Environmental forcing, invasion and control of ecological and epidemiological systems.

Destabilising a biological system through periodic or stochastic forcing can lead to significant changes in system behaviour. Forcing can bring about coexistence when previously there was exclusion; it can excite massive system response through resonance, it can offset the negative effect of apparent competition and it can change the conditions under which the system can be invaded. Our main focus is on the invasion properties of continuous time models under periodic forcing. We show that invasion is highly sensitive to the strength, period, phase, shape and configuration of the forcing components. This complexity can be of great advantage if some of the forcing components are anthropogenic in origin. They can be turned into instruments of control to achieve specific objectives in ecology and disease management, for example. Culling, vaccination and resource regulation are considered. A general analysis is presented, based on the leading Lyapunov exponent criterion for invasion. For unstructured invaders, a formula for this exponent can typically be written down from the model equations. Whether forcing hinders or encourages invasion depends on two factors: the covariances between invader parameters and resident populations and the shifts in average resident population levels brought about by the forcing. The invasion dynamics of a structured invader are much more complicated but an analytic solution can be obtained in quadratic approximation for moderate forcing strength. The general theory is illustrated by a range of models drawn from ecology and epidemiology. The relationship between periodic and stochastic forcing is also considered.