The effect of long time delays in predator-prey systems

Past studies have indicated that a time delay longer than the natural period of a system will generally cause instability; however here it is shown that including long maturational time delays in a general predator-prey model need not have this effect. In each of the three cases studied (a predator delay, a prey delay, and both), local stability can persevere despite the presence of arbitrarily long time delays. This perseverence depends upon an interaction between delayed and undelayed features of the model. Delayed processes always act to destabilize the model. For example, prey self-regulation, usually a source of stability, becomes destabilizing if subject to a long delay. However, the effect of such a delay is offset by undelayed regulatory processes, such as a stabilizing functional response. In addition, the adverse effects of delayed predator recruitment can be reduced by the nonreproductive component of the numerical response, a feature not usually involved in determining stability. Finally, it is shown that long time delays are not necessarily more disruptive than short delays; it cannot be assumed that lengthening a time delay progressively reduces stability.