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Population dynamics in a periodically varying environment

Nisbet, R.M. and Gurney, William (1976) Population dynamics in a periodically varying environment. Journal of Theoretical Biology, 56 (2). pp. 459-475. ISSN 0022-5193

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We have investigated numerically the effect of periodic variation in the values of the intrinsic growth rate, r, and of the carrying capacity, K, for a species described by the logistic equation with a time delay, τ. If in the absence of driving oscillations the system would have a stable equilibrium, then oscillations in K normally cause population oscillations of the same frequency, accompanied by a drop in the time average population. Large amplitude oscillations in K may cause population cycles whose frequency is a subharmonic of the driver. The qualitative behaviour of the system is unchanged by oscillations in r alone, while the effect of simultaneous oscillations in r and K depends critically on the phase difference between them. If the population would, in the absence of driving oscillations, be in a limit cycle, then this limit cycle is normally robust against K-oscillations at frequencies very different from the limit cycle frequency. If the limit cycle frequency is close to the driving frequency or to one of its subharmonics, then the system may synchronize to the driver or to that subharmonic. Oscillations in r have little effect on a limit cycling system unless they are of large enough amplitude to decrease the effective value of the product rτ sufficiently to destroy the limit cycle. On the basis of the above results we comment on the analysis of cycles in field and laboratory situations, especially those where synchronization may occur. We finally use our results to discuss the direction of evolution of r in a regularly varying environment.