those dynamics.  Here, we investigate the effects of genetic variability in predator search efficiency and in prey defense against predation on the stability and dynamics of a predator-prey system.  In particular, we examine the impact of varied vs. fixed cost of anti-predator defense on population dynamics.  Varied cost implies that the cost of anti-predator defense varies with population size, so that, at low prey density, defense is without cost.  We assume that the tradeoff for investment in defense is a decrease in fecundity.  We present an ordinary differential equation model with four equations: two which describe the change in population densities and two which track the mean values of prey and predator traits using a quantitative trait approach. Bifurcation analysis is done through the use of a MATLAB continuation software package, CL MATCONT. 
Results/Conclusions

We find that when the cost for prey defense varies with population size, the system is much more stable than for the fixed cost model. In other words, the region of parameters that give rise to cycling is much smaller for variable vs. fixed cost. In fact the cycling region for the fixed cost model is very similar to that of the 2D Rosenzweig-MacArthur model with no evolution. We also find that if we “slow” down the rate of evolution the fixed model approaches the Rosenzweig-MacArthur model. In contrast the change in the variable cost model is non-monotonic. More specifically, the variable cost model is most stable for very slow or very fast rates of evolution, and least stable for intermediate rates of evolution. These results might help us to understand why natural populations do not exhibit cycling as often, or for as prolonged a period of time as would be predicted by a purely ecological model.